Low energy levels of harmonic maps into analytic manifolds

TL;DR

This paper investigates the energy spectrum of harmonic maps into analytic manifolds, proving that for generic 3-manifolds, the spectrum's accumulation points are excluded, and establishing obstructions to harmonic sphere gluing.

Contribution

It demonstrates that the lowest potential accumulation point of the energy spectrum is not realized for generic 3-manifolds and introduces new obstructions to harmonic sphere gluing.

Findings

Excludes the possibility of energy spectrum accumulation at 2 E_{min} for generic 3-manifolds.

Establishes obstructions to gluing harmonic spheres.

Provides Lojasiewicz-estimates for almost harmonic maps.

Abstract

We consider the energy spectrum $\Xi_E(N)$ of harmonic maps from the sphere into a closed Riemannian manifold $N$. While a well known conjecture asserts that $\Xi_E(N)$ is discrete whenever $N$ is analytic, for most analytic targets it is only known that any potential accumulation point of the energy spectrum must be given by the sum of the energies of at least two harmonic spheres. The lowest energy level that could hence potentially be an accumulation point of $\Xi_E$ is thus $2 E_{min}$. In the present paper we exclude this possibility for generic 3 manifolds and prove additional results that establish obstructions to the gluing of harmonic spheres and Lojasiewicz-estimates for almost harmonic maps.

Equations

$$E(u)=\frac{1}{2}{\int }_{S^2}|\nabla u{|}^{2}d{v}_{g_{S^2}}.$$

$${\Xi }_E(N,g_N):=\{E(u):u:S^2\to N\text{ harmonic, not constant }\}$$

$$\Vert {\tau }_{g_{S^2}}(u_j){\Vert }_{L^2(S^2,g_{S^2})}\to 0$$

$$u_{j}(x)-\sum\big{(}\omega_{k}(\mu_{j}^{k}(x-p_{j}^{k}))-\omega_{k}(\infty)\big{)}-\omega_{0}(x)\to 0\text{ strongly in }L^{\infty}\cap H^{1}(B_{r}(p)).$$

$$|E(u)-E(u^{\ast })|\le C\Vert {\tau }_{g_{S^2}}(u){\Vert }_{L^2(S^2,g_{S^2})}^{\gamma }\text{ for some }\gamma =\gamma (u^{\ast })\in (1,2]$$

$$\mathcal{U}(u^{\ast })=\{u:S^2\to N:\Vert u-u^{\ast }{\Vert }_{L^{\infty }}<\epsilon \text{ and }\Vert u-u^{\ast }{\Vert }_{H^1}<\epsilon \}.$$

$$u_j(z)-{\omega }_0(z)-[{\omega }_1({\mu }_j(z-z_j))-{\omega }_1(\infty )]\to 0\text{ in }{\dot{H}}^1\cap L^{\infty }(\mathbb{C}).$$

$$|E(u_j)-[E({\omega }_0)+E({\omega }_1)]|\le C\min (\Vert dE(u_j){\Vert }_{\ast },\log {\mu }_j\Vert dE(u_j){\Vert }_{\ast }^2),$$

$$u_{j}(z)=U^{*}\big{(}q_{0}^{j}(z)+q_{1}^{j}(\tfrac{1}{\mu_{j}z})\big{)}$$

$${\partial }_{t}u={\tau }_{g_{S^2}}(u).$$

$$|E(u_j)-E^{\ast }|\le C\Vert {\tau }_{g_{S^2}}(u){\Vert }_{L^2(S^2,g_{S^2})}^2.$$

$${\mu }_j^{-\min (n_0^{\ast },n_1^{\ast })}\le C\Vert dE(u_j){\Vert }_{\ast },$$

$$|E(u_j)-[E({\omega }_0)+E({\omega }_1)]|\le C\Vert dE(u_j){\Vert }_{\ast }.$$

$$|E(u_j)-[E({\omega }_0)+E({\omega }_1)]|\le C\Vert {\tau }_{g_{S^2}}(u_j){\Vert }_{L^2(S^2,g_{S^2})}^{\gamma }$$

$$\ker (L_{{\omega }_i})=\{W\circ q^{\ast },W\in \ker (L_{U_i^{\ast }})\}+T_{{\omega }_i}\{U_i^{\ast }\circ q:q:S^2\to S^2\text{ rational }\},$$

$$\tau (U\circ q)=\frac{1}{2}|\nabla q{|}^2\tau (U\circ q)$$

$${\mu }_j^{-\min (n_0^{\ast },n_1^{\ast })}\le C\Vert dE(u_j){\Vert }_{\ast }$$

$$|E(u)-[E({\omega }_1)+E({\omega }_0)]|\le C\Vert dE(u_j){\Vert }_{\ast }.$$

$$Q(\mathfrak{z},y_{\mathfrak{z}}):=\frac{\|dE(u)\|_{*}\big{(}\|dE(u)\|_{*}+\|d^{2}E(u)(y_{\mathfrak{z}},\cdot)\|_{*}\big{)}}{dE(u)(y_{\mathfrak{z}})}\text{ is small .}$$

$$\mathcal{R}:=\{q:\hat{}\mathbb{C}\to \hat{}\mathbb{C}\text{ rational with }q(0)=0\}.$$

$$\mathcal{Z}:=\{{\mathfrak{z}}_0\circ R_p:p\in S^2,{\mathfrak{z}}_0\in {\mathcal{Z}}_0\},$$

$${\mathcal{H}}^{\sigma }({\omega }_i):=\{U\circ q:U\in {\mathcal{H}}_1^{\sigma }(U_i^{\ast }),q\in {\mathcal{R}}^{\sigma }(q_i^{\ast })\}$$

$${\mathcal{R}}^{\sigma }(q_i^{\ast }):\{q\in \mathcal{R}:\Vert \pi \circ q\circ {\pi }^{-1}-\pi \circ q_i^{\ast }\circ {\pi }^{-1}{\Vert }_{C^k(S^2)}<\sigma \}$$

$${\mathcal{H}}_1^{\sigma }(U_i^{\ast }):=\{U_i^{\ast }(\cdot +c):c\in \mathbb{C},|c|<\sigma \}.$$

$${\mathcal{H}}_1^{\sigma }(U_i^{\ast }):=\{U(\cdot +c):c\in \mathbb{C},|c|<\sigma ,U\in {\mathcal{H}}_0(U_i^{\ast })\text{ with }\Vert (U-U_i^{\ast })\circ {\pi }^{-1}{\Vert }_{C^k(S^2)}<\sigma \}$$

$$T_{U_i^{\ast }}\{U\circ M:U\in {\mathcal{H}}_0(U_i^{\ast }),M\text{ Mo¨bius transform}\}=\ker (L_{U_i^{\ast }})$$

$$\Vert {\tau }_{g_{S^2}}(U\circ {\pi }^{-1}){\Vert }_{C^k(S^2)}\le C\Vert {\tau }_{g_{S^2}}(U\circ {\pi }^{-1}){\Vert }_{L^2(S^2,g_{S^2})}$$

$$\frac{d}{d\epsilon }{|}_{\epsilon =0}E(U_{\epsilon })\ge \Vert {\tau }_{g_{S^2}}(U\circ {\pi }^{-1}){\Vert }_{L^2(S^2,g_{S^2})},$$

$${\mu }_0:=|a_{n_0^{\ast }}(q_0){|}^{-1/n_0^{\ast }}\text{ and }{\mu }_1:=\mu |a_{n_1^{\ast }}(q_1){|}^{-1/n_1^{\ast }}.$$

$$A={\mathbb{D}}_{r_0}\setminus {\mathbb{D}}_{r_1},A^{\ast }:=({\mathbb{D}}_{2r_1}\setminus {\mathbb{D}}_{r_1})\cup ({\mathbb{D}}_{r_0}\setminus {\mathbb{D}}_{\frac{1}{2}{r}_0})\text{ and }\hat{A}={\mathbb{D}}_{2\hat{r}}\setminus {\mathbb{D}}_{\frac{1}{2}\hat{r}}$$

Full text

1. Introduction

Let $(N,g_{N})$ be a closed Riemannian manifold which we can assume without loss of generality to be embedded in a suitable Euclidean space ${\mathbb{R}}^{N}$ using Nash’s embedding theorem. We recall that a map $u:S^{2}\to N$ is called harmonic if it is a critical point of the Dirichlet energy

[TABLE]

Such harmonic maps are characterised by $\tau_{g_{S^{2}}}(u)=0$ where the tension field is given by $\tau_{g_{S^{2}}}(u)=-\nabla^{L^{2}}E(u)=\Delta_{g_{S^{2}}}u+A(u)(\nabla u,\nabla u)$, $A$ the second fundamental form of $N\hookrightarrow{\mathbb{R}}^{N}$.

We recall that any harmonic map from $S^{2}$ is (weakly) conformal and hence that the set of harmonic maps from the sphere coincides with the set of (weakly) conformal parametrisations of minimal spheres in $N$. In particular, the energy spectrum

[TABLE]

of harmonic spheres agrees with the areas (counted with multiplicity) of (possibly branched) immersed minimal spheres.

While it is easy to construct smooth manifolds $(N,g_{N})$ for which $\Xi_{E}(N,g_{N})$ has accumulation points, a well known conjecture of Leon Simon and of Fang-Hua Lin [5] asserts that the energy spectrum of harmonic maps from $S^{2}$ into any closed analytic manifold is discrete.

In some very special cases this follows as a consequence of much stronger results such as an explicit characterisation of all minimal spheres in $N$ and the resulting explicit knowledge of $\Xi_{E}(N)$. E.g. if the target is the round sphere then all harmonic spheres are described in stereographic coordinates $z\in{\mathbb{C}}$ by rational maps in either $z$ or $\bar{z}$ and hence $\Xi(S^{2},g_{S^{2}})$ is made up by the multiples of $4\pi$.

Conversely, for more general analytic targets very little is known about $\Xi_{E}$ with the few existing results following already from the compactness theory of (almost) harmonic maps developed in the 1990s in [14, 1, 10, 4] and the seminal result of Simon [13] from 1983.

To describe the known properties of $\Xi_{E}$ we first recall that sequences $u_{j}$ of maps with bounded energy which are almost harmonic in the sense that

[TABLE]

subconverge to a bubble tree without loss of energy or formation of necks: That is, there exists a harmonic base map $\omega_{0}:S^{2}\to N$ and a finite set $S\subset S^{2}$ so that $u_{j}\to\omega_{0}$ strongly in $H_{loc}^{2}(S^{2}\setminus S)$ and weakly in $H^{1}(S^{2})$ and so that near each point $p\in S$ the $u_{j}$ are essentially described by a collection of highly concentrated harmonic spheres. Namely, for each such $p$ there exists a finite collection of harmonic maps $\omega_{k}:{\mathbb{R}}^{2}\to N$, points $p_{j}^{k}\to p$ and scales $\mu_{j}^{k}\to\infty$ so that, working in stereographic coordinates centred at $p$,

[TABLE]

We recall that the conformal invariance of the energy and the result [12] of Sacks-Uhlenbeck imply that a finite energy map $u:{\mathbb{R}}^{2}\to N$ is harmonic if and only if $u\circ\pi^{-1}:S^{2}\to N$ is a harmonic map from the sphere, $\pi(x)=(\frac{2x}{1+|x|^{2}},\frac{1-|x|^{2}}{1+|x|^{2}}):{\mathbb{R}}^{2}\cup\{\infty\}\to S^{2}$ the inverse stereographic projection. In the following we can thus switch viewpoint and work on ${\mathbb{R}}^{2}$, or equivalently on $\hat{}{\mathbb{C}}={\mathbb{C}}\cup\{\infty\}$, whenever this is more convenient and, except in the trivial case where $E(u_{j})\to 0$, can pull-back a sequence of maps $u_{j}$ satisfying (1.2) by suitable Möbius transforms to ensure that the base map $\omega_{0}$ is non-constant.

If no bubbles form, i.e. if the maps converge strongly in $H^{2}(S^{2})$, and if $N$ is analytic then the seminal results [13] of Simon are applicable. These ensure that for every harmonic map $u^{*}:S^{2}\to N$ there exists an $\varepsilon=\varepsilon(u^{*})>0$ so that a Łojasiewicz-estimate of the form

[TABLE]

holds true for all maps in the neighbourhood

[TABLE]

Conversely these results cannot be applied to study sequences of harmonic maps that undergo bubbling as such maps will not be $H^{1}$ close to a fixed harmonic map, or indeed to any set of harmonic maps for which the results of [13] yield Łojasiewicz-estimates on a uniform $\varepsilon$-neighbourhood, compare Remark 4.3. To the authors knowledge the only properties of the energy spectrum of harmonic maps into general analytic manifolds that are currently known are the ones that follow from the above mentioned compactness theory and from the work of Simon, i.e. that

• •

The lowest non-trivial energy level $E_{\text{min}}:=\inf\Xi_{E}(N)$ is achieved and isolated in the energy spectrum.

• •

Any potential accumulation point of $\Xi_{E}(N)$ must be of the form $\bar{E}=\sum_{i=0}^{n}E(\omega_{i})$ for some non-trivial harmonic spheres $\omega_{i}$, $i=0,\ldots,n\geq 1$, and such an accumulation point must be due to the existence of harmonic maps $u_{j}$ with energies $\bar{E}\neq E(u_{j})\to\bar{E}$ that converge to a non-trivial bubble tree.

The lowest level which might potentially be an accumulation point of the energy spectrum is hence $2E_{\text{min}}$ and so a natural starting point to develop a better understanding of $\Xi_{E}$, and to gain new insight into the above mentioned conjecture, is to investigate whether energies of harmonic maps might accumulate at $2E_{\text{min}}$. In the present paper we exclude this possibility for $3$ manifolds under some natural assumptions on the minimal energy harmonic spheres

Theorem 1.1**.**

*Let $(N,g_{N})$ be any analytic $3$ manifold and let $E_{\text{min}}=\min\Xi_{E}(N,g_{N})$ be the minimal energy of a non-trivial harmonic sphere in $N$. Suppose that harmonic spheres $\omega:S^{2}\to N$ with this energy $E_{\text{min}}$ are unbranched, non-degenerate critical points and that any intersection or self-intersection between such minimal spheres is transversal in the sense that the corresponding tangent spaces do not coincide.

Then $2E_{\text{min}}$ cannot be an accumulation point of the energy spectrum $\Xi_{E}(N,g_{N})$.*

As the energy is invariant under conformal changes we know that any vector field of the form $Y=\tfrac{d}{d\varepsilon}|_{\varepsilon=0}\,\omega\circ M_{\varepsilon}$, $M_{\varepsilon}$ a family of Möbius transforms with $M_{\varepsilon=0}=\mathrm{Id}$, is a Jacobi field along $\omega$, i.e. so that $d^{2}E(\omega)(Y,\cdot)=0$. We hence say that $\omega$ is non-degenerate if all Jacobi-fields along $\omega$ are generated in this way.

We recall that the results of Gulliver, Osserman and Royden [2] ensure that minimial surfaces in $3$-manifolds that are locally area minimising cannot have any true branch points. Any harmonic sphere with energy $E_{\text{min}}$ that is a local area minimiser will hence automatically be unbranched. Of course the example of an equatorial sphere in $S^{3}$ shows that harmonic spheres of minimal energy might not be local minimisers of the Area and to the author’s knowledge it is not known whether there any minimal energy harmonic spheres in a $3$ manifold that are branched.

The above theorem is valid also for manifolds which are not analytic but for which it is known that prime harmonic spheres of energy $2E_{\text{min}}$ are non-degenerate critical points.

As the bumpy metric theorems obtained by White in [19, 20, 21] and by Moore in [8] and in Theorems 5.1.1 and 5.1.2 of [9], ensure that for generic metrics on manifolds of dimension at least $3$ all prime harmonic maps are non-degenerate critical points, not branched and that all potential intersections and self-intersections are transverse we hence obtain

Corollary 1.2**.**

For generic $3$ manifolds the possibility that $2E_{\text{min}}$ is an accumulation point of the energy spectrum $\Xi_{E}(N)$ is excluded and Łojasiewicz-estimates as stated in Theorems 1.3, 1.5 and 1.7 below hold true for any sequence of almost harmonic maps that converges to a bubble tree with a single bubble.

We remark that while standard arguments show that the Łojasiewicz-estimate (1.3) holds true (with $\gamma=2$) in an $\varepsilon=\varepsilon(u^{*})>0$ neighbourhood of any non-degenerate harmonic map $u^{*}$ into a smooth manifold, these arguments cannot be used to obtain the above result as sequences of harmonic maps undergoing bubbling will not be in a neighbourhood of a fixed such map.

To prove Theorem 1.1 and the above corollary we need to exclude the possibility that there exists a sequence of harmonic maps $u_{j}:S^{2}\to N$ with energy $2E_{\text{min}}\neq E(u_{j})\to 2E_{\text{min}}$. We know that such a sequence cannot converge strongly as that would contradict [13], so (after pull-back by suitable Möbius tranforms) must converge to a bubble tree with a base map $\omega_{0}$ and a bubble $\omega_{1}$ of energy $E(\omega_{0})=E(\omega_{1})=E_{\text{min}}$.

We hence need to ask whether it is possible to glue increasingly concentrated harmonic spheres onto harmonic base maps in a way that results in a harmonic map whose energy is close, but not equal, to $E(\omega_{0})+E(\omega_{1})$. We will not only exclude this for harmonic spheres as considered in Theorem 1.1, but will establish obstructions to gluing harmonic spheres that apply in more general settings, including situations where the involved harmonic spheres are branched and have non-trivial, and even non-integrable, Jacobi-fields.

It is natural to distinguish between bubble tree for which the bubble $\omega_{1}$ and base $\omega_{0}$

1. (1)

parametrise the same minimal surface with the same orientation 2. (2)

parametrise the same minimal surface with the opposite orientation 3. (3)

parametrise transversally intersecting minimal surfaces.

We note that the no-neck property of the convergence to a bubble tree ensures that we only have to consider maps $\omega_{0}$ and $\omega_{1}$ for which $\omega_{0}(p)=\omega_{1}(\infty)$, $p$ the point at which the bubble forms. After rotating the domain, we can assume without loss of generality that the bubble is attached at the north pole, so in stereographic coordinates at $z=0$. In the following we hence only have to consider maps $u_{j}:\hat{}{\mathbb{C}}\to N$ for which there exist $\mu_{j}\to\infty$ and $z_{j}\to 0$ for which

[TABLE]

Our analysis is not restricted to bubble trees for which $\omega_{0}$ and $\omega_{1}$ are unbranched but we can more generally consider bubble trees with base maps and bubbles which are obtained as a composition of a harmonic sphere $U^{*}:\hat{}{\mathbb{C}}\to N$ with $dU^{*}(q^{*}(0))\neq 0$ and a rational map $q^{*}:\hat{}{\mathbb{C}}\to\hat{}{\mathbb{C}}$ of arbitrary degree, i.e. a map of the form $q^{*}(z)=\frac{p_{1}(z)}{p_{2}(z)}$, $p_{1,2}$ polynomials. As we can precompose $U^{*}$ with a rotation we can restrict our attention to rational maps $q^{*}(z)$ which map [math] to [math] and so will always ask that $dU^{*}(0)\neq 0$.

We note that the conformal invariance of the energy ensures that any map of the form $U\circ q$, $U$ harmonic and $q$ rational, will be a harmonic map and that any vector field of the form $Y=\tfrac{d}{d\varepsilon}|_{\varepsilon=0}(U\circ q_{\varepsilon})$, $q_{\varepsilon}$ rational maps, is a Jacobi field along $U\circ q_{\varepsilon=0}$. For higher order coverings of harmonic spheres it is hence natural to say that

Definition 1**.**

The second variation of the energy is non-degenerate along a harmonic map $\omega$ of the form $\omega=U\circ q$ if every Jacobi-field $Y$ along $\omega$ is generated by a variation of rational maps $q_{\varepsilon}$, i.e. given by $Y=\tfrac{d}{d\varepsilon}|_{\varepsilon=0}(U\circ q_{\varepsilon})$.

We first consider the case where the base and the bubble parametrise the same minimal surface with the same orientation. Here we include settings in which $U^{*}(0)$ is a point of higher multiplicity of an immersed minimal sphere $U^{*}(\hat{}{\mathbb{C}})$ provided $\omega_{0}$ and $\omega_{1}(\frac{1}{z})$ parametrise the same leaf of $U^{*}(\hat{}{\mathbb{C}})$ near $z=0$, but will instead include settings in which they parametrise transversally intersecting leafs in the third case, compare Theorem 1.7 below.

So suppose that $\omega_{0}$ and $\omega_{1}$ are obtained by composing the same harmonic $U^{*}:\hat{}{\mathbb{C}}\to N$ with rational maps $q_{0}^{*}(z)$ and $q_{1}^{*}(\frac{1}{z})$ for which $q_{0}^{*}(0)=q_{1}^{*}(0)=0$. We want to show that the only sequences of harmonic maps $u_{j}$ that converge to such a bubble tree are higher degree coverings of $U^{*}:S^{2}\to N$.

This does not follow from the existing theory since the size of the neighbourhoods on which Łojasiewicz-estimates are known to hold true shrink to zero as the maps undergo bubbling, compare Remark 4.3. If $u_{j}$ is a sequence of harmonic maps that converges to such a bubble tree then the compactness theory from [14, 1, 10, 4] ensures that $\|u_{j}-U^{*}\circ r_{j}\|_{L^{\infty}\cap\dot{H}^{1}}\to 0$ for some rational maps $r_{j}$. However the existing theory does not provide the quantitative estimates on the rate of this convergence which would allow one to know that the maps $u_{j}$ are in the smaller and smaller neighbourhoods $\mathcal{U}(U^{*}\circ r_{j})$ for which the results of Simon apply.

Conversely, our method allows us to prove that Łojasiewicz-estimates indeed hold true on a uniform $\varepsilon$ neighbourhood around such a non-compact set of harmonic maps $\{U^{*}\circ r\}$ and hence that such sequences of harmonic maps cannot be responsible for an accumulation point of the energy spectrum. To be more precise, we show

Theorem 1.3**.**

*Let $(N,g_{N})$ be a smooth Riemannian manifold of any dimension. Suppose that $U^{*}:\hat{}{\mathbb{C}}\to N$ is a harmonic map with $dU^{*}(0)\neq 0$ and ${q_{0}^{*}},{q_{1}^{*}}:\hat{}{\mathbb{C}}\to\hat{}{\mathbb{C}}$ are rational maps with $q_{1}^{*}(0)=q_{0}^{*}(0)=0$ so that $d^{2}E$ is non-degenerate at $\omega_{0}(z):=U^{*}\circ{q_{0}}^{*}$ and $\omega_{1}(z):=U^{*}\circ{q_{1}}^{*}(\frac{1}{z})$ in the sense of Definition 1.

Let $(u_{j})$ be any sequence of almost harmonic maps which converges to a bubble tree with base map $\omega_{0}$ and bubble $\omega_{1}$ as described in (1.1) and (1.4). Then there exists a constant $C$ so that*

[TABLE]

*for all sufficiently large $j$ and for $\mu_{j}\to\infty$ as in (1.4).

Furthermore, if the maps $u_{j}$ are harmonic, then, after precomposing with a suitable rotation $R_{j}$ of the domain, they can be written as*

[TABLE]

for rational maps $q_{i}^{j}$ which converge smoothly to $q_{i}^{*}$ when viewed as maps from $S^{2}$ to $S^{2}$.

Here and in the following $\|\cdot\|_{*}$ denotes a weighted $H^{-1}$ norm which is defined in Remark 4.1 and which is chosen so that the above estimate (1.5) is invariant under pull-back by Möbius transforms. We note that $\|\tau_{g_{S^{2}}}(u)\|_{L^{2}(S^{2},g_{S^{2}})}$, which dominates $\|dE(u)\|_{*}$, does not have this property. However while it is natural work with $\|dE(u)\|_{*}$ to analyse the energy spectrum, it also of interest to obtain $L^{2}$-Łojasiewicz-estimates e.g. to analyse the asymptotic behaviour of the harmonic map heat flow

[TABLE]

We shall hence also prove

Corollary 1.4**.**

For sequences of maps $u_{j}$ as in Theorem 1.3 we can bound

[TABLE]

A consequence of this corollary is that solutions $u(t)$ of the flow (1.6) for which sequences $u(t_{j})$, $t_{j}\to\infty$, are known to subconverge to a bubble tree as considered in Theorem 1.3 must indeed converge exponentially fast to a unique base map $u_{\infty}:S^{2}\to N$ as $t\to\infty$ away from a unique point $p_{\infty}\in S^{2}$ in the sense of both $L^{2}(S^{2},g_{S^{2}})$ and $C^{k}_{loc}(S^{2}\setminus\{p_{\infty}\})$.

In the one bubble case this extends results that were previously obtained by Topping [15] in the case where $N$ is the round $2$-sphere and that were later extended by Liu and Yang in [6] to compact Kähler manifolds with nonnegative holomorphic bisectional curvature. We note that the results from [15, 6] are not restricted to the one bubble case but apply to sequences of maps and solutions of the flow which converge to any bubble tree for which the base map and the bubbles are all parametrisations of the same minimal spheres with the same orientation.

The above corollary suggest that while the special geometric structure of the target is crucially used in the proofs of [15, 6], analogue results might be valid for more general targets and one would expect that the main property needed is the non-degeneracy of the second variation of the energy at the underlying harmonic spheres $U^{*}\circ q_{i}^{*}$, which for $N=S^{2}$ follows from [3].

Next we want to show that it is impossible to glue a highly concentrated bubble onto a base map which parametrises the same minimal surface, but with opposite orientation. Indeed, we will prove more generally that for almost harmonic maps that converge to such a bubble tree the energy defect and the bubble scale are controlled in terms of $\|dE(u)\|_{*}$.

Theorem 1.5**.**

Let $(N,g_{N})$ be a smooth Riemannian manifold of any dimension, let $U^{*}:\hat{}{\mathbb{C}}\to N$ be any harmonic map with $dU^{*}(0)\neq 0$ and let $q_{0}^{*},q_{1}^{*}:\hat{}{\mathbb{C}}\to\hat{}{\mathbb{C}}$ be any rational maps with ${q_{1}^{*}}(0)={q_{0}^{*}}(0)=0$ so that so that $d^{2}E$ is non-degenerate at $\omega_{0}(z):=U^{*}\circ{q_{0}}^{*}$ and $\omega_{1}(z):=U^{*}\circ{q_{1}}^{*}(\frac{1}{\bar{z}})$ in the sense of Definition 1.

Let $(u_{j})$ be any sequence of almost harmonic maps which converges to a bubble tree with base map $\omega_{0}$ and bubble $\omega_{1}$ as described in (1.1) and (1.4). Then the bubble scale must satisfy

[TABLE]

$n_{i}^{*}\in{\mathbb{N}}$* the order of the zero of $q_{i}^{*}$ at $z=0$, while the energy defect is controlled by*

[TABLE]

In particular, no sequence of harmonic maps can converge to such a bubble tree.

For maps to the round sphere $L^{2}$-Łojasiewicz-estimates with exponent $\gamma=2$ and exponential bounds on the bubble scale were proven by Topping in [16]. In this setting there are only two types of harmonic spheres, namely holomorphic or antiholomorphic maps. The results of [16] apply for maps that converge to any bubble tree which is so that for each singular point $p$ the bubbles forming at $p$ are all of the same type, where for points $p$ at which bubbles form that have a different type than the base map one needs to impose additionally that the base map is not branched.

Topping already observed in [17] that even for maps into $(S^{2},g_{S^{2}})$ one cannot expect to prove $L^{2}$-Łojasiewicz-estimates with exponent $\gamma=2$ if the base map is branched at a point where a bubble of a different type forms.

At the same time, any $L^{2}$-Łojasiewicz-estimate with an exponent $\gamma>1$ is sufficient to prove convergence results for solutions of harmonic map flow. As this is our main motivation to also prove Łojasiewicz-estimates that involve $\|\tau_{g_{S^{2}}}(u)\|_{L^{2}(S^{2},g_{S^{2}})}$ rather than the weaker, and scaling invariant $\|dE(u)\|_{*}$, we shall hence simply show

Corollary 1.6**.**

For any sequence $(u_{j})$ of almost harmonic maps as in Theorem 1.5 we can bound

[TABLE]

for an exponent $\gamma>1$ that only depends on $U^{*}$ and $q_{0,1}^{*}$.

We finally want to show that for maps into $3$ manifolds it is impossible to glue two harmonic spheres $\omega_{0}$ and $\omega_{1}$ which are so that $\omega_{0}$ and $\omega_{1}(\frac{1}{z})$ parametrise transversally intersecting minimal surfaces, respectively transversally intersecting leafs of the same minimal surface, near $z=0$.

Our result in this setting also applies for harmonic spheres which are degenerate critical points, i.e. which have Jacobi-fields that are not generated by variations of rational maps. Indeed, we can even deal with harmonic spheres $\omega_{i}$ which have non-integrable Jacobi-fields, i.e. for which $d^{2}E(\omega_{i})$ has null directions that are not tangential to the set of harmonic maps near $\omega_{i}$.

The one thing we want to ask is that for maps $\omega_{i}$ that are obtained as higher order coverings of a map $U_{i}^{*}$ this structure is reflected also at the level of Jacobi-fields. Namely, if $q_{i}^{*}$ has degree strictly greater than $1$ then we ask that all Jacobi-fields along $\omega_{i}=U_{i}^{*}\circ q^{*}$ are generated by a combination of Jacobi-fields along $U_{i}^{*}$ and Jacobi-fields that are generated by variations of the rational maps, i.e.

[TABLE]

$L_{u}$ the Jacobi operator along $u$. Here we note that the fact that $W\circ q^{*}$ is a Jacobi-field along $U^{*}\circ q^{*}$ whenever $W$ is a Jacobi-field along $U^{*}$ follows from the conformal invariance of the energy and the resulting formula

[TABLE]

for the transformation of the tension under conformal changes.

As our final main result we prove

Theorem 1.7**.**

Let $(N,g_{N})$ be any analytic $3$ manifold and let $U_{0,1}^{*}:\hat{}{\mathbb{C}}\to N$ be any harmonic spheres with $U_{1}^{*}(0)=U_{0}^{*}(0)$ and $dU_{0,1}^{*}(0)\neq 0$ which are so that the tangent spaces $T_{U_{i}^{*}(0)}U_{i}^{*}(\hat{}{\mathbb{C}})$ to the corresponding minimal surfaces $U_{i}(\hat{}{\mathbb{C}})$ do not coincide. Let $q_{0}^{*}$, $q_{1}^{*}$ be any rational maps with $q_{0}^{*}(0)=q_{1}^{*}(0)=0$, assumed to satisfy (1.9) if their degree is at least $2$.

Then for any sequence $(u_{j})$ of almost harmonic maps that converges to a bubble tree with base map $\omega_{0}=U_{0}^{*}\circ q_{0}^{*}$ and bubble $\omega_{1}(z)=U_{1}^{*}(q_{1}^{*}(\frac{1}{z}))$ as described in (1.1) and (1.4) we can bound the bubble scale by

[TABLE]

and we have a Łojasiewicz-estimate of the form

[TABLE]

In particular, no sequence of harmonic maps can converge to such a bubble tree.

Remark 1.8**.**

If $U_{0,1}^{*}$ are non-degenerate critical points, then the above theorem remains valid also without the assumption that $(N,g_{N})$ is analytic.

We note that in contrast to Theorems 1.3 and 1.5 which are valid for target manifolds of arbitrary dimension, here we have to impose that the target is $3$ dimensional. We cannot expect to obtain the same repulsion effect if we were to consider maps into higher dimensional manifolds such as $N=S^{2}\times S^{2}\hookrightarrow{\mathbb{R}}^{6}$ where e.g. $u_{j}(\cdot):=(\pi(\cdot),\pi(\frac{1}{\mu_{j}\cdot})):{\mathbb{R}}^{2}\to N$, $\mu_{j}\to\infty$, are harmonic maps which converge to a bubble tree for which the tangent spaces of the base $\omega_{0}=(\pi,\pi(\infty))$ and the bubble $\omega_{1}=(\pi(\infty),\pi(\frac{1}{\cdot}))$ intersect transversally in a point.

**Outline of the paper:

**To prove our main results we use a general method that was first developed in the joint work [7] of Malchiodi, Sharp and the author on $H$-surfaces. A crucial aspect of this method is that it allows us to prove Łojasiewicz estimates for sequences of (almost) critical points of an energy $E$ that converge to a singular limit, here a bubble tree, without having to analyse the properties of such general (almost) critical points.

Instead, if we can construct a suitable finite dimensional (non-compact) manifold $\mathcal{Z}$ of singularity models so that the restriction of the energy to $\mathcal{Z}$ has the right properties, then this method allows us to obtain Łojasiewicz-estimates that are valid on a uniform $\varepsilon$ neighbourhood of $\mathcal{Z}$ based on the properties of the energy and its variation on $\mathcal{Z}$.

These singularity models will in general not be critical points of the energy, but will always be so that $\|dE(\mathfrak{z})\|_{*}$ is small as they serve as models for the behaviour of almost critical points. The key properties $\mathcal{Z}$ needs to satisfy are that

• (i)

The second variation of the energy is uniformly definite orthogonal to $\mathcal{Z}$

• (ii)

For each $\mathfrak{z}\in\mathcal{Z}$ which is not a critical point of $E$ we can identify a direction on $\mathcal{Z}$ in which the variation of $E$ has a sign and suitable scaling. To be more precise, we need that for each such $\mathfrak{z}\in\mathcal{Z}$ there exists a unit element $y_{\mathfrak{z}}\in T_{\mathfrak{z}}\mathcal{Z}$ so that

[TABLE]

Our proofs hence consist of three main steps: the construction of the manifold $\mathcal{Z}$ of singularity models, the analysis of the energy and its variations on $\mathcal{Z}$ and finally the arguments of how these properties of $E$ on $\mathcal{Z}$ yield our main results. After explaining the construction of the singularity models in Section 2 we state the relevant properties of $E$ on $\mathcal{Z}$ in a series of lemmas in the subsequent Section 3, but postpone the rather technical proofs until Sections 5 and 6 as an understanding of these proofs is not required to complete the proofs of the main results which are carried out in Section 4.

2. Construction of the singularity models

In this section we construct the singularity models which are maps $\mathfrak{z}:S^{2}\to N$ that model the behaviour of sequences of almost harmonic maps that converge to a bubble tree with base $\omega_{0}=U_{0}^{*}\circ q_{0}^{*}$ and bubble $\omega_{1}(z)=U_{1}^{*}\circ q_{1}^{*}(\frac{1}{z})$ where here and in the following $U_{i}^{*}:\hat{}{\mathbb{C}}\to N$ always denote harmonic maps with $dU_{i}^{*}(0)\neq 0$ and $q_{i}^{*}$ are elements of

[TABLE]

It suffices to construct singularity models $\mathfrak{z}:S^{2}\to N$ for which the bubble is attached at the north pole and we denote the corresponding set of maps by $\mathcal{Z}_{0}$. Once we have constructed $\mathcal{Z}_{0}$ we can then obtain the full manifold of singularity models as

[TABLE]

where $R_{p}\in SO(3)$ denotes the rotation which maps the plane containing [math], $p$ and the north pole $\pi(0)$ to itself and which is so that $R_{p}(p)=\pi(0)$ (with the convention that $R_{p}=\mathrm{Id}$ if $p=\pi(0)$).

To construct these maps $\mathfrak{z}\in\mathcal{Z}_{0}$ we work in fixed stereographic coordinates where the base and bubble will be represented by harmonic maps $U_{i}^{*}:\hat{}{\mathbb{C}}\to N$ and rational maps $q_{i}^{*}:\hat{}{\mathbb{C}}\to\hat{}{\mathbb{C}}$. As discussed in the introduction we need to analyse the three cases where either $U_{0}^{*}=U_{1}^{*}$ or $U_{0}^{*}(\bar{z})=U_{1}^{*}(z)$ or where $U_{0,1}^{*}$ parametrise minimal spheres in a 3-manifold that intersect transversally in $U_{1}^{*}(0)=U_{0}^{*}(0)$.

We can assume without loss of generality that the rational maps $q_{0,1}^{*}$ are so that the leading order coefficients in $q_{i}^{*}(z)=a_{n_{i}^{*}}(q_{i}^{*})z^{n_{i}^{*}}+\sum_{j>n_{i}^{*}}a_{j}(q_{i}^{*})z^{j}$ are so that $|a_{n_{i}^{*}}(q_{i}^{*})|=1$. Indeed, to ensure this for $q_{0}^{*}$ we can simply replace $U_{0}^{*}$ by $U_{0}^{*}(c\cdot)$ for a suitable $c>0$, while replacing the chosen scales $\mu_{i}$ in the bubble tree convergence (1.4) by $c\mu_{i}$ allows us to assume that this holds true for $q_{1}^{*}$ while still preserving the relations $U_{0}^{*}=U_{1}^{*}$ respectively $U_{0}^{*}(z)=U_{1}^{*}(\bar{z})$.

To obtain a manifold $\mathcal{Z}$ which is so that $d^{2}E$ is definite orthogonal to $\mathcal{Z}$ we first need to determine suitable sets $\mathcal{H}_{1}^{\sigma}(U_{i}^{*})$ of maps which are so that the tangent space to

[TABLE]

at $\omega_{i}=U_{i}^{*}\circ q_{i}^{*}$ coincides with the space $\ker(L_{\omega_{i}})$ of Jacobi-fields along $\omega_{i}$. Here and in the following $\sigma>0$ denotes a small constant that is chosen later and that is in particular small enough so that all maps in

[TABLE]

have the same degree as $q_{i}^{*}$ and are so that $|a_{n_{i}^{*}}(q_{i})|\in(\frac{1}{2},2)$. Here and in the following we can choose $k\geq 2$ to be any fixed number.

If $\omega_{i}$ is non-degenerate, i.e. if all Jacobi fields are generated by variations of rational maps, then we can simply choose

[TABLE]

More generally if all Jacobifields along $U_{i}^{*}$ are integrable we first choose a manifold $\mathcal{H}_{0}(U_{i}^{*})$ of harmonic maps of dimension $\dim(\ker L_{U_{i}^{*}})-6$ which is so that each harmonic map close to $U_{i}^{*}$ can be written uniquely in the form $U\circ M$ for a Möbiustransform $M$ and an element $U$ of $\mathcal{H}_{0}(U_{i}^{*})$. We then set

[TABLE]

and note that (1.9) ensures that the resulting $\mathcal{H}^{\sigma}(\omega_{i})$ is indeed so that $T_{\omega_{i}}\mathcal{H}^{\sigma}(\omega_{i})=\ker(L_{\omega_{i}})$.

In the more involved case where $U_{i}^{*}$ has non-integrable Jacobi-fields we need to additionally include certain non-harmonic maps in $\mathcal{H}^{\sigma}(\omega_{i})$, and hence in $\mathcal{H}_{1}^{\sigma}(U_{i}^{*})$ and $\mathcal{H}_{0}(U_{i}^{*})$, to ensure that also these non-integrable Jacobi-fields correspond to directions that are tangent to $\mathcal{H}^{\sigma}(\omega_{i})$. In this case, we first choose $\mathcal{H}_{0}(U_{i}^{*})$ as described in detail in Appendix A of [11] as a manifold of smooth maps with

[TABLE]

which has the key properties that for every $U\in\mathcal{H}_{0}(U_{i}^{*})$ we can bound

[TABLE]

and find a variation $U_{\varepsilon}$ in $\mathcal{H}_{0}(U_{i}^{*})$ of $U$ with $\|\partial_{\varepsilon}(U_{\varepsilon}\circ\pi^{-1})\|_{C^{k}}\leq C$ for which

[TABLE]

see Lemma 2.1 of [11].

We then again define $\mathcal{H}_{1}^{\sigma}(U_{i}^{*})$ by (2.5) and note that this manifold inherit the properties (2.7) and (2.8), while (1.9) ensures that $T_{\omega_{i}}\mathcal{H}^{\sigma}(\omega_{i})=\ker(L_{\omega_{i}})$ for $\mathcal{H}^{\sigma}(\omega_{i})$ defined by (2.3).

Having chosen $\mathcal{H}_{1}^{\sigma}(U_{i}^{*})$ in this way we now obtain our singularity models by carefully gluing a highly concentrated copy of an element of $\mathcal{H}_{1}^{\sigma}(U_{1}^{*})$ onto an element of $\mathcal{H}_{1}^{\sigma}(U_{0}^{*})$. That is we want to construct the elements $\mathfrak{z}$ of $\mathcal{Z}_{0}$ in a way that $\mathfrak{z}\approx U_{0}(q_{0}(z))$ near $z=0$ while $\mathfrak{z}(z)\approx U_{1}\circ q_{1}(\frac{1}{\mu z})$ away from zero for maps $U_{i}\in\mathcal{H}_{1}^{\sigma}(U_{i}^{*})$, $q_{i}\in\mathcal{R}^{\sigma}(q^{*}_{i})$, $i=0,1$ and a large scaling factor $\mu$.

Remark 2.1**.**

We will always consider elements that are obtained from maps in $\mathcal{H}_{1}^{\sigma}(\omega_{i})$ and scaling factors $\mu>\bar{\mu}$ for a sufficiently large number $\bar{\mu}$ and a sufficiently small number $\sigma>0$ (depending only on $U_{i}^{*}$, $q_{i}^{*}$) and in the following all claims and estimates are to be understood to hold true after increasing $\bar{\mu}$ and decreasing $\sigma$ if necessary.

To describe this construction we first assign to each $q_{0}\in\mathcal{R}^{\sigma}(q_{0}^{*})$ and each $q_{1}(\frac{1}{\mu z})$ that we obtain from an element $q_{1}\in\mathcal{R}^{\sigma}(q_{1}^{*})$ and a scaling factor $\mu>\bar{\mu}$ the numbers

[TABLE]

We note that while there are multiple ways of representing the same rational map $q_{1}(\frac{1}{\mu z})$ using an element of $\mathcal{R}^{\sigma}(q_{1}^{*})$ and a factor $\mu>\bar{\mu}$, the number $\mu_{1}$ is uniquely determined by $q_{1}(\frac{1}{\mu z})$ and while $\mu_{0}$ is of order $1$, $\mu_{1}$ will be of order $\mu$.

Some of the gluing construction below will be carried out on annuli

[TABLE]

whose radii $\mu^{-1}\ll r_{1}\ll\hat{r}\ll r_{0}\ll 1$ are determined by

[TABLE]

Here $\mu\mapsto f_{\mu}$ is a fixed non-decreasing function which is so that

[TABLE]

for $\mu>\bar{\mu}$ and a small constant $\sigma_{1}\in(0,\frac{1}{4}]$. We use the same convention for $\sigma_{1}$ as for $\sigma$, i.e. ask in the following that all claims hold provided ${f_{\mu}}$ satisfies this assumption for a sufficiently small number $\sigma_{1}>0$ that only depends on $U_{0,1}^{*}$ and $q_{0,1}^{*}$.

The precise choice of ${f_{\mu}}$ is not important in the construction and it will be useful later on that we can fix ${f_{\mu}}$ according to our needs as for the proof of Theorems 1.7 it will be convenient to work with ${f_{\mu}}=\log\mu$ while in the proof of Corollary 1.6 we will want to choose ${f_{\mu}}=\mu^{\sigma_{1}}$.

Remark 2.2**.**

We note that the choice of the above radii and annuli is so that the construction is invariant under a rescaling $z\mapsto\lambda z$ of the stereographic coordinates and symmetric with respect to interchanging the roles of the base and the bubble as we change the coordinates according to $z\mapsto\frac{\mu_{0}}{\mu_{1}z}$.

To construct our singularity models we will first modify the maps

[TABLE]

on $A$ so that they agree upto first order errors in $|z|+|\mu z|^{-1}$, then further modify the resulting maps on all of ${\mathbb{R}}^{2}$ so that they agree upto second order before gluing these maps together on the central annulus $\hat{A}$ and projecting onto $N$.

To this end we let $\hat{\gamma}=\hat{\gamma}_{{U_{1}}(0),{U_{0}}(0)}:[0,1]\to N$ be the shortest geodesic from $\hat{\gamma}(0)=U_{1}(0)$ to $\hat{\gamma}(1)=U_{0}(0)$. We note that $\hat{\gamma}$ is well defined as $\delta({U_{0}},{U_{1}}):=\text{dist}_{N}({U_{0}}(0),{U_{1}}(0))$ is small for all elements of $\mathcal{H}^{\sigma}(U_{i}^{*})$ as such maps are $C^{k}$ close to $U_{i}^{*}$ and $U_{0}^{*}(0)=U_{1}^{*}(0)$.

To transition between $U_{1}(0)$ and $U_{0}(0)$ on $A$ we use

[TABLE]

where $\phi_{r_{0},r_{1}}=\phi_{r_{0},r_{1}}(|x|):{\mathbb{R}}^{2}\to[0,1]$ is defined by

[TABLE]

for ${c_{r_{0},r_{1}}}:=(\log(\frac{r_{0}}{r_{1}}))^{-1}$ and a fixed cut-off function $\phi\in C_{c}^{\infty}([0,2),[0,1])$ with $\phi\equiv 1$ on $[0,1]$.

We note that $\phi_{r_{0},r_{1}}$ is chosen so that $\Delta\phi_{r_{0},r_{1}}\equiv 0$ outside of $A^{*}$ and so that $\phi_{r_{0},r_{1}}|_{\mathbb{D}_{r_{1}}}\equiv 0$ while $\phi_{r_{0},r_{1}}|_{(\mathbb{D}_{r_{0}})^{c}}\equiv 1$. Hence $\gamma:{\mathbb{R}}^{2}\to N$ is given by a harmonic map outside of $A^{*}$ and transitions between $\gamma|_{\mathbb{D}_{r_{1}}}\equiv U_{1}(0)$ and $\gamma|_{\mathbb{D}_{r_{0}}^{c}}\equiv U_{0}(0)$. Setting

[TABLE]

we can hence change our original maps $u_{0,1}$ into maps $u_{1}(z)+{\tilde{\gamma}_{{U_{1}}}}(z)$ and $u_{0}(z)+{\tilde{\gamma}_{{U_{0}}}}(z)$ that agree upto error terms of order $O(|z|+\frac{1}{\mu|z|})$ on $A$. Next we set

[TABLE]

to obtain maps

[TABLE]

and

[TABLE]

which now agree upto a second order error term $v_{0}-v_{1}={\beta}({q_{\mu}})-{\beta}(0)-d{\beta}(0)({q_{\mu}}).$

We finally interpolate between these maps on the central annulus $\hat{A}$ and project onto $N$. That is we define our singularity model as

[TABLE]

where we set $\varphi_{\hat{r}}(z):=\bar{\varphi}(\hat{r}^{-1}|z|)$ for a fixed smooth function $\bar{\varphi}:[0,\infty)\to[0,1]$ which is so that $\bar{\varphi}(t)=0$ for $t\leq\frac{1}{2}$ while $\bar{\varphi}(t)=1$ for $t\geq 2$.

We note that these maps are well defined for $U_{0,1}\in\mathcal{H}^{\sigma}(U_{0,1}^{*})$ and $q_{0,1}\in\mathcal{R}^{\sigma}(q_{0,1}^{*})$ as the image of $(1-\varphi_{\hat{r}})v_{1}+\varphi_{\hat{r}}v_{0}$ will be contained in a small tubular neighbourhood of $N$ where the nearest point projection is well defined.

Remark 2.3**.**

An important feature of our construction is that the modifications to the maps $u_{i}$ are not just supported on an annulus, but rather on all of ${\mathbb{R}}^{2}$. In contrast to maps we would get from a more ad hoc gluing construction this means that $dE(\mathfrak{z})$ has a significant part that is supported on $\mathbb{D}_{r_{1}}$ and $\mathbb{D}_{r_{0}}^{c}$. This will be crucial in the proof of our main results later on as variations of elements of $\mathcal{Z}_{0}$ induced by variations of one of the rational maps are mainly concentrated on $\mathbb{D}_{r_{1}}$ respectively $\mathbb{D}_{r_{0}}^{c}$ and hence can only be used to obtain directions $y_{\mathfrak{z}}\in T_{\mathfrak{z}}\mathcal{Z}$ satisfying (1.11) if $dE(\mathfrak{z})$ is so that it interacts suitably with such directions.

3. Key lemmas

Here we collect the key properties on the behaviour of the energy and its variations on $\mathcal{Z}$ that we need to prove our main results. We postpone the proofs of these lemmas to Sections 5 and 6, as an understanding of these technical proofs is not required for the proofs of our main results.

As these estimates will all be invariant under precomposition with a rotation, it suffices to state and prove them for elements of the set $\mathcal{Z}_{0}$ of singularity models for which the bubble is attached at the north pole. We can furthermore assume that $q_{1}(\frac{1}{\mu z})$ is represented by a map $q_{1}$ for which $|a_{n_{1}^{*}}(q_{1})|=1$ and hence use that $\mu_{1}$ and $\mu_{0}$ defined in (2.9) are so that $\mu_{1}=\mu$ while $\mu_{0}\sim 1$.

Given $\mathfrak{z}=\mathfrak{z}(U_{0}\circ q_{0},U_{1}\circ q_{1}(\frac{1}{\mu\cdot}))\in\mathcal{Z}_{0}$ we will work with respect to the weighted $H^{1}$-norm $\|\cdot\|_{\mathfrak{z}}$ and the corresponding inner product $\langle\cdot,\cdot\rangle_{\mathfrak{z}}$ which are given in stereographic coordinates by

[TABLE]

$\pi_{\lambda}(z):=\pi(\lambda z)$. We note that this definition respects the symmetry of the construction, compare Remark 2.2, and that we define $\|\cdot\|_{\mathfrak{z}}$ for general elements $\mathfrak{z}=\mathfrak{z}_{0}\circ R_{p}$ of $\mathcal{Z}$ by the analogue formula in stereographic coordinates centred at $p$.

We recall that the second variation of the energy is given by

[TABLE]

compare e.g. [11, Lemma 3.1], and note that $d^{2}E$ is uniformly bounded with respect to this norm, i.e. so that

[TABLE]

compare Section 6. Here and in the following we write $T_{1}\lesssim T_{2}$ if we can bound $T_{1}\leq CT_{2}$ for a constant $C$ that only depends on the limiting configuration, i.e. on $U_{0,1}^{*}$ and $q_{0,1}^{*}$ while we write $T_{1}\sim T_{2}$ if both $T_{1}\lesssim T_{2}$ and $T_{2}\lesssim T_{1}$.

The first key property we need is that the choice of $\mathcal{H}_{1}^{\sigma}(U^{*}_{0,1})$ in the construction of $\mathcal{Z}_{0}$ ensures that the second variation is uniformly definite orthogonal to $\mathcal{Z}$, i.e. that

Lemma 3.1**.**

There exist $c_{0}>0$ depending only on $U_{0,1}^{*}$ and $q_{0,1}^{*}$ so that for every $\mathfrak{z}\in\mathcal{Z}_{0}$ we can split $T_{\mathfrak{z}}^{\perp}\mathcal{Z}$ orthogonally into subspaces $\mathcal{V}_{\mathfrak{z}}^{\pm}$ which are so that

[TABLE]

To state all other properties of the energy $E$ on $\mathcal{Z}$ we now associate to each $\mathfrak{z}=\mathfrak{z}(U_{i},q_{i},\mu)$ the quantities

[TABLE]

where we set $\mathcal{T}_{i}:=\|\tau_{g_{S^{2}}}({U_{1}}\circ\pi^{-1})\|_{L^{2}(S^{2},g_{S^{2}})}$ and write $q_{i}(z)=\sum_{j}a_{j}(q_{i})z^{j}$ near $z=0$ to define

[TABLE]

$n_{0,1}^{*}\geq 1$ the order of the zero of the fixed rational maps $q_{0,1}^{*}$ at $z=0$. We remark that these quantities respect the symmetry discussed in Remark 2.2 and recall furthermore that $\mathcal{T}_{i}$ also controls the $C^{k}$-norm of the tension, compare (2.7).

The first quantitative estimate we need is a bound on the dual norm $\|dE(\mathfrak{z})\|_{*}:=\sup\{dE(\mathfrak{z})(w):\|w\|_{\mathfrak{z}}=1\}$ of the first variation and in Section 6 we shall prove

Lemma 3.2**.**

For any $\mathfrak{z}\in\mathcal{Z}_{0}$ we have

[TABLE]

Here and in the following we compute the norms of $\beta=U_{0}-U_{1}$ and $\partial_{\varepsilon}\beta$ as norms of the corresponding maps from $S^{2}$ i.e. set

[TABLE]

In the following we consider variations $\mathfrak{z}_{\varepsilon}=\mathfrak{z}(U_{0,1}^{(\varepsilon)},q_{0,1}^{(\varepsilon)},\mu)$ in $\mathcal{Z}$ which are induced by variations of $U_{i}^{(\varepsilon)}$ in $\mathcal{H}_{1}^{\sigma}(U_{i}^{*})$ and of $q_{i}^{(\varepsilon)}$ in $\mathcal{R}^{\sigma}(q_{i}^{*})$. We do not impose the assumption that $|a_{n_{i}}^{*}(q_{i}^{(\varepsilon)})|=1$ for $\varepsilon\neq 0$ but allow for general variations of $q_{i}$ in $\mathcal{R}^{\sigma}(q_{i})$ as this allows us to represent variations of $\mu$ such as $\mu_{\varepsilon}=\mu(1+\varepsilon)$ as variations of $q_{1}$ of the form $q_{1}^{(\varepsilon)}(z)=q_{1}(\frac{1}{1+\varepsilon}z)$. In the following we hence do not need to consider variations of $\mu$ separately.

To simplify the notation we will drop the index $\varepsilon$ whenever there is no room for confusion and use the convention that $\partial_{\varepsilon}$ is evaluated at $\varepsilon=0$ unless specified otherwise. We will only ever consider variations for which

[TABLE]

as these corresponds to variations in $\mathcal{Z}_{0}$ for which $\|\partial_{\varepsilon}\mathfrak{z}\|_{\mathfrak{z}}\lesssim 1$. We note that for such variations also $\|\partial_{\varepsilon}\mathfrak{z}\|_{L^{\infty}}\lesssim 1$ while $|\partial_{\varepsilon}\mu_{i}|\lesssim\mu_{i}$.

We associate to each such variation the quantities

[TABLE]

and furthermore set $\eta_{rat}=0$ whenever $\partial_{\varepsilon}q_{0}=\partial_{\varepsilon}q_{1}=0$ while otherwise we let $\eta_{rat}:=1$ and choose $j^{*}\geq 1$ so that both $\partial_{\varepsilon}q_{0}$ and $\partial_{\varepsilon}q_{1}$ are of order $O(z^{j^{*}})$ near $z=0$.

The dual norm $\|d^{2}E(\mathfrak{z})(\partial_{\varepsilon}\mathfrak{z},\cdot)\|_{*}:=\sup\{\partial_{\varepsilon}dE(\mathfrak{z}_{\varepsilon})(w):w\in\Gamma(\mathfrak{z}^{*}TN),\|w\|_{\mathfrak{z}}=1\}$ of the second variation of $E$ along such variations is controlled by the following lemma which we will prove in Section 6.

Lemma 3.3**.**

For any variation $\mathfrak{z}_{\varepsilon}$ for which (3.7) holds we can bound

[TABLE]

We note that for changes of $U_{i}$ that are induced by translations on the domain we can bound $\eta_{\tau}\lesssim\mathcal{T}$ since $\mathcal{T}_{i}$ controls $C^{k}$ norm of the tension of $U_{i}$, compare (2.7). For more general variations in $\mathcal{H}_{1}^{\sigma}(U_{i}^{*})$ we can instead use the continuity of the Jacobi operator and the fact that elements of $T_{U_{i}^{*}}{\mathcal{H}_{0}}({U_{i}^{*}})$ are Jacobifields along ${U_{i}^{*}}$ to see that

[TABLE]

will be smaller than any given positive constant if $\sigma>0$ is chosen sufficiently small.

In addition we need to understand the behaviour of $E$ on $\mathcal{Z}$ and we will show

Lemma 3.4**.**

Let $U_{0,1}^{*}$ be any harmonic maps from $S^{2}$ into a smooth Riemannian manifold $N$ of any dimension, let $q_{0,1}^{*}:\hat{}{\mathbb{C}}\to\hat{}{\mathbb{C}}$ be any rational maps with $q_{0,1}^{*}(0)=0$ and let $\mathcal{Z}_{0}$ be the corresponding set of singularity models defined in Section 2. Then we have

[TABLE]

for any variation $\mathfrak{z}_{\varepsilon}$ in $\mathcal{Z}_{0}$ which satisfies (3.7). Here for $\Omega_{1}:=\mathbb{D}_{\hat{r}}=\Omega_{0}^{c}$, $c_{r_{0},r_{1}}=(\log(\frac{r_{0}}{r_{1}}))^{-1}$ and err denotes an error term that is bounded by $|\text{err}|\lesssim R_{0}$ for

[TABLE]

As $c_{r_{0},r_{1}}\sim(\log\mu)^{-1}$ and as we shall see that $|\int_{\Omega_{i}}j_{i}\partial_{\varepsilon}\Delta u_{i}|\lesssim\|\beta\|(\nu_{i}^{*}+\mu^{-\frac{1}{2}}{\bar{\nu}})$ for $\nu_{1^{*}}=\nu_{0}$ and $\nu_{0^{*}}=\nu_{1}$ this ensures in particular that

[TABLE]

for any variation $\mathfrak{z}_{\varepsilon}$ in $\mathcal{Z}$ for which $\|\partial_{\varepsilon}\mathfrak{z}\|\lesssim 1$. For variations induced by only a change of $q_{1}$ we obtain an improved bound of

[TABLE]

The same bound also applies to $\mu|\partial_{\mu}E(\mathfrak{z}(U_{0,1},q_{0,1},\mu)|$ since variations of $\mu$ can be represented by variations of $q_{1}$. On the other hand, for variations of $U_{i}$ that are induced by translations on the domain we obtain that

[TABLE]

To prove our main results we now want to identify variations for which $dE(\mathfrak{z})(\partial_{\varepsilon}\mathfrak{z})$ scales like the dominating term in the above estimates.

To state these lemmas in a way that makes them applicable to all our main results we recall that we always deal with settings for which the following assumption is satisfied.

Assumption 1**.**

We ask that $U_{0,1}^{*}:\hat{}{\mathbb{C}}\to N$ are harmonic maps with $U_{0}^{*}(0)=U_{1}^{*}(0)$ and $dU_{0,1}^{*}(0)\neq 0$ for which one of the following holds

1. (i)

*We have $U_{0}^{*}=U_{1}^{*}$ or $U_{0}^{*}(z)=U_{1}^{*}(\bar{z})$ and these maps are non-degenerate critical points *

or 2. (ii)

The target manifold $N$ is $3$ dimensional and the tangent spaces $T_{U_{i}^{*}(0)}U_{i}^{*}(\hat{}{\mathbb{C}})$ do not coincide.

For elements $\mathfrak{z}\in\mathcal{Z}_{0}$ for which $\|\beta\|\nu_{0}$, respectively $\|\beta\|\nu_{1}$, is large compared to the error terms that involve $\delta$ and $\mathcal{T}$, we will want to consider variations that are induced by a suitable variations of the rational map $q_{1}$, respectively $q_{0}$. This case only occurs if $U_{1}^{*}\neq U_{0}^{*}$ and in Section 5.3 we shall prove

Lemma 3.5**.**

Let $U_{0,1}^{*}$ be harmonic maps with $U_{0}^{*}\neq U_{1}^{*}$ for which Assumption 1 holds, let $q_{0,1}^{*}\in\mathcal{R}$ and let $\mathcal{Z}_{0}$ be the corresponding set of singularity models defined in Section 2.

Then there exists a constant $c_{1}>0$ so that for every $\mathfrak{z}\in\mathcal{Z}_{0}$ and each $i\in\{0,1\}$ there is a variation $\mathfrak{z}_{\varepsilon}$ that is induced by a variation of $q_{i}$ which satisfies (3.7) for which

[TABLE]

where we write for short $\nu_{0^{*}}=\nu_{1}$ and $\nu_{1^{*}}=\nu_{0}$.

Conversely, if $\frac{\delta}{\log\mu}$ is large compared to $\|\beta\|{\bar{\nu}}$ and $\mathcal{T}$ we want to consider variations which reduce $\delta$. To this end we can use

Lemma 3.6**.**

Let $U_{i}^{*}$ be as in Assumption 1 and let $U_{0,1}\in\mathcal{H}_{1}^{\sigma}(U_{0,1}^{*})$. Then there exists a variation of the form $U_{i,\varepsilon}=U_{i}(\cdot+\varepsilon a)$, $a\in{\mathbb{R}}^{2}$ with $|a|\lesssim 1$ so that

[TABLE]

where for settings for which the first part of Assumption 1 is satisfied, we can freely choose which of $U_{0}$ or $U_{1}$ we want to vary, while for settings satisfying the second part of Assumption 1 the above holds true for at least one of $i\in\{0,1\}$.

For such variations we then obtain

Lemma 3.7**.**

Let $U_{0,1}^{*}$ be as in Assumption 1, let $q_{0,1}^{*}$ be any maps in $\mathcal{R}$ and let $\mathcal{Z}_{0}$ be the corresponding set of singularity models defined in Section 2. Then

[TABLE]

for the variation $\mathfrak{z}_{\varepsilon}$ in $\mathcal{Z}_{0}$ that is induced by changing $U_{i}$ as described in Lemma 3.6.

Finally, in non-integrable settings where we have to deal with maps $U_{i}$ which are not harmonic, we also want to consider variations which reduce $E(U_{i})$ and hence show

Lemma 3.8**.**

Let $U_{0,1}^{*}:S^{2}\to N$ be harmonic maps into an analytic Riemannian manifold of any dimension with $U_{0}^{*}(0)=U_{1}^{*}(0)$. Then for any $\mathfrak{z}\in\mathcal{Z}_{0}$ there exists a variation $\mathfrak{z}_{\varepsilon}$ in $\mathcal{Z}_{0}$ induced by a variation of $U_{i}$ in $\mathcal{H}_{1}^{\sigma}(U_{i}^{*})$ which satisfies (3.7) so that

[TABLE]

4. Proof of the main theorems

Here we give the proof of our main results based on the lemmas stated in the previous section. So let $(u_{j})$ be a sequence of almost harmonic maps from $S^{2}$ to $N$ that converges to a bubble tree with bubble $\omega_{1}(z)=U_{1}^{*}(q_{1}^{*}(\frac{1}{z}))$ and base $\omega_{0}=U_{0}^{*}\circ q_{0}^{*}$ and let $\mathcal{Z}=\{\mathfrak{z}:S^{2}\to N\}$ be the corresponding set of singularity models constructed in Section 2.

For sufficiently large $j$ our construction and (1.4) ensure that the function $\mathfrak{z}\mapsto\|u_{j}-\mathfrak{z}\|_{\mathfrak{z}}$ achieves its minimum on $\mathcal{Z}$ and that the corresponding $\mathfrak{z}_{j}\in\mathcal{Z}$ with $\|u_{j}-\mathfrak{z}_{j}\|_{\mathfrak{z}_{j}}=\text{dist}(u_{j},\mathcal{Z}):=\inf_{z\in\mathcal{Z}}\|u_{j}-\mathfrak{z}\|_{\mathfrak{z}}$ are obtained from maps $U_{0,1}^{(j)}\to U_{0,1}^{*}$, rational functions $q_{0,1}^{(j)}\to q_{0,1}^{*}$ and scales $\mu(\mathfrak{z}_{j})$ that are comparable to the numbers $\mu_{j}$ from (1.4). Furthermore, (1.4) can be used to see that not only $\|u_{j}-\mathfrak{z}_{j}\|_{\mathfrak{z}_{j}}\to 0$ but also $\|u_{j}-\mathfrak{z}_{j}\|_{L^{\infty}(S^{2})}\to 0$.

It hence suffices to prove that there exists a constant $\varepsilon>0$ so that the claims of our main results hold true for all maps $u$ for which

[TABLE]

where after a rotation of the domain we can assume without loss of generality that $u$ is so that the above holds for a $\mathfrak{z}\in\mathcal{Z}_{0}$.

Here and in the following we continue to use the convention that all claims are to be understood to hold true provided the numbers $\sigma,\sigma_{1}>0$ and $\bar{\mu}$ in the construction of $\mathcal{Z}$ are chosen sufficiently small respectively large.

Remark 4.1**.**

Given any $u$ for which (4.1) holds we use the convention that

[TABLE]

is to be computed with respect to the weighted norm $\|\cdot\|_{\mathfrak{z}}$ defined in (3.1) for a $\mathfrak{z}\in\mathcal{Z}$ for which (4.1) holds.

We do not claim that $\mathfrak{z}$ is uniquely determined by this relation but observe that $\|\cdot\|_{\tilde{\mathfrak{z}}}\sim\|\cdot\|_{\mathfrak{z}}$ if (4.1) is satisfied for both $\mathfrak{z}$ and $\tilde{\mathfrak{z}}$ and hence that it does not matter which such element of $\mathcal{Z}$ we use to define $\|dE(u)\|_{*}$ in our main results.

We will combine the results from the previous section with the following lemma that describes the basic estimates that we can obtain from our method of proof, compare also [7, Section 2].

Lemma 4.2**.**

Let $U_{0,1}^{*}:S^{2}\to N$ be any given harmonic maps into a smooth Riemannian manifold, let $q_{0,1}^{*}\in\mathcal{R}$ and let $\mathcal{Z}$ be the corresponding set of singularity models. Then there exist constants $\varepsilon>0$, $\bar{\Lambda}<\infty$ and $C$ so that the following holds true for any $u\in H^{1}(S^{2},N)$ with $\|\tau_{g_{S^{2}}}(u)\|_{L^{2}(S^{2},g_{S^{2}})}\leq 1$ for which there exists $\mathfrak{z}\in\mathcal{Z}$ so that (4.1) is satisfied.

We can bound $w:=u-\mathfrak{z}$ by

[TABLE]

and furthermore get that

[TABLE]

for any $u,\mathfrak{z}$ for which there is a unit element $y_{\mathfrak{z}}\in T_{\mathfrak{z}}\mathcal{Z}$ with

[TABLE]

For $C^{2}$-energies that are defined on a fixed Hilbert-space, the analogue of the above lemma can be obtained from a simple argument that is based on the fundamental theorem of calculus, compare [7]. In more involved settings such as the present problem where we consider an energy on a suitable Hilbert-manifold of maps and where we work with distances that are calculated using a weighted norm that depends on the corresponding element of $\mathcal{Z}$ one needs to proceed with more care. For almost harmonic maps from higher genus surfaces which are close to a simple bubble tree a similar result was obtained by the author in Section 4 of [11], and most of the arguments from [11] are applicable also in the present situation. For the convenience of the reader we hence include a sketch of the proof of this lemma, but omit the technical details except to explain why the estimate (4.3) is simpler than the one obtained in [11, Lemma 4.4].

Sketch of the proof of Lemma 4.2.

Given $u$, $\mathfrak{z}$ and $w=u-\mathfrak{z}$ as in the lemma we set $w_{\mathfrak{z}}:=P_{\mathfrak{z}}w$, $P_{p}$ the projection onto $T_{p}N$. We first note that (4.1) ensures that $\|w-w_{\mathfrak{z}}\|_{\mathfrak{z}}\leq C\|w\|_{L^{\infty}}\|w\|_{\mathfrak{z}}$ where and in the following all $L^{p}$ norms are computed over $(S^{2},g_{S^{2}})$ unless indicated otherwise.

We can easily check that the variation of the weight $\rho_{\mathfrak{z}}$ in (3.1) is controlled by $\|\partial_{\varepsilon}\rho_{\mathfrak{z}}\|_{L^{2}}\lesssim 1$ whenever $\|\partial_{\varepsilon}\mathfrak{z}\|_{\mathfrak{z}}\leq 1$. As $\mathfrak{z}$ minimises the distance to $u$ and thus

[TABLE]

we hence get that $\|P^{T_{\mathfrak{z}}\mathcal{Z}}w_{\mathfrak{z}}\|_{\mathfrak{z}}\leq C\|w\|_{L^{\infty}}\|w\|_{\mathfrak{z}}$, i.e. that $w_{\mathfrak{z}}$ is nearly orthogonal to $T_{\mathfrak{z}}\mathcal{Z}$.

We then set $\tilde{w}_{\mathfrak{z}}:=P^{\mathcal{V}_{\mathfrak{z}}^{+}}w_{\mathfrak{z}}-P^{\mathcal{V}_{\mathfrak{z}}^{-}}w_{\mathfrak{z}},$ where $P^{\mathcal{V}_{\mathfrak{z}}^{\pm}}:\Gamma(\mathfrak{z}^{*}TN)\to\mathcal{V}_{\mathfrak{z}}^{\pm}$ denote the $\langle\cdot,\cdot\rangle_{\mathfrak{z}}$ orthogonal projections onto the subspaces $\mathcal{V}_{\mathfrak{z}}^{\pm}$ of $T_{\mathfrak{z}}^{\perp}\mathcal{Z}$ from Lemma 3.3. The uniform definiteness of $d^{2}E$ on these spaces ensures that

[TABLE]

where we can analyse the resulting error term as explained in detail in the proof of Lemma 4.4 of [11], resulting in an estimate of

[TABLE]

The only difference to the proof in [11] is that in the present situation

[TABLE]

and that we can hence bound the second term on the right hand side of (4.7) by a multiple of the first term. This was not possible in [11], as the correct weighted norm in that context does not control the $H^{1}$ norm in a uniform way.

For $\varepsilon>0$ sufficiently small we hence conclude that

[TABLE]

which yields the first claim of the lemma.

If there exists a unit element $y_{\mathfrak{z}}$ for which (4.5) is satisfied then we can use that

[TABLE]

for an error of order $|\text{err}_{2}|\lesssim\|w\|_{\mathfrak{z}}^{2}$. We hence conclude that

[TABLE]

for constants $C$ that only depend on $U_{i}^{*}$ and $q_{i}^{*}$. For $\bar{\Lambda}=\bar{\Lambda}(U_{i}^{*},q_{i}^{*})$ sufficiently large this yields the second claim of the lemma. ∎

To prove our main results we will now apply the above lemma in directions $y_{\mathfrak{z}}$ that correspond to variations $\mathfrak{z}_{\varepsilon}$ as considered in one of the three Lemmas 3.5, 3.7, 3.8.

4.1. Proofs of Theorem 1.3 and Corollary 1.4

We first consider settings in which the base and bubble are given by parametrisations of the same non-degenerate minimal sphere with the same orientation.

So let $U^{*}:{\mathbb{R}}^{2}\to N$, $q_{0,1}^{*}\in\mathcal{R}$ and $\omega_{0,1}$ be as in Theorem 1.3, let $\mathcal{Z}$ be the corresponding set of singularity models and let $u$ be a map for which (4.1) holds true for $\varepsilon>0$ chosen as in Lemma 4.2 and $\mathfrak{z}\in\mathcal{Z}_{0}$.

We then consider the variation $\mathfrak{z}_{\varepsilon}$ of $\mathfrak{z}$ that we obtain by fixing $U_{0}$, $q_{0,1}$ and $\mu$ and varying $U_{1}$ according to $U_{1,\varepsilon}=U_{1}(\cdot+\varepsilon a)$ for $|a|\sim 1$ chosen as in Lemma 3.6. As $\|\partial_{\varepsilon}\mathfrak{z}_{\varepsilon}\|_{\mathfrak{z}}\sim 1$ we obtain from Lemma 3.7 that

[TABLE]

for the corresponding unit element $y_{\mathfrak{z}}=\frac{\partial_{\varepsilon}\mathfrak{z}_{\varepsilon}}{\|\partial_{\varepsilon}\mathfrak{z}_{\varepsilon}\|_{\mathfrak{z}}}$ and for constants $C,c>0$ that only depend on $U^{*}$ and $q_{0,1}^{*}$. Here we use that $\mathcal{T}=0$ as the maps $U_{i}$ are given by $U_{i}(\cdot)=U^{*}(\cdot+c_{i})$ for some $|c_{i}|<\sigma$ since $U^{*}$ is assumed to be non-degenerate. As $dU^{*}(0)\neq 0$ this furthermore implies that $\delta\sim\|\beta\|$ as both of these quantities scale like $|c_{0}-c_{1}|$. As ${\bar{\nu}}\lesssim\mu^{-1}\ll(\log\mu)^{-1}$ we hence deduce from (4.10) that

[TABLE]

while Lemmas 3.2 and 3.3 ensure that

[TABLE]

again for constants that only depend on the limiting configuration.

We hence deduce that

[TABLE]

$\bar{\Lambda}$ as in Lemma 4.2 since ${f_{\mu}}$ satisfies (2.12) for a constant $\sigma_{1}=\sigma(U_{i}^{*},q_{i}^{*})>0$ that can still be reduced if necessary. This lemma hence implies that

[TABLE]

and

[TABLE]

If $u$ is harmonic then (4.13) ensures that $\delta(\mathfrak{z})=0$ and thus that $\mathfrak{z}(z)=U^{*}(q_{0}(z)+q_{1}(\frac{1}{\mu z})+c_{0})$. Thus $dE(\mathfrak{z})=0$ so (4.12) ensures that $u=\mathfrak{z}$ and we obtain the claim about harmonic maps made in the theorem.

To prove the claims about almost harmonic maps made in the theorem and the subsequent Corollary 1.4 we now show that

[TABLE]

where we write for short $E^{*}=E(\omega_{0})+E(\omega_{1})=[\deg(q_{0}^{*})+\deg(q_{1}^{*})]E(U^{*})$.

To see this we first use (4.11) and (4.12) to bound

[TABLE]

To obtain (4.14) it hence suffices to check that

[TABLE]

If $\delta=0$, i.e. if $c_{0}=c_{1}$, this is trivially true. Conversely if $c_{0}\neq c_{1}$ then we can interpolate between $\mathfrak{z}$ and $\hat{\mathfrak{z}}=U^{*}\circ(q_{\mu}+c_{0})$, which satisfies $E(\hat{\mathfrak{z}})=E^{*}$, by considering the family $(\mathfrak{z}_{t})_{t\in[0,|c_{1}-c_{0}|]}$ that is generated by $U_{1,t}=U^{*}(\cdot+c_{1}+t\tfrac{c_{0}-c_{1}}{|c_{1}-c_{0}|})$. As $\|\partial_{t}\mathfrak{z}_{t}\|\lesssim 1$ and $\|\beta(\mathfrak{z}_{t})\|\lesssim\delta(\mathfrak{z}_{t})\lesssim\delta:=\delta(\mathfrak{z})$ we can use (3.15) to bound $|\partial_{t}E(\mathfrak{z}_{t})|\lesssim\tfrac{\delta}{\log\mu}$ which, when integrated over $0\leq t\leq|c_{0}-c_{1}|\sim\delta$, yields (4.15) and hence (4.14).

From (4.13) and (4.14) we now immediately deduce that

[TABLE]

which completes the proof of Theorem 1.3.

It remains to prove the $L^{2}$-Łojasiewicz-estimate (1.7) claimed in Corollary 1.4. To this end we first combine (4.13) with the fact that $\|\cdot\|_{L^{2}(S^{2},g_{S^{2}})}\lesssim\|\cdot\|_{\mathfrak{z}}$, and hence $\|dE(u)\|_{*}\lesssim\|\tau_{g_{S^{2}}}(u)\|_{L^{2}(S^{2},g_{S^{2}})}$, to bound

[TABLE]

As $y_{\mathfrak{z}}$ is obtained from a variation of just the bubble $U_{1}$, it is mainly concentrated on a small ball around the point where the bubble is attached and hence its $L^{2}$ norm is small compared to $\|y_{\mathfrak{z}}\|_{\mathfrak{z}}=1$. To be more precise, for variations satisfying (3.7) for which $q_{0}$ and $U_{0}$ are fixed we shall see in (5.19) that

[TABLE]

In the present situation where $\eta_{0}\sim 1$ we hence get that $\|y_{\mathfrak{z}}\|_{L^{2}(S^{2},g_{S^{2}})}\lesssim(\log\mu)^{-1}f_{\mu}^{-1}$ so (4.13) allows us to conclude that

[TABLE]

Combined with (4.14) this immediately yields the claim $|E(u)-E^{*}|\lesssim\|\tau_{g_{S^{2}}}(u)\|_{L^{2}(S^{2},g_{S^{2}})}^{2}$ made in Corollary 1.4.

Remark 4.3**.**

As noted in the introduction, the reason we cannot obtain these results from the existing theory is that the spectral gap at zero for the Jacobi-operator at $U^{*}(q_{\mu})$ tends to zero as $\mu\to\infty$ and that the size of the neighbourhoods on which the existing theory yields Łojasiewicz-estimates scales like this spectral gap. To be more precise, as the energy defect scales like $\frac{\delta^{2}}{\log\mu}$ we can see that the Jacobi-operator must have an eigenvalue that scales like $\frac{1}{\log\mu}$. If we did not include these directions in our set of singularity models $\mathcal{Z}$ this would mean that the argument of Lemma 4.2 would break down even for harmonic maps $u$ unless we knew a priori that $\|w\|_{\mathfrak{z}}$ is bounded by a small enough multiple of $\frac{1}{\log\mu}$, compare (4.6).

4.2. Proofs of Theorems 1.5 and 1.7 and of Corollary 1.6

In this section we always consider settings for which the assumptions of either Theorem 1.5 or of Theorem 1.7 are satisfied and let $\mathcal{Z}$ be the corresponding set of singularity models. We note that in these settings $|d\beta(0)|$ and $\|\beta\|$ are of order $1$ and that we have

Lemma 4.4**.**

The energy defect of elements $\mathfrak{z}\in\mathcal{Z}$ is bounded by

[TABLE]

where $\gamma_{0,1}\in(1,2]$ are the exponents for which the classical Łojasiewicz-Simon estimate (1.3) is valid near $U_{0,1}^{*}$ while $E^{*}:=E(\omega_{0})+E(\omega_{1})=\deg(q_{0})E(U_{0}^{*})+\deg(q_{1})E(U_{1}^{*})$.

Proof of Lemma 4.4.

It suffices to carry out the proof for elements of $\mathcal{Z}_{0}$ and using the symmetry described in Remark 2.2 we can assume without loss of generality that $\nu_{0}\leq\nu_{1}$.

We will first prove the lemma in the special case where $\delta=0$. To this end, we consider the family $\mathfrak{z}(t)$, $t>0$, which we obtain by varying the bubble scale according to $\mu(t)=\mu(\mathfrak{z})e^{t}$ while keeping the maps $U_{i}$ and $q_{i}$ fixed and use that (3.14) ensures that

[TABLE]

As $\nu_{i}(t)=\max_{j\leq n_{i}^{*}}|a_{j}(q_{i})|\mu(t)^{-j}\lesssim e^{-t}\nu_{i}(0)$ and as $E(\mathfrak{z}(t))\to\deg(q_{0})E(U_{0})+\deg(q_{1})E(U_{1})$ for $t\to\infty$ we hence deduce that

[TABLE]

which, combined with (1.3), yields the claim in this special case where $\delta=0$.

So suppose that $\delta=\delta(U_{1},U_{0})\neq 0$. In this case we can choose $U_{i,t}=U_{i}(\cdot+c_{i}(t))$, $t\in[0,\delta]$ with $|\dot{c}_{i}(t)|\lesssim 1$ in a way that $\delta(U_{0,t},U_{1,t})=\delta-t$, compare Lemma 3.6. The resulting family $\mathfrak{z}_{t}\in\mathcal{Z}$ satisfies

[TABLE]

compare (3.15), so turns $\mathfrak{z}$ into an element $\hat{\mathfrak{z}}=\mathfrak{z}_{t=\delta}$ with $\delta(\hat{\mathfrak{z}})=0$ and

[TABLE]

As ${\bar{\nu}}\log\mu\lesssim\mu^{-1}\log\mu\lesssim\mu^{-\frac{1}{2}}$ and as we have already shown that (4.18) holds for $\hat{\mathfrak{z}}$ we hence obtain the claimed bound on the energy defect also for general elements of $\mathcal{Z}_{0}$. ∎

Let now $u$ be a map with $\|\tau_{g_{S^{2}}}(u)\|_{L^{2}(S^{2},g_{S^{2}})}\leq 1$ which satisfies (4.1) for $\varepsilon>0$ as in Lemma 4.2.

We can combine the above Lemma 4.4 with the estimate

[TABLE]

that we obtain from (4.3) and Lemma 3.2 to see that

[TABLE]

for any such $u$ and for $\gamma:=\min(\gamma_{1},\gamma_{2})\in(1,2]$.

In order to prove the bounds on $|E(u)-E^{*}|$ claimed in Theorems 1.5 and 1.7 we hence need to show that all these quantities are controlled by $\|dE(u)\|_{*}$. For this it is convenient to choose ${f_{\mu}}:=\log\mu$ which of course satisfies (2.12) for $\bar{\mu}$ sufficiently large.

We distinguish between three different cases, depending on whether terms involving $\delta$, $\mathcal{T}$ or ${\bar{\nu}}$ dominate, beginning with

**Case 1: ** Suppose that $\mathfrak{z}$ is so that

[TABLE]

In this case we let $y_{\mathfrak{z}}$ be the unit element in the direction of a variation $\partial_{\varepsilon}\mathfrak{z}_{\varepsilon}$ as obtained in Lemma 3.7. Inserting (4.20) into the estimates obtained in Lemmas 3.2, 3.3 and 3.7 gives

[TABLE]

This ensures that $Q(\mathfrak{z},y_{\mathfrak{z}})\lesssim\frac{\log\log\mu}{\log\mu}$ is small so Lemma 4.2 yields

[TABLE]

Combined with (4.20) this immediately implies that $\mu^{-\min(n_{0}^{*},n_{1}^{*})}\lesssim{\bar{\nu}}\lesssim\|dE(u)\|_{*}$ and inserting these estimates into (4.19) gives

[TABLE]

Having thus seen that the claims of Theorems 1.5 and 1.7 hold in this case where $\delta$ dominates we now consider

Case 2: Suppose that $\mathfrak{z}$ is so that

[TABLE]

In this situation (4.19) ensure that $|E(u)-E^{*}|\lesssim\mathcal{T}$ and we construct $y_{\mathfrak{z}}$ using a variation as considered in Lemma 3.8 for $i$ so that $\mathcal{T}_{i}=\mathcal{T}$. Lemmas 3.8 and 3.2 yield

[TABLE]

while Lemma 3.3 and (3.10) ensure that we can make $\|d^{2}E(\mathfrak{z})(y_{\mathfrak{z}},\cdot)\|_{*}$ smaller than any given constant by reducing $\sigma$ accordingly. Hence we can ensure that $Q(\mathfrak{z},y_{\mathfrak{z}})\lesssim\bar{\Lambda}^{-1}$ and apply Lemma 4.2 to get

[TABLE]

Combined with (4.19) and (4.22) this immediately yields the claims of the theorems.

Finally, if $\mathfrak{z}$ is so that neither of the above cases applies then we must have

Case 3:

[TABLE]

In this case we can choose a variation as in Lemma 3.5 to obtain $y_{\mathfrak{z}}$ with

[TABLE]

At the same time Lemma 3.2 and (4.24) ensure that $\|dE(\mathfrak{z})\|_{*}\leq{\bar{\nu}}\log\mu$ and hence that $Q(\mathfrak{z},y_{\mathfrak{z}})\lesssim\mu^{-1}(\log\mu)^{2}$ will be small.

Lemma 4.2 hence implies that ${\bar{\nu}}\lesssim\|dE(u)\|_{*}$. This immediately yields the claimed bound on the bubble scale and, as (4.24) ensures that all terms that involve $\delta$ or $\mathcal{T}$ in (4.19) are small compared to ${\bar{\nu}}$, also that $|E(u)-E^{*}|\lesssim\|dE(u)\|_{*}$.

Having thus completed the proofs of both Theorems 1.5 and 1.7 we finally explain how the above arguments have to be modified to obtain the

Proof of Corollary 1.6.

Here it is convenient to choose ${f_{\mu}}:=\mu^{\sigma_{1}}$ where we fix $\sigma_{1}>0$ small enough so that all results in Section 3 hold for this $\sigma_{1}$ and for all sufficiently small $\sigma$ and large $\bar{\mu}$. Compared to the previous proof this choice of ${f_{\mu}}$ gives us more flexibility with regards to which type of variation we can choose as we obtain better bounds on the error terms in the energy expansions. This is useful as variations of the bubble are preferable to variations of the base when proving $L^{2}$-Łojasiewicz-estimates since they result in elements $y_{\mathfrak{z}}$ with smaller $L^{2}$-norm.

To prove this corollary we first note that in this setting (4.19) reduces to

[TABLE]

as all elements of $\mathcal{H}_{1}^{\sigma}(U_{i}^{*})$ are harmonic and as $\|dE(u)\|_{*}\lesssim\|\tau_{g_{S^{2}}}(u)\|_{L^{2}(S^{2},g_{S^{2}})}$.

We fix $c_{1}=\frac{\sigma_{1}}{4}\in(0,\frac{1}{16})$ and distinguish between cases dependent on whether $\mu^{c_{1}}{\bar{\nu}}\leq\tfrac{\delta}{\log\mu}.$

If $\mu^{c_{1}}{\bar{\nu}}\leq\tfrac{\delta}{\log\mu}$ then we can argue exactly as in the proof of Corollary 1.4 to see that

[TABLE]

compare (4.17). As ${\bar{\nu}}\gtrsim\mu^{-n^{*}}$, $n^{*}:=\min(n_{1}^{*},n_{0}^{*})$, we furthermore know that in this case

[TABLE]

and hence immediately obtain from (4.25) that

[TABLE]

It hence remains to consider the case $\tfrac{\delta}{\log\mu}\leq\mu^{c_{1}}{\bar{\nu}}$ where (4.25) and Lemma 3.2 give

[TABLE]

while Lemmas 3.5 and 3.3 give

[TABLE]

for the unit element $y_{\mathfrak{z}}$ that is generated by a variation of $q_{i}$ as considered in Lemma 3.5.

If $\mu^{c_{1}}\nu_{0}\geq\nu_{1}$ we can vary the rational map of the bubble to get a $y_{\mathfrak{z}}$ with

[TABLE]

compare (4.16) and (4.27). Hence $Q(\mathfrak{z})(y_{\mathfrak{z}})\lesssim\mu^{-1+3c_{1}}\log\mu$ is small and we can apply the estimate (4.4) from Lemma 4.2 to see that

[TABLE]

We can thus bound

[TABLE]

and (4.26) implies that the claim of the corollary holds true for any such exponent.

In the final case where $\nu_{0}\leq\mu^{-c_{1}}\nu_{1}$ and $\tfrac{\delta}{\log\mu}\leq\mu^{c_{1}}{\bar{\nu}}$ we instead consider a variation of $q_{0}$ and use the above estimates and Lemma 3.2 to see that ${\bar{\nu}}\lesssim\nu_{1}\lesssim\|dE(u)\|_{*}\lesssim\|\tau_{g_{S^{2}}}(u)\|_{L^{2}(S^{2},g_{S^{2}})}$. From (4.26) we hence obtain that

[TABLE]

for any $\alpha\in(1,1+\frac{c_{1}}{n^{*}})$. ∎

5. Variations of the energy along $\mathcal{Z}_{0}$

In this section we carry out the proofs of the key lemmas on the behaviour of the energy on the manifold $\mathcal{Z}_{0}$ of singularity models, i.e. of Lemmas 3.4, 3.5, 3.7 and 3.8, that we stated in Section 3.

Before we turn to these proofs we collect a number of estimates, which will be used both throughout this section as well as in the subsequent Section 6 where we will prove the properties of $dE(\mathfrak{z})$ and $d^{2}E(\mathfrak{z})$ claimed in Lemmas 3.1, 3.2 and 3.3.

5.1. Technical estimates

Given any $\mathfrak{z}\in\mathcal{Z}_{0}$ we work in fixed stereographic coordinates which we can assume to be scaled in a way that the corresponding function $q_{0}$ satisfies $|q_{n_{0}^{*}}(q_{0})|=1$. As we represent $q_{1}(\frac{1}{\mu\cdot})$ for any fixed $\mathfrak{z}$ by a function $q_{1}$ satisfying $|a_{n_{1}^{*}}(q_{1})|=1$ we hence get $\mu_{1}(\mathfrak{z})=\mu$, $\mu_{0}(\mathfrak{z})=1$ and

[TABLE]

To simplify the notation we will in the following write for short $\phi_{\mu}={\phi_{r_{0},r_{1}}}$ for the cut-off function (2.15) used in the definition of $\mathfrak{z}$ and also use the slight abuse of notation of writing $\partial_{\varepsilon}\phi_{\mu}:=\partial_{\varepsilon}|_{\varepsilon=0}\phi_{r_{0}(\mathfrak{z}_{\varepsilon}),r_{1}(\mathfrak{z}_{\varepsilon})}$ even though (5.1) is only applicable at $\varepsilon=0$.

For most of the proofs it will be more convenient to view the rational maps $q_{i}$ as maps into the sphere $S^{2}_{p^{*}}:=S^{2}-(0,0,1)$, which is shifted in a way that elements of $\mathcal{R}$ map [math] to the origin in ${\mathbb{R}}^{3}$. We hence associate to any given $U:\hat{}{\mathbb{C}}\to N$ and any rational map $q:\hat{}{\mathbb{C}}\to\hat{}{\mathbb{C}}$ the maps $\hat{U}:=U\circ\pi_{p^{*}}^{-1}:S^{2}_{p^{*}}\to N$ and $\hat{q}=\pi_{p^{*}}\circ q:\hat{}{\mathbb{C}}\to S^{2}_{p^{*}}$ for $\pi_{p^{*}}:=\pi-(0,0,1):{\mathbb{R}}^{2}\to S^{2}_{p^{*}}$.

As at least one of $q_{0}(z)$ or $q_{1,\mu}(z):=q_{1}(\frac{1}{\mu z})$ will be small for every $z\in{\mathbb{C}}$ we can easily check that the map $\hat{q}_{\mu}=\pi_{p^{*}}\circ(\hat{q}_{0}\circ\pi_{p^{*}}^{-1}+\hat{q}_{1,\mu}\circ\pi_{p^{*}}^{-1})$ that represents $q_{\mu}=q_{0}+q_{1,\mu}$ is so that

[TABLE]

Here and in the following we use the convention that unless specified otherwise all quantities are computed with respect to the euclidean metric on ${\mathbb{R}}^{2}$, rather than with respect to the metric that is induced by $g_{S^{2}}$.

We can hence estimate the maps $\tilde{u}_{i}:=u_{i}-U_{i}(0)=\hat{U}_{i}\circ\hat{q}_{\mu}-U_{i}(0)$ by

[TABLE]

where the last estimate follows as the maps $\hat{q}_{i}$ are harmonic maps into the sphere.

Since the maps $\hat{q}_{i}\circ\pi_{p^{*}}^{-1}:S_{p^{*}}^{2}\to S_{p^{*}}^{2}$ and their variations are uniformly bounded in $C^{k}$, we have

[TABLE]

and

[TABLE]

In particular we can always estimate $|\nabla\hat{q}_{\mu}|\lesssim\rho_{\mathfrak{z}}$, though will later need more refined bounds in settings where we deal with rational maps $q_{0,1}^{*}$ that are branched at $z=0$, compare (5.45) and (5.46) below.

The above estimates imply in particular that

[TABLE]

where we recall that the annuli $A$, $\hat{A}$ and $A^{*}$ are as defined in (2.10). Similarly we have

[TABLE]

for the cut-off function $\phi_{\mu}$ that is defined in (2.15) and that satisfies

[TABLE]

As $\hat{\gamma}$ is a geodesic with $\hat{\gamma}(0)=U_{1}(0)$ and $|\hat{\gamma}^{\prime}|=\delta$ we have

[TABLE]

and

[TABLE]

so

[TABLE]

As ${c_{\mu}}\sim(\log\mu)^{-1}$ we hence get

[TABLE]

To obtain suitable bounds on the variations of ${\phi_{\mu}}$ we recall that $|\partial_{\varepsilon}\mu_{i}|\lesssim\mu_{i}$ and that $|\partial_{\mu}{f_{\mu}}|\mu\leq{f_{\mu}}$. This ensures that

[TABLE]

which in turn imply that

[TABLE]

We also observe that the definition of ${\phi_{\mu}}$ and the above estimates ensure that

[TABLE]

As $|\partial_{\varepsilon}\hat{\gamma}|+|\partial_{\varepsilon}\nabla\hat{\gamma}|\lesssim{\eta_{0}}$, $\eta_{0}$ defined in (3.8), we can furthermore bound

[TABLE]

and

[TABLE]

We also note that $|q_{0}(z)|\lesssim|z|$ is small on the set $\{z:|z|\leq 2\mu^{-\frac{1}{2}}\}$ where we consider $v_{1}$ and hence $j_{1}=d\beta(0)(q_{0}(z))$. On this set we can hence bound $|q_{0}(z)|\lesssim|\hat{q}_{0}(z)|$ and thus have

[TABLE]

as well as

[TABLE]

Analogue estimates hold for ${j_{0}}$ on the set $|z|\geq\frac{1}{2}\mu^{-\frac{1}{2}}$ where $v_{0}$, and hence $j_{0}$, are considered.

We also observe that we can combine the above estimates with (5.57) to see that if $\mathfrak{z}_{\varepsilon}$ is a variation along which $q_{0}$ and $U_{0}$ are fixed then

[TABLE]

as claimed in the proofs of Corollaries 1.4 and 1.6.

5.2. General variations of the energy along $\mathcal{Z}$

The purpose of this section is to derive the expression for $\partial_{\varepsilon}E(\mathfrak{z}_{\varepsilon})$ claimed in Lemma 3.4.

Thanks to the symmetry of the construction and the fact that on $\Omega_{1}=\{z:|z|\leq\mu^{-\frac{1}{2}}\}$ our map $\mathfrak{z}$ is essentially described by $\mathfrak{z}_{1}=\pi_{N}(v_{1})$ the main step in this proof is to show

Lemma 5.1**.**

If $\mathfrak{z}_{\varepsilon}$ is a variation in $\mathcal{Z}_{0}$ so that (3.7) is satisfied then

[TABLE]

for an error term which is bounded by $|\text{err}_{1}|\leq R_{1}$ for

[TABLE]

Here $\delta,\tau,\eta_{0}$ and $\eta_{\tau}$ are as in (3.5) and (3.8) and the integrals $I_{\cdot}=I_{\cdot}(q_{0},q_{1,\mu})$ are as in

Lemma 5.2**.**

For any variations $q_{0,1}^{(\varepsilon)}$ in $\mathcal{R}^{\sigma}(q_{0,1}^{*})$ satisfying (3.7) we can bound

[TABLE]

where we write for short for $\rho_{q}:=|\nabla{\hat{q}_{0}}|+|\nabla{\hat{q}_{1,\mu}}|.$

This lemma ensures that $R_{1}$ defined by (5.21) is bounded by the quantity $R_{0}$ defined in (3.12) and hence that $\text{err}_{1}$ can be included in the error term in Lemma 3.4.

Similarly the contributions of $\mathfrak{z}-\mathfrak{z}_{1}$ to $\partial_{\varepsilon}E(\mathfrak{z}_{\varepsilon})$ are of lower order, namely

Lemma 5.3**.**

For any variation in $\mathcal{Z}_{0}$ as considered above we can bound

[TABLE]

Combining these three lemmas with their analogues on $\Omega_{0}$, which follow by symmetry, we get

[TABLE]

for an error term that satisfies the required bound $|\text{err}_{2}|\lesssim R_{0}$.

As $|\hat{\gamma}^{\prime}|=\delta$ we can use (5.15) to see that

[TABLE]

for an error term that is bounded by $|\text{err}_{3}|\lesssim\tfrac{\delta(\delta+{\eta_{0}})}{(\log\mu)^{2}}\lesssim R_{0}$.

Finally, we can use that

[TABLE]

where we use (5.5) to bound

[TABLE]

and set $\text{err}_{4}:=\int_{\Omega_{1}}\partial_{\varepsilon}{u_{1}}\tau({u_{1}})-\partial_{\varepsilon}(\hat{U}_{1}\circ{\hat{q}_{1,\mu}})\tau(\hat{U}_{1}\circ{\hat{q}_{1,\mu}})$. To bound this term we can use that $|\partial_{\varepsilon}({u_{1}}-\hat{U}_{1}\circ{\hat{q}_{1,\mu}})|\lesssim|{\hat{q}_{0}}|+|\partial_{\varepsilon}{\hat{q}_{0}}|$, while (1.10) ensures that

[TABLE]

Combined with (5.4) and (5.5) this yields

[TABLE]

This reduces the proof of Lemma 3.4 to the proofs of the three lemmas stated above, which we carry out in the remainder of this section.

To prove Lemma 5.1, and later on also Lemmas 3.2 and 3.3, it is useful to expand

[TABLE]

and note that the resulting error term can be written as

[TABLE]

and hence bounded by

[TABLE]

since $d\pi_{N}(p)$ agrees with the projection $P_{p}$ onto the tangent space $T_{p}N$ whenever $p\in N$.

Proof of Lemma 5.1.

Writing $\mathfrak{z}_{1}$ as in (5.31) and using that $\Delta{j_{1}}=0$ we find that

[TABLE]

for $T_{2}:=\int_{\Omega_{1}}\partial_{\varepsilon}(P_{u_{1}}{j_{1}}\Delta{u_{1}})=\int_{\Omega_{1}}\partial_{\varepsilon}({j_{1}}\tau({u_{1}}))$ and

[TABLE]

We can rewrite the third term in (5.34) using integration by parts, which results in a boundary term of the form

[TABLE]

to obtain the desired expression

[TABLE]

This reduces the proof of the lemma to showing that all error terms $T_{1}^{(i)}$, $T_{2}$ and $B_{1}$ obtained above are controlled by the quantity $R_{1}$ defined in (5.21).

To bound $T_{1}^{(1)}$ we consider $A(p)$ as a bilinear form $A(p):{\mathbb{R}}^{N}\times{\mathbb{R}}^{N}\to T_{p}^{\perp}N$ on the whole space which vanishes in normal directions and write

[TABLE]

Combined with (1.10), (5.9) and (5.16) we can hence bound

[TABLE]

where we use (5.6) and (5.7) in the last step. Hence $T_{1}^{(1)}\lesssim R_{1}$ as required.

Next we use that that the formula (5.32) for $\text{err}_{u_{1}}$, combined with (5.16) and (5.18), gives

[TABLE]

Using additionally (5.3), (5.6) and (5.7) we hence see that

[TABLE]

and thus that $T_{1}^{(2)}\lesssim R_{1}$.

We then split

[TABLE]

As $|\Delta{u_{1}}|+|\nabla{u_{1}}|^{2}\lesssim|\nabla{\hat{q}_{\mu}}|^{2}\lesssim\rho_{q}^{2}$ and as $\Delta j_{1}=0$ we can certainly estimate

[TABLE]

To obtain an improved bound on the tangential part of this quantity we write $P_{{u_{1}}}^{\perp}{j_{1}}=\sum_{j}\langle{j_{1}},\nu_{u_{1}}^{j}\rangle\nu_{u_{1}}^{j}$ to see that $|P_{{u_{1}}}(\Delta(P_{{u_{1}}}^{\perp}{j_{1}})|\lesssim|{j_{1}}|\rho_{q}^{2}+\rho_{q}|P_{u_{1}}^{\perp}(\nabla{j_{1}})|,$ and write

[TABLE]

to see that

[TABLE]

Combined we hence get

[TABLE]

On the other hand, while we can only bound

[TABLE]

compare (5.38) and (5.16), we get a stronger bound of

[TABLE]

Combined we hence get

[TABLE]

and thus $T_{1}^{(3)}\lesssim R_{1}$. Here we simply use that $\|{\hat{q}_{0}}\|_{L^{\infty}(\Omega_{1})}+\|\nabla{\hat{q}_{0}}\|_{L^{2}(\Omega_{1})}\lesssim\mu^{-\frac{1}{2}}$ to deal with the terms involving $\delta$ as these only give lower order contributions.

Next we estimate

[TABLE]

To deal with the first term we exploit that $A(x)=-dP_{x}$ and thus that

[TABLE]

Combined with (5.40) this allows us to bound

[TABLE]

while the formula (5.32) for $\text{err}_{u_{1}}$ allows us to check that

[TABLE]

Combined we hence get

[TABLE]

where we also use (5.6), (5.7), (5.8) and (5.12) in the second step. Thus also $T_{1}^{(4)}\lesssim R_{1}$.

Next we estimate

[TABLE]

where we used (5.12), $|u_{1}-\mathfrak{z}_{1}|\leq\mu^{-\frac{1}{2}}+\delta$ and (5.7) in the last step. Using furthermore that $|u_{1}-\gamma|\lesssim|u_{1}-U_{1}(0)|+|{\tilde{\gamma}_{u_{1}}}|\leq{f_{\mu}}^{-1}+\delta{\phi_{\mu}}$ on $A$ we hence get

[TABLE]

Finally, to bound $T_{1}^{(6)}$ we write ${\mathfrak{z}_{1}}$ instead as

[TABLE]

and use (5.6) and (5.16) to see that

[TABLE]

As $\|\Delta\gamma\|_{L^{1}}\lesssim\delta(\log\mu)^{-1}$ we hence immediately deduce that

[TABLE]

Since $T_{2}=\int_{\Omega_{1}}\partial_{\varepsilon}({j_{1}}\tau(\hat{U}_{1})\circ{\hat{q}_{\mu}}|\nabla{\hat{q}_{\mu}}|^{2})$, compare (1.10), we furthermore have

[TABLE]

Finally, to estimate $B_{1}$ we use (5.39) to see that

[TABLE]

Since $|\nabla u_{1}|+|\partial_{\varepsilon}\nabla u_{1}|\lesssim 1$ on $\partial\Omega_{1}$ and hence $|P_{u_{1}}(\partial_{n}P_{u_{1}}^{\perp}{j_{1}})|\lesssim|\partial_{n}{u_{1}}||P_{u_{1}}^{\perp}{j_{1}}|\lesssim|P_{u_{1}}^{\perp}{j_{1}}|$ we can hence bound

[TABLE]

where the last step follows since $|{\hat{q}_{0}}|$ and $|\partial\Omega_{1}|$ are of order $O(\mu^{{-\frac{1}{2}}})$.

We have thus shown that all error terms are controlled by $R_{1}$ which completes the proof of Lemma 5.1. ∎

Proof of Lemma 5.2.

To prove this lemma we will use that

[TABLE]

and for $q_{0}\in\mathcal{R}^{\sigma}(q_{0}^{*})$, $\hat{q}_{0}=\pi_{p^{*}}^{-1}\circ q_{0}$. To see this we note that our choice of $\sigma$ ensures that the right hand side is at least $\frac{1}{2}|z|^{n_{0}^{*}}$ and that (5.45) is hence trivially satisfied outside of any fixed size disc $\mathbb{D}_{\bar{r}}(0)$ since the maps $\hat{q}_{0}$ satisfy uniform $C^{k}$ bounds. As $q_{0}^{*}(0)=0$ and as the maps $\hat{q}_{0}$ are uniformly close to $\hat{q}_{0}^{*}$ we can fix $\bar{r}=\bar{r}(q_{0}^{*})>0$ small enough so that $|q_{0}(z)|\leq 1$ for all $|z|\leq 2\bar{r}$. Standard estimates from complex analysis then ensure that $|a_{j}(q_{0})|\lesssim(2\bar{r})^{-j}$ for all $j$ and hence that $\sum_{j\geq n_{0}^{*}+1}j|a_{j}(q_{0})||z|^{j}\lesssim|z|^{n_{0}^{*}}$ for all $|z|\leq\bar{r}$ which ensures that (5.45) also holds on $\mathbb{D}_{\bar{r}}(0)$.

Similarly, given $q_{1}\in\mathcal{R}^{\sigma}(q_{i}^{*})$ and $\mu>\bar{\mu}$ we can use that $\hat{q}_{1,\mu}=\pi_{p^{*}}\circ q_{1}(\tfrac{1}{\mu\cdot})$ is so that

[TABLE]

The proof of all claims made in the lemma now follow from short explicit calculation based these two estimates (5.45) and (5.46). To begin with we note that

[TABLE]

where here and in the following we only need to sum over $j=1,\ldots,{n_{0}^{*}}$ and $k=1,\ldots,{n_{1}^{*}}$ and can use that the corresponding coefficients $a_{j}(q_{i})$ are uniformly bounded as we only consider elements of $\mathcal{R}^{\sigma}(q_{i}^{*})$. As we furthermore have

[TABLE]

so in particular for $\alpha=1$, we obtain the claimed estimate (5.22) on $I_{1}$.

Using (5.45), (5.46) and (5.4) we then get

[TABLE]

As the other contributions to $I_{2}$ are bounded by

[TABLE]

and

[TABLE]

compare (5.47) and (5.48), we hence obtain the claimed bound (5.23) on $I_{2}$.

Combining (5.48) with

[TABLE]

then shows that $I_{3}$ satisfies (5.24), while (5.4) and (5.5) give the claimed bound on

[TABLE]

Finally, as $|{\hat{q}_{0}}|^{2}+|{\hat{q}_{1,\mu}}|^{2}\lesssim{\bar{\nu}}$ on $\partial\Omega_{1}=\partial\mathbb{D}_{\mu^{-\frac{1}{2}}}$, we get that $|I_{\partial\Omega_{1}}|\leq|\partial\Omega_{1}|{\bar{\nu}}\lesssim{\bar{\nu}}\mu^{-\frac{1}{2}}$. ∎

In Section 6 we will also use that a very similar calculation gives

[TABLE]

We finally need to prove that the contribution of the error terms $\text{err}_{\varphi}=\mathfrak{z}-\mathfrak{z}_{1}$ to $\partial_{\varepsilon}E(\mathfrak{z}_{\varepsilon})$ is controlled as described in Lemma 5.3.

As $\text{err}_{\varphi}$ is supported on $\hat{A}$ we first collect some estimates that are valid on this set where $|z|\sim\mu^{-\frac{1}{2}}$ and where we can hence bound

[TABLE]

and

[TABLE]

Since $v_{0}-v_{1}=\beta({q_{\mu}})-\beta(0)-d\beta(0)({q_{\mu}})$ we have

[TABLE]

as well as

[TABLE]

We furthermore note that $\gamma$ is harmonic on ${\hat{A}}$ so we can bound

[TABLE]

Since the error term is given by

[TABLE]

for a function $\varphi$ that vanishes for $|z|\leq\frac{1}{2}\mu^{-\frac{1}{2}}$ and that satisfies $|\nabla\varphi|^{2}+|\Delta\varphi|\lesssim\mu$ we have

[TABLE]

As $|\partial_{\varepsilon}\mu_{i}|\lesssim\mu_{i}$ we have

[TABLE]

allowing us to bound

[TABLE]

We will later also use that the above estimates also allow us to bound

[TABLE]

Based on these estimates we can now complete the

Proof of Lemma 5.3.

Since $|\partial_{\varepsilon}\mathfrak{z}_{1}|\lesssim\mu^{-\frac{1}{2}}+\delta+{\eta_{0}}$ on ${\hat{A}}$ and since $|{\hat{A}}|\lesssim\mu^{-1}$ we can bound

[TABLE]

To bound this last term we split $\partial_{\varepsilon}\text{err}_{\varphi}\Delta\mathfrak{z}_{1}$ into the contributions of the tangential and the normal parts. Since $P_{\mathfrak{z}_{1}}=d\pi(\mathfrak{z}_{1})$ and since $|v_{1}-\mathfrak{z}_{1}|\leq\delta+\|\beta\|{\bar{\nu}}^{\frac{1}{2}}$ and $|\partial_{\varepsilon}v_{1}|\lesssim\eta_{0}+\mu^{-\frac{1}{2}}$ on ${\hat{A}}$ we get

[TABLE]

and can thus in particular bound $|P_{\mathfrak{z}_{1}}^{\perp}(\partial_{\varepsilon}\text{err}_{\varphi})|\lesssim\mu^{-1}(\|\beta\|+\delta)$.

Since (5.54) ensures that $|\Delta\mathfrak{z}_{1}|=|\Delta(\pi_{N}(v_{1}))|\lesssim\mu({\bar{\nu}}+\delta^{2})$ on ${\hat{A}}$ and since $|{\hat{A}}|\lesssim\mu^{-1}$ this suffices to estimate

[TABLE]

and hence to see that this term is bounded by the right hand side of (5.27).

From (5.31) we furthermore get that

[TABLE]

As (5.58) ensures that $|\partial_{\varepsilon}\text{err}_{\varphi}|\lesssim(\|\beta\|+\|\partial_{\varepsilon}\beta\|)\mu^{-1}$ and as $|{\hat{A}}|\lesssim\mu^{-1}$ we immediately conclude that also $\int|\partial_{\varepsilon}\text{err}_{\varphi}||P_{u_{1}}\Delta\mathfrak{z}_{1}|$ is controlled by the right hand side of (5.27), which completes the proof of the lemma. ∎

5.3. Dominating terms in the energy expansions for variations of the rational maps

In this section we explain how the general energy expansion proven in the previous section can be used to identify a variation for which Lemma 3.5 holds true.

By symmetry it suffices to prove the claim of the lemma for $i=1$, i.e. to identify a variation of $q_{1}$ so that (3.16) holds for the resulting family $\mathfrak{z}_{\varepsilon}$ for which we keep $U_{0.1}$, $q_{1}$ and $\mu$ fixed.

For such variations we know from Lemma 3.4 that

[TABLE]

now for an error that is bounded by $|\text{err}_{1}|\lesssim R_{2}$ for

[TABLE]

as $\eta_{0}=\eta_{\tau}=0$. As $\partial_{\varepsilon}U_{0}=0$ and as $|\partial_{\varepsilon}\hat{q}_{\mu}|\lesssim|\partial_{\varepsilon}q_{1,\mu}|$ is small on $\Omega_{0}$ we can easily check that

[TABLE]

To prove the lemma it hence suffices to show that we can find a variation of $q_{1}$ for which

[TABLE]

for an error term that is bounded by $|\text{err}_{2}|\lesssim R_{2}$.

To this end we will use that away from $z=0$ the functions $2(\mu z)^{-j}$ are well approximated by the first two components $h_{j,\mu}(z):=\tfrac{2(\mu\bar{z})^{j}}{1+|\mu\bar{z}|^{2j}}$ of the harmonic map $\bar{h}_{j,\mu}(z):=\pi((\mu\bar{z})^{j}):\hat{}{\mathbb{C}}\to S^{2}$, where we continue to identify $(x,y)\in{\mathbb{R}}^{2}$ with $x+iy\in{\mathbb{C}}$ whenever convenient. We hence write $u_{1}(z)=U_{1}(q_{0}(z)+q_{1,\mu}(z))$ as

[TABLE]

for

[TABLE]

As $\Delta j_{1}=0$ we get that $T_{B}^{i}:=\int_{\Omega_{1}}j_{1}\Delta\partial_{\varepsilon}f_{i}=\int_{\partial\Omega_{1}}{j_{1}}\partial_{n}\partial_{\varepsilon}f_{i}-\partial_{n}j_{1}\partial_{\varepsilon}f_{i}dS$ and will use this to show that all these terms are controlled by $R_{2}$.

We can use that $|\partial\Omega_{1}|\sim\mu^{-\frac{1}{2}}$ and that the rational functions are bounded by (5.50) and their variations by (5.51) at points with $|z|\sim\mu^{-\frac{1}{2}}$. This immediately implies that

[TABLE]

while $|T_{B}^{3}|\lesssim|d\beta(0)|\mu^{-\frac{1}{2}}\nu_{{\hat{q}_{0}}}\lesssim R_{2}$ follows since $|\partial_{\varepsilon}f_{3}|+\mu^{-\frac{1}{2}}|\partial_{\varepsilon}\nabla f_{3}|\lesssim\mu^{-({n_{0}^{*}}+1)/2}\lesssim\mu^{-\frac{1}{2}}{\bar{\nu}}^{\frac{1}{2}}$.

We can now use that terms of the form $d\beta(0)(a_{j}z^{j})$, $a_{j}\in{\mathbb{C}}$, and their radial derivatives are $L^{2}(\partial\mathbb{D}_{r})$ orthogonal to terms of the form $dU_{1}(0)(b_{k}(\bar{z})^{k})$, $b_{k}\in{\mathbb{C}}$, and their radial derivatives on any circle $\partial\mathbb{D}_{r}=\partial\mathbb{D}_{r}(0)$ if $j\neq k$. This implies that

[TABLE]

At this stage we hence know that

[TABLE]

for $|\text{err}_{3}|\lesssim R_{2}$ since we know that $\Delta q_{0}=0$.

As $\bar{h}_{j,\mu}$ is a harmonic map into $S^{2}$ and as

[TABLE]

we can write

[TABLE]

for $H_{j}(t):=\tfrac{4j^{2}t^{2j-2}}{(1+t^{2j})^{3}}$. The orthogonality of the different Fourier modes on circles hence allows us to write the main terms in (5.65) as

[TABLE]

for \alpha_{j}=\alpha_{j}(\partial_{\varepsilon}q_{1},q_{0}):=j^{-1}\big{[}\text{Arg}(a_{j}(\partial_{\varepsilon}q_{1}))-\text{Arg}(a_{j}(q_{0}))\big{]},

[TABLE]

and

[TABLE]

We note that the above calculation applies for any variation of $q_{i}$ that satisfies (3.7) and hence shows that $\partial_{\varepsilon}E(\mathfrak{z})$ is always controlled by (3.14).

To find a specific variation of $q_{1}$ for which Lemma 3.5 holds we now use the following lemma, a proof of which is included below.

Lemma 5.4**.**

Let $U_{0,1}^{*}:\hat{}{\mathbb{C}}\to N$ be distinct harmonic maps which satisfy Assumption 1. Then there exists $\alpha^{*}\in[0,2\pi)$ and $c^{*}>0$ so that $-I_{j}^{\theta}(\alpha^{*},U_{0}^{*},U_{1}^{*})>2c^{*}$ for all $j\in{\mathbb{N}}$ and hence so that

[TABLE]

Given a rational map $q_{0}$ we will apply this lemma for $j_{0}\in\{1,\ldots,{n_{0}^{*}}\}$ chosen so that

[TABLE]

where $\Lambda_{1}\geq 1$ is a large number that only depends on $U_{0,1}^{*}$ and $q_{0,1}^{*}$ and that is fixed below.

Writing ${q_{1}}$ in the form $q_{1}=\tfrac{r_{1}}{r_{2}}$ for polynomials $r_{1,2}$ which are normalised by $r_{2}(0)=1$, we finally define the desired variation of $q_{1}$ by

[TABLE]

where we choose $\theta^{*}:=\alpha^{*}+\frac{1}{j_{0}}\text{Arg}(a_{{j_{0}}})$ so that $\alpha_{j_{0}}(\partial_{\varepsilon}q_{1},q_{0})$ is given by the number $\alpha^{*}$ from Lemma 5.4.

As $a_{j}(\partial_{\varepsilon}q_{1})=0$ for $j<j_{0}$ while $|a_{j_{0}}(\partial_{\varepsilon}q_{1})|=1$ we hence get from (5.65) and (5.66) that

[TABLE]

for a constant $c>0$ that only depends on $U_{i}^{*}$ and $q_{0,1}^{*}$, where the last step holds true provided $\Lambda_{1}=\Lambda_{1}(U_{i}^{*},q_{i}^{*})$ in (5.69) is chosen sufficiently large.

Since (5.69) ensures that $\nu_{q_{0}}\lesssim|a_{j_{0}}(q_{0})|\mu^{-j_{0}}$ this completes the proof of (5.64) and hence of Lemma 3.5 up to the

Proof of Lemma 5.4.

We use that harmonic spheres are weakly conformal and that $dU_{i}^{*}(0)\neq 0$.

If $U_{1}^{*}(z)=U_{0}^{*}(\bar{z})$ we can hence choose coordinates on the target so that $dU_{0,1}^{*}(0)$ are given by the matrices with columns $\tfrac{1}{\sqrt{2}}|dU_{0}^{*}(0)|(1,0,0_{N-2})^{T}$ and $\tfrac{1}{\sqrt{2}}|dU_{0}^{*}(0)|(0,\pm 1,0_{N-2})^{T}$ when viewed as maps from ${\mathbb{R}}^{2}$ to ${\mathbb{R}}^{N}$. Setting $\alpha^{*}:=\pi$ hence gives

[TABLE]

which establishes the claim in this case.

So suppose instead that $N$ is three dimensional and that $U_{0,1}^{*}$ are minimal spheres whose tangent spaces at $U_{0}^{*}(0)=U_{1}^{*}(0)$ intersect transversally. In this case we choose the coordinates on the domain so that $\partial_{x_{1}}U_{1}^{*}(0)$ is contained in the intersection of these tangent spaces and then use the conformality of $U_{1}^{*}$ to choose coordinates on the target so that $\partial_{x_{1,2}}U_{1}^{*}=c_{U_{1}^{*}}e_{1,2}$ and so that $T_{U_{0}^{*}(0)}N={\mathbb{R}}^{3}\times\{0\}$. Here $e_{i}$ is the standard basis of ${\mathbb{R}}^{N}$ and $c_{U_{1}^{*}}:=\tfrac{1}{\sqrt{2}}|dU_{1}^{*}(0)|$.

The matrix $A=c_{U_{0}^{*}}^{-1}d{U_{0}}^{*}(0):{\mathbb{R}}^{2}\to{\mathbb{R}}^{N}$, whose columns are orthonormal, is hence so that $A_{31}=0$, $A_{32}\neq 0$ and so that $A_{ij}=0$ for $i\geq 4$. As

[TABLE]

for any $\alpha^{*}$ we hence get that

[TABLE]

If $A_{11}\neq A_{22}$ we can hence choose $\alpha^{*}$ to be either [math] or $\pi$. Conversely, if $A_{11}=A_{22}$ then we cannot have that also $A_{12}=-A_{21}$ as both columns of $A$ have unit length and as $A_{31}=0$ while $A_{32}\neq 0$. In this second case we can thus choose $\alpha^{*}$ as either $\frac{\pi}{2}$ or $\tfrac{3\pi}{2}$. ∎

5.4. Dominating terms in the energy expansion for variations of the underlying maps $U_{i}$

We now want to prove that we can always find variations of the maps $U_{i}$ in $\mathcal{H}_{1}(U_{i}^{*})$ so that the induced variation in $\mathcal{Z}_{0}$ is as described in Lemmas 3.7 respectively 3.8.

For such a variation we can always bound $|\partial_{\varepsilon}\Delta u_{1}|\lesssim\rho_{q}^{2}$ and hence have

[TABLE]

compare Lemma 5.2, while the analogue estimate for $|\int_{\Omega_{0}}j_{0}\partial_{\varepsilon}\Delta u_{0}|$ follows by symmetry.

The general formula (3.11) for the variation of the energy on $\mathcal{Z}_{0}$ hence tells us that

[TABLE]

for $|\text{err}_{4}|\lesssim\|\beta\|{\bar{\nu}}+\frac{\delta}{\log\mu}[\delta+{f_{\mu}}^{-1}+(\log\mu)^{-1}]+\mathcal{T}(\log\mu)^{-1}{f_{\mu}}^{-1}$.

As $c_{\mu}\sim(\log\mu)^{-1}$ we immediately get that Lemma 3.8 holds true if we choose a variation $U_{i,\varepsilon}$ in $\mathcal{H}_{1}^{\sigma}(U_{i}^{*})$ for which (2.8) holds.

On the other hand, if we consider variations of $U_{i}$ that are induced by translations on the domain then we know that $\tfrac{d}{d\varepsilon}|_{\varepsilon=0}E(U_{i})=0$ as the energy is conformally invariant. To prove Lemma 3.7 it hence suffices to prove Lemma 3.6, i.e. that there always exist variations of the form $U_{i,\varepsilon}=U_{i}(\cdot+\varepsilon e^{i\theta_{0}})$ so that

[TABLE]

If we are in a non-degenerate setting for which $U_{0}^{*}=U_{1}^{*}$ then we know that $U_{i}(z)=U_{0}^{*}(z+c_{i})$ for some $|c_{i}|\leq\sigma$. If $c_{0}=c_{1}$ then $\delta=0$ so there is nothing to prove, while for $c_{0}\neq c_{1}$ we can consider $U_{1,\varepsilon}(z)=U_{0}^{*}(z+c_{1}+\varepsilon\tfrac{c_{0}-c_{1}}{|c_{0}-c_{1}|})=U_{1}(z+\varepsilon\tfrac{c_{0}-c_{1}}{|c_{0}-c_{1}|})$ and use that $dU_{0}^{*}(0)\neq 0$ to get a variation for which (5.71) holds. The analogue argument also applies in settings where $U_{0}^{*}(z)=U_{1}^{*}(\bar{z})$ and in both these cases we could just as well have constructed a variation of $U_{0}$.

So suppose that the maps $U_{0,1}^{*}$ are instead as in part (ii) of Assumption 1. In this case we can exploit that the tangent spaces $T_{U_{i}^{*}(0)}U_{i}^{*}(\hat{}{\mathbb{C}})$, which are 2 dimensional subspaces of the same 3 dimensional space $T_{U_{i}^{*}(0)}N$, intersect transversally and that the length $\delta$ of the geodesic $\hat{\gamma}_{U_{1},U_{0}}$ that connects the points $U_{i}(0)$ can be made smaller than any given constant by reducing $\sigma$.

We hence obtain that at least one of the angles $\alpha_{i}\in[0,\frac{\pi}{2}]$ at which $\hat{\gamma}$ intersects $T_{U_{i}(0)}U_{i}(\hat{}{\mathbb{C}})$ must be so that $|\frac{\pi}{2}-\alpha_{i}|\geq\tfrac{1}{3}\bar{\alpha}$, where $\bar{\alpha}\in(0,\frac{\pi}{2}]$ denotes the angle between the spaces $T_{U_{i}^{*}(0)}U_{i}^{*}(\hat{}{\mathbb{C}})$. For this $i$ we hence know that $\|P^{T_{U_{i}(0)}U_{i}(\hat{}{\mathbb{C}})}(\tfrac{\hat{\gamma}^{\prime}(i^{*})}{\|\hat{\gamma}^{\prime}(i^{*})\|})\|=\cos(\alpha_{i})\geq\sin(\bar{\alpha}/3)$ is bounded away from zero, where we recall that $\hat{\gamma}(i^{*})=U_{i}(0)$, $0^{*}=1,1^{*}=0$.

As $U_{i}^{*}$ is weakly conformal and $dU_{i}^{*}(0)\neq 0$ and as the maps $U_{i}$ are $C^{k}$ close to $U_{i}^{*}$ we hence deduce that $U_{i}(\cdot+\varepsilon e^{i\theta_{0}})$ has the desired property (5.71) if $\theta_{0}$ is chosen so that $\partial_{\varepsilon}U_{i}(0)(e^{i\theta_{0}})$ points in the direction of $\pm P^{T_{U_{i}(0)}U_{i}(\hat{}{\mathbb{C}})}(\tfrac{\hat{\gamma}^{\prime}(i^{*})}{\|\hat{\gamma}^{\prime}(i^{*})\|})$ .

This completes the proof of Lemma 3.6 and hence the proof of Lemma 3.7.

6. Estimates on the first and second variation of the energy

We conclude this paper with the proofs of the claims on the first and second variation of the energy at points $\mathfrak{z}\in\mathcal{Z}_{0}$ made in Lemmas 3.1-3.3. We will continue to work in stereographic coordinates which are scaled so that $|a_{n_{0}^{*}}(q_{0})|=1$ and hence so that $\mu_{0}=1$, but we now need to consider variations in general directions $w\in\Gamma(\mathfrak{z}^{*}TN)$. It is hence useful to observe that for any such $w$ we can bound

[TABLE]

as $\|\nabla w\|_{L^{2}(\mathbb{D}_{1})}$, $\int_{S^{1}}|w(e^{i\theta})|$ and $\int_{S^{1}}|w(\mu^{-1}e^{i\theta})|$ are all controlled by $\|w\|_{\mathfrak{z}}$. In the following we can thus use that

[TABLE]

which in particular implies that

[TABLE]

and we note that combined with (3.2) this immediately gives (3.3).

In the proofs of Lemmas 3.2 and 3.3 below we will furthermore use that (6.1) ensures that

[TABLE]

6.1. Uniform definiteness of the second variation orthogonal to $\mathcal{Z}$

In this section we want to prove that the second variation is uniformly definite orthogonal to our set of singularity models as claimed in Lemma 3.1.

The main step in the proof of Lemma 3.1 is to show that there exists a constant $c_{0}>0$ so that for any $w\in T_{\mathfrak{z}}^{\perp}\mathcal{Z}$ with $\|w\|_{\mathfrak{z}}=1$ there is an element $v\in T_{\mathfrak{z}}^{\perp}\mathcal{Z}$ with $\|v\|_{\mathfrak{z}}=1$ so that

[TABLE]

Once we have shown this we can define $\mathcal{V}_{\mathfrak{z}}^{\pm}$ as the span of the eigenfunctions to positive respectively negative eigenvalues of the corresponding Jacobi operator

[TABLE]

which is characterised by

[TABLE]

and obtain the claim of the lemma from the spectral theorem for selfadjoint Fredholm operators.

To prove this claim (6.5) we will relate elements $\mathfrak{z}$ of $\mathcal{Z}_{0}$ and vector fields $v$ and $w$ along $\mathfrak{z}$ to the corresponding elements $\hat{\mathfrak{z}}=U_{0}\circ q_{0}$ of $\mathcal{H}^{\sigma}(\omega_{0})$ and to vector fields $\hat{v}$ and $\hat{w}$ along $\hat{\mathfrak{z}}$.

For such vector fields we will always work with respect to the inner product

[TABLE]

which (upto a factor $2$) corresponds to the $H^{1}$ inner product on the sphere and which approximates $\langle\cdot,\rangle_{\mathfrak{z}}$ well away from the origin.

We note that $d^{2}E(\omega_{0})$ is uniformly definite on $T_{\omega_{0}}^{\perp}\mathcal{H}^{\sigma}(\omega_{0})$ since $T_{\omega_{0}}\mathcal{H}^{\sigma}(\omega_{0})$ coincides with the kernel of the corresponding Jacobi-operator $L_{\omega_{0}}:\Gamma(\omega_{0}^{*}TN)\to\Gamma(\omega_{0}^{*}TN)$ and since the eigenvalues of this self-adjoint Fredholm operator tend to infinity. As $\mathcal{H}^{\sigma}(\omega_{0})$ is a finite dimensional manifold which is contained in a small $C^{k}$ neighbourhood of $\omega_{0}$ we can then use the continuity of the Jacobi-operator to deduce that the analogue statement is also true for general elements of $\mathcal{H}^{\sigma}(\omega_{0})$ (provided $\sigma>0$ is chosen sufficiently small).

I.e. we can use that there exists $c>0$ so that for any $\hat{\mathfrak{z}}\in\mathcal{H}^{\sigma}(\omega_{0})$ and any $\hat{w}\in T_{\hat{\mathfrak{z}}}^{\perp}\mathcal{H}^{\sigma}(\omega_{0})$ with $\|\hat{w}\|_{\pi}=1$ there exists a unit element $\hat{v}\in T_{\hat{\mathfrak{z}}}^{\perp}\mathcal{H}(\omega_{0})$ so that

[TABLE]

In the proof below we will furthermore use that variations $\mathfrak{z}_{\varepsilon}$ in $\mathcal{Z}_{0}$ induced by variations of only $U_{0}$ and $q_{0}$ are well approximated by the corresponding variations $\hat{\mathfrak{z}}_{\varepsilon}=U_{0}^{(\varepsilon)}\circ q_{0}^{(\varepsilon)}$ in $\mathcal{H}^{\sigma}(\omega_{0})$. To be more precise we can easily check that

[TABLE]

where here and in the following we write that a quantity is given by $o(1)$ if we can ensure that it is smaller than any given positive number by choosing $\bar{\mu}$ sufficiently large and $\sigma$ and $\sigma_{1}$ sufficiently small. Here we use that $|1-\phi_{\mu}|\lesssim\tfrac{\log{f_{\mu}}}{\log\mu}=o(1)$ for $|z|\geq r_{0}^{2}={f_{\mu}}^{-2}$ to bound the $L^{\infty}$ norm of $\partial_{\varepsilon}(\mathfrak{z}-\hat{\mathfrak{z}})$ and we note that we furthermore have

[TABLE]

where $\|\cdot\|_{\pi,\Omega}$ and $\|\cdot\|_{\mathfrak{z},\Omega}$ are defined as in (3.1) and (6.6) but with the integrals taken over $\Omega$.

With these preparations in place we now turn to the proof of (6.5).

So let $\mathfrak{z}\in\mathcal{Z}_{0}$ and let $w\in T_{\mathfrak{z}}^{\perp}\mathcal{Z}$ be so that $\|w\|_{\mathfrak{z}}=1$. We first remark that (6.2) and (6.3) ensure that $\|w\|_{\mathfrak{z},A}+\||\nabla\mathfrak{z}|w\|_{L^{2}(A)}=o(1)$ and hence that

[TABLE]

The claim (6.5) is thus trivially true (for $v=w$) if $\|w\|_{\mathfrak{z},{\mathbb{R}}^{2}\setminus A}$ is small and by symmetry we can hence assume that $\|w\|_{\mathfrak{z},{\mathbb{R}}^{2}\setminus\mathbb{D}_{r_{0}}}^{2}\geq c_{4}$ for a small, but fixed constant $c_{4}>0$.

Given such a $w$ we first want to construct a $\hat{w}\in T_{\hat{\mathfrak{z}}}^{\perp}\mathcal{H}^{\sigma}(\omega_{0})$, so that

[TABLE]

for some $s_{0}\in[2r_{0}^{2},r_{0}]$. To this end we initially construct a function $w_{1}$ which vanishes on $\Omega_{1}=\{z:|z|\leq\mu^{-\frac{1}{2}}\}$ and for which there is a radius $s_{0}\in[2r_{0}^{2},r_{0}]$ so that

[TABLE]

To obtain such a function $w_{1}$ we use that $\mathbb{D}_{r_{0}}\setminus\mathbb{D}_{r_{0}^{2}}$ contains $J\gtrsim\log(r_{0}^{-1})=\log{f_{\mu}}$ disjoint annuli of the form $\mathbb{D}_{4r}\setminus\mathbb{D}_{\tfrac{1}{4}r}$ to select $s_{0}\in[2r_{0}^{2},\frac{1}{2}r_{0}]$ for which

[TABLE]

On $\mathbb{D}_{s_{0}}\setminus\mathbb{D}_{s_{0}/2}$ we then define $w_{1}$ using a standard interpolation between $w|_{\partial\mathbb{D}_{s_{0}}}$ and the mean value $\bar{w}$ over this circle which, thanks to (6.1), is bounded by $|\bar{w}|\lesssim(\log{f_{\mu}})^{\frac{1}{2}}$. Combined with (6.11) and $\int_{A}\rho_{\mathfrak{z}}^{2}\lesssim{f_{\mu}}^{-2}$ this ensures that $\|w_{1}\|_{\mathfrak{z},\mathbb{D}_{2s_{0}}\setminus\mathbb{D}_{s_{0}^{2}}}=o(1)$. On $\mathbb{D}_{s_{0}/2}\setminus\mathbb{D}_{\mu^{-\frac{1}{2}}}$ we can then choose $w_{1}$ as the harmonic function which transitions between [math] and $\bar{w}$ and note that this map has energy of order $O(\tfrac{\log{f_{\mu}}}{\log\mu})=o(1)$ and weighted $L^{2}$ norm of order $O((\log{f_{\mu}})^{\frac{1}{2}}{f_{\mu}}^{-1})=o(1)$.

Given such a function $w_{1}$ we then set $w_{2}:=P_{\hat{\mathfrak{z}}}w_{1}$ to obtain a vector field along $\hat{\mathfrak{z}}$. We note that (6.8) ensures that $\|w_{1}-w_{2}\|_{\pi,{\mathbb{R}}^{2}\setminus\mathbb{D}_{s_{0}}}\sim\|w_{1}-w_{2}\|_{\mathfrak{z},{\mathbb{R}}^{2}\setminus\mathbb{D}_{s_{0}}}=o(1)$ and that we still have $\|w_{2}\|_{\mathfrak{z},\mathbb{D}_{s_{0}}}=o(1)$.

Given a variation $\hat{\mathfrak{z}}_{\varepsilon}$ in $\mathcal{H}^{\sigma}(\omega_{0})$ with $\|\partial_{\varepsilon}\hat{\mathfrak{z}}\|_{\pi}=1$ we can hence consider the corresponding variation $\mathfrak{z}_{\varepsilon}$ of $\mathfrak{z}$ in $\mathcal{Z}$ and use (6.8) and (6.9) as well as that $w\in P_{\mathfrak{z}}^{\perp}\mathcal{Z}$ to see that

[TABLE]

We thus deduce that $\|P^{T_{\hat{\mathfrak{z}}}\hat{\mathcal{Z}}}w_{2}\|_{\pi}=o(1)$ and hence that $\hat{w}:=P^{T_{\hat{\mathfrak{z}}}^{\perp}\hat{\mathcal{Z}}}w_{2}$ satisfies (6.10).

As $\|\hat{w}\|_{\pi}^{2}\geq c_{4}-o(1)$ is bounded away from zero we can hence apply (6.7) to obtain a vector-field $\hat{v}\in T_{\hat{\mathfrak{z}}}\mathcal{H}^{\sigma}(\omega_{0})$ with $\|\hat{v}\|_{\pi}\lesssim 1$ so that

[TABLE]

and we will obtain the required element $v$ of $T_{\mathfrak{z}}^{\perp}\mathcal{Z}$ by modifying this vector field $\hat{v}$.

To this end we repeat the argument from above to obtain a $v_{1}$ with

[TABLE]

except that we work with a radius $\hat{s}_{0}\in[s_{0},2s_{0}]$ with $\int_{S^{1}}|\partial_{\theta}\hat{v}(\hat{s}_{0}e^{i\theta})|^{2}\lesssim 1$ rather than with $s_{0}$.

As $\hat{v}$ is tangential to $N$ along $\hat{\mathfrak{z}}$ and supported on $\Omega_{0}$ where $\rho_{\mathfrak{z}}\sim|\nabla\pi|$ we can combine (6.8) and (6.13) to see that

[TABLE]

where we use that $|\nabla\mathfrak{z}-\nabla\hat{\mathfrak{z}}|\lesssim o(1)(\rho_{\mathfrak{z}}+|\nabla{\phi_{\mu}}|)$ on ${\mathbb{R}}^{2}\setminus\mathbb{D}_{r_{0}^{2}}$ to deal with the last term.

As $\hat{v}$ is orthogonal to $T_{\hat{\mathfrak{z}}}\mathcal{H}^{\sigma}(\omega_{0})$ we can combine this bound with (6.8) and (6.9) to see that $\langle\partial_{\varepsilon}\mathfrak{z}_{\varepsilon},P_{\mathfrak{z}}v_{1}\rangle_{\mathfrak{z}}=o(1)$ for all variations $\mathfrak{z}_{\varepsilon}$ in $\mathcal{Z}$ that are induced by variations of only $U_{0}$ and $q_{0}$. Conversely, variations of $U_{1}$ and $q_{1}$ result in variations $\mathfrak{z}_{\varepsilon}$ for which $\|\partial_{\varepsilon}\mathfrak{z}_{\varepsilon}\|_{\mathfrak{z},\Omega_{0}}=o(1)$ and for which we hence trivially know that $\langle\partial_{\varepsilon}\mathfrak{z}_{\varepsilon},P_{\mathfrak{z}}v_{1}\rangle_{\mathfrak{z}}=o(1)$.

All in all this ensures that $v:=P^{T^{\perp}_{\mathfrak{z}}\mathcal{Z}}(P_{\mathfrak{z}}v_{1})\in T_{\mathfrak{z}}^{\perp}\mathcal{Z}$ is so that

[TABLE]

We now write for short $II_{\mathfrak{z}}(w,v)=\nabla w\nabla v-A(\mathfrak{z})(\nabla\mathfrak{z},\nabla\mathfrak{z})A(\mathfrak{z})(w,v)$ for the integrand that appears in the formula (3.2) for $d^{2}E$ and note that $\int_{\Omega}|II_{\mathfrak{z}}(w,v)|\lesssim\|w\|_{\mathfrak{z},\Omega}\|v\|_{\mathfrak{z},\Omega}+o(1)$ for every $\Omega\subset{\mathbb{R}}^{2}$ where the $o(1)$ term comes from the contribution of $\nabla\gamma$ to $\nabla\mathfrak{z}$, compare (6.3). We hence get

[TABLE]

where the last inequality holds true provided $\sigma$ and $\sigma_{1}$ are sufficiently small and $\bar{\mu}$ is sufficently large. As $\|v\|_{\mathfrak{z}}\lesssim 1$ we hence obtain that (6.5) holds true for the corresponding unit element of $T_{\mathfrak{z}}^{\perp}\mathcal{Z}$ which completes the proof of Lemma 3.1.

6.2. Proofs of Lemmas 3.2 and 3.3

We finally turn to the proofs of the estimates on the norms of the first and second variation of $E$ claimed in Lemmas 3.2 and 3.3. To simplify the notation we write for short

[TABLE]

and

[TABLE]

and recall that to prove Lemma 3.2 we need to show that

[TABLE]

while Lemma 3.3 asserts that

[TABLE]

By symmetry it suffices to bound the corresponding integrals over $\Omega_{1}$ where $\mathfrak{z}=\mathfrak{z}_{1}+\text{err}_{\varphi}$ for $\mathfrak{z}_{1}=\pi_{N}({v_{1}})$ and for $\text{err}_{\varphi}$ which is defined by (5.55) and supported on ${\hat{A}}$.

To prove both lemmas we write the tension of $\mathfrak{z}$ on $\Omega_{1}$ as

[TABLE]

and use that $\Delta j_{1}=0$ to write

[TABLE]

where we split the contribution of $\Delta u_{1}$ into the terms

[TABLE]

and write for short

[TABLE]

In the analysis of the resulting integrals we will use that

[TABLE]

which, thanks to (5.47), (5.48) and (5.7), ensures that

[TABLE]

Similarly, since we can bound

[TABLE]

compare also (5.16), we can use these estimates (5.47), (5.48) and (5.7) together with (5.49) to see that

[TABLE]

To analyse terms that involve $\gamma$ we instead write $v_{1}=\gamma+\tilde{u}_{1}+j_{1}$ to see that

[TABLE]

and hence that also

[TABLE]

Finally since $|P_{\mathfrak{z}_{1}}\nabla v_{1}-\nabla v_{1}|\lesssim|u_{1}-\mathfrak{z}_{1}|\rho_{q}+{\|\beta\|}|\nabla{\hat{q}_{0}}|+|v_{1}-\gamma||\nabla\gamma|$ we also get that

[TABLE]

With these estimates in place we can now show that all contributions to $dE(\mathfrak{z})(w)$ are controlled by $R_{3}+\mathcal{T}$ and all contributions to $d^{2}E(\mathfrak{z})(\partial_{\varepsilon}\mathfrak{z},w)$ are controlled by $R_{4}+\mathcal{T}+\eta_{\mathcal{T}}$.

To begin with we use that $|\tau(u_{1})|\lesssim\mathcal{T}\rho_{q}^{2}\lesssim\mathcal{T}\rho_{\mathfrak{z}}^{2}$ to see that

[TABLE]

while we can use that $|\partial_{\varepsilon}T_{u_{1}}^{\tau}|\lesssim|\tau(u_{1})|+|\partial_{\varepsilon}\tau(u_{1})|\lesssim\mathcal{T}\rho_{q}\rho_{\mathfrak{z}}+\eta_{\tau}\rho_{q}^{2}\lesssim(\mathcal{T}+\eta_{\tau})\rho_{\mathfrak{z}}^{2}$ to bound

[TABLE]

As $A(u_{1})(\nabla u_{1},\nabla u_{1})\in T_{u_{1}}^{\perp}N$ we can then write

[TABLE]

and hence use (6.21) to bound

[TABLE]

From (6.26) we see that |\partial_{\varepsilon}T_{u_{1}}^{A}|\lesssim\big{[}|u_{1}-v_{1}|+|\partial_{\varepsilon}(u_{1}-v_{1})|\big{]}\rho_{q}\cdot\rho_{\mathfrak{z}} which we can combine with (6.21) and (6.23) to see that also

[TABLE]

Similarly, we can write

[TABLE]

and use (5.11), (6.3), (6.4) and (6.24) to see that

[TABLE]

As

[TABLE]

we can furthermore use (6.3), (6.4), (6.24) and (5.11) to see that

[TABLE]

where the penultimate step follows from (6.4).

Finally we use (5.41) to write

[TABLE]

From (6.3), (6.21), (6.24) and (6.25) we hence get that also

[TABLE]

From (6.30) we see that

[TABLE]

As we can write $P_{\mathfrak{z}_{1}}^{\perp}\nabla v_{1}=(P_{\mathfrak{z}_{1}}^{\perp}-P_{u_{1}}^{\perp})(\nabla u_{1})+(P_{\mathfrak{z}_{1}}^{\perp}-P_{\gamma}^{\perp})(\nabla\gamma)+(P_{\mathfrak{z}_{1}}^{\perp}-P_{U_{1}(0)}^{\perp})\nabla j_{1}$ we hence also get that $|\int\partial_{\varepsilon}T_{\pi}w|\lesssim R_{4}$.

All in all we hence get that

[TABLE]

To complete the proofs of Lemmas 3.2 and 3.3 it remains to bound the contributions of the last two terms in (6.16). These are supported on $\hat{A}$ where $\rho_{\mathfrak{z}}\lesssim 1$, $\tau(\gamma)=0$ and hence

[TABLE]

so in particular $|\Delta\mathfrak{z}_{1}|\lesssim 1+\tfrac{\delta^{2}}{(\log\mu)^{2}}\mu$. Combined with (5.56) and (6.4) this shows that

[TABLE]

only gives a lower order contribution, while (5.56) and (6.4) also allow us to bound

[TABLE]

This completes the proof of Lemma 3.2

Combining (6.31) with (5.58) gives a pointwise bound of $|\partial_{\varepsilon}\text{err}_{\varphi}||\Delta\mathfrak{z}_{1}|\lesssim R_{4}$ while combining $|\partial_{\varepsilon}\Delta\mathfrak{z}_{1}|\lesssim\mu({\bar{\nu}}+\eta_{rat}\mu^{-j^{*}})+\mu\tfrac{\delta^{2}+\delta\eta_{0}}{(\log\mu)^{2}}$ with (5.56) ensures that also $|\text{err}_{\varphi}||\partial_{\varepsilon}\Delta\mathfrak{z}_{1}|\lesssim R_{4}$ on $\hat{A}$. Combined with (6.4) we can hence obtain that

[TABLE]

just gives a lower order contribution to $d^{2}E(\mathfrak{z})(\partial_{\varepsilon}\mathfrak{z},w)$.

Finally (5.56), (5.59) and (6.4) allow us to also bound

[TABLE]

which completes the proof of Lemma 3.3.