Prescribing Morse scalar curvatures: blow-up analysis

TL;DR

This paper analyzes the blow-up behavior of solutions to prescribed Morse scalar curvature problems, establishing precise blow-up rates in the subcritical case and excluding tower bubbles across all dimensions.

Contribution

It provides a detailed blow-up rate analysis for subcritical solutions and rules out tower bubbles, advancing understanding of scalar curvature prescription problems.

Findings

Precise blow-up rates for subcritical solutions

Exclusion of tower bubbles in all dimensions

Foundation for future existence results

Abstract

We study finite-energy blow-ups for prescribed Morse scalar curvatures in both the subcritical and the critical regime. After general considerations on Palais-Smale sequences we determine precise blow up rates for subcritical solutions: in particular the possibility of tower bubbles is excluded in all dimensions. In subsequent papers we aim to establish the sharpness of this result, proving a converse existence statement, together with a one to one correspondence of blowing-up subcritical solutions and {\em critical points at infinity}. This analysis will be then applied to deduce new existence results for the geometric problem.

Equations

$$R_{g_u}{u}^{\frac{n+2}{n-2}}=L_{g_0}u,$$

$$L_{g_0}u=-c_n{\Delta }_{g_0}u+R_{g_0}u,c_n=\frac{4(n-1)}{n-2}$$

$$L_{g_0}u=K{u}^{\frac{n+2}{n-2}},u>0$$

$${\int }_{S^n}⟨\nabla K,\nabla f{⟩}_{g_{S^n}}{u}^{\frac{2n}{n-2}}d{\mu }_{g_{S^n}}=0$$

$$\{\nabla K=0\}\cap \{\Delta K=0\}=\varnothing ,$$

$$x\text{mbox}saddlepointforK\wedge \underset{t}{\inf }K(x(t))\le K(x)<K(x_0)\Rightarrow \Delta K(x)>0.$$

$$\sum _{x\in \{\nabla K=0\}\cap \{\Delta K<0\}}(-1{)}^{m(x,K)}\ne -1.$$

$$c_q=♯\{x\in M:\nabla K(x)=0,\Delta K(x)<0\text{ and }m(K,x)=3-q\}$$

$$\sum _{x\in \{\nabla K=0\}\cap \{\Delta K<0\}}(-1{)}^{m(x,K)}\ne (-1{)}^n.$$

$$-c_n{\Delta }_{g_0}u+R_{g_0}u=K{u}^{\frac{n+2}{n-2}-\tau },0<\tau \ll 1.$$

$$J(u)=\frac{{\int }_M\left(c_n|\nabla u{|}_{g_0}^2+R_{g_0}{u}^2\right)d{\mu }_{g_0}}{(\int K{u}^{\frac{2n}{n-2}}d{\mu }_{g_0}{)}^{\frac{n-2}{n}}},$$

$${\int }_{S^n}K({\delta }_{a_1,\lambda }+{\delta }_{a_2,\lambda }{)}^{\frac{2n}{n-2}}d{\mu }_{g_{S^n}}\simeq K(a_1)+K(a_2)+\frac{c_1}{{\lambda }^{n-2}},{\int }_{S^n}K{\delta }_{a_i,\lambda }^{\frac{2n}{n-2}}d{\mu }_{g_{S^n}}\simeq c_{2}K(a_i)-\frac{c_3}{{\lambda }^2}\Delta K(a_i)$$

$$\bigg{\|}u_{m}-\sum_{j=1}^{q}\alpha_{j,m}\delta_{\lambda_{j,m},a_{j,m}}\bigg{\|}_{W^{1,2}(M,g_{0})}\longrightarrow 0\quad\text{ as }\quad m\longrightarrow\infty$$

$${\alpha }_{j,m}=\frac{\Theta }{K(x_j{)}^{\frac{n-2}{4}}}+o(1),a_{j,m}⟶x_j\text{ and }{\lambda }_{j,m}\simeq {\lambda }_{{\tau }_m}={\tau }_m^{-\frac{1}{2}},$$

$${\lambda }_{j,m}=c_1\sqrt{\frac{\Delta K(x_j)}{K(x_j)\tau }},a_{j,m}=c_2({\nabla }^{2}K(x_j){)}^{-1}\frac{\nabla \Delta K(x_j)}{{\lambda }_{j,m}^3},{\alpha }_j=\Theta \cdot \sqrt[p-1]{\frac{{\lambda }_j^{\theta }}{K(a_{j,m})}}$$

$$J_{\tau }({\delta }_{a,\lambda })\simeq \frac{1}{K^{\frac{n-2}{n}}(a)}({\lambda }^{\tau }-\frac{\Delta K(a)}{K(a){\lambda }^2}),$$

$$J_{\tau }({\delta }_{a_1,\lambda }+{\delta }_{a_2,\lambda })\simeq \frac{c_0}{K^{\frac{n-2}{n}}(a_1)}({\lambda }^{\tau }-\frac{\Delta K(a_1)}{K(a_1){\lambda }^2})-c_1\frac{1}{d^{n-2}{\lambda }^{n-2}}\simeq (c_2-c_3{d}^2)\left({\lambda }^{\tau }+c_4{\lambda }^{-2}\right)-c_1\frac{1}{d^{n-2}{\lambda }^{n-2}}$$

$$\frac{1}{{\lambda }^2}\simeq \tau +\frac{1}{(\lambda d{)}^{n-2}}\text{ and }d\simeq \frac{1}{{\lambda }^{n-2}{d}^{n-1}}.$$

$$\begin{array}{rl}Y(M,g_0)=& \underset{\mathcal{A}}{\inf }\frac{\int \left(c_n|\nabla u{|}_{g_0}^2+R_{g_0}{u}^2\right)d{\mu }_{g_0}}{(\int u^{\frac{2n}{n-2}}d{\mu }_{g_0}{)}^{\frac{n-2}{n}}},c_n=4\frac{n-1}{n-2}.\end{array}$$

$$\begin{array}{r}{L}_{g_0}=-c_n{\Delta }_{g_0}+R_{g_0}\end{array}$$

$$G_{g_0}:M\times M\setminus \Delta ⟶{\mathbb{R}}_{+}$$

$$\begin{array}{r}d{\mu }_{g_u}=u^{\frac{2n}{n-2}}d{\mu }_{g_0}\text{ and }R=R_{g_u}=u^{-\frac{n+2}{n-2}}(-c_n{\Delta }_{g_0}u+R_{g_0}u)=u^{-\frac{n+2}{n-2}}{L}_{g_0}u.\end{array}$$

$$c\Vert u{\Vert }_{W^{1,2}(M,g_0)}^2\le \int u{L}_{g_0}ud{\mu }_{g_0}=\int \left(c_n|\nabla u{|}_{g_0}^2+R_{g_0}{u}^2\right)d{\mu }_{g_0}\le C\Vert u{\Vert }_{W^{1,2}(M,g_0)}^2.$$

$$p=\frac{n+2}{n-2}-\tau \text{ and }0\le \tau ⟶0$$

$$J_{\tau }(u)=\frac{{\int }_M\left(c_n|\nabla u{|}_{g_0}^2+R_{g_0}{u}^2\right)d{\mu }_{g_0}}{(\int K{u}^{p+1}d{\mu }_{g_0}{)}^{\frac{2}{p+1}}},u\in \mathcal{A}.$$

$$r=r_u=\int Rd{\mu }_{g_u}=\int u{L}_{g_0}ud{\mu }_{g_0},$$

$$J_{\tau }(u)=\frac{r}{k_{\tau }^{\frac{2}{p+1}}}\text{ with }{k}_{\tau }=\int K{u}^{p+1}d{\mu }_{g_0}.$$

$$\partial J_{\tau}(u)v=\frac{2}{k_{\tau}^{\frac{2}{p+1}}}\big{[}\int L_{g_{0}}uvd\mu_{g_{0}}-\frac{r}{k_{\tau}}\int Ku^{p}vd\mu_{g_{0}}\big{]};$$

$$\begin{split}\partial^{2}J_{\tau}(u)vw=&\frac{2}{k_{\tau}^{\frac{2}{p+1}}}\big{[}\int L_{g_{0}}vwd\mu_{g_{0}}-p\frac{r}{k_{\tau}}\int Ku^{p-1}vwd\mu_{g_{0}}\big{]}\\ &-\frac{4}{k_{\tau}^{\frac{2}{p+1}+1}}\big{[}\int L_{g_{0}}uvd\mu_{g_{0}}\int Ku^{p}wd\mu_{g_{0}}+\int L_{g_{0}}uwd\mu_{g_{0}}\int Ku^{p}vd\mu_{g_{0}}\big{]}\\ &+\frac{2(p+3)r}{k_{\tau}^{\frac{2}{p+1}+2}}\int Ku^{p}vd\mu_{g_{0}}\int Ku^{p}wd\mu_{g_{0}}.\end{split}$$

$$U_{ϵ}=\{u\in \mathcal{A}\mid ϵ<\Vert u\Vert ,J_{\tau }(u)\le {ϵ}^{-1}\}.$$

Full text

Key Words: Conformal geometry, sub-critical approximation, blow-up analysis.

1. 1 Introduction

2. 2 Variational setting and preliminaries

3. 3 Blow-up analysis

4. 4 Reduction and v-part estimates

5. 5 The functional and its derivatives

6. 6 Gradient bounds

7. 7 Appendix

8. 7.1 Interactions

9. 7.2 Derivatives

10. 7.3 List of constants

1 Introduction

The problem of prescribing the scalar curvature of a manifold conformally has a long history, starting from [33], see also [31], [32]. In case of the round sphere, this is known as Nirenberg’s problem.

Given a closed manifold $(M,g_{0})$ of dimension $n\geq 3$ and a conformal metric $g=u^{\frac{4}{n-2}}g_{0}$ for a positive function $u>0$ on $M$, the conformal change of the scalar curvature is given by

[TABLE]

where by definition

[TABLE]

is the conformal Laplacian, while $\Delta_{g_{0}}$ is the Laplace-Beltrami operator with respect to $g_{0}$. Thus, in order to prescribe a function $K$ on $M$ as the scalar curvature with respect to $g$, one needs to solve

[TABLE]

pointwise on $M$, see [3]. The exponent on the right-hand side is critical with respect to Sobolev’s embedding, which makes the problem particularly challenging. In contrast to the Yamabe problem, which amounts to finding a constant scalar curvature metric, for $K$ varying on $M$ there are obstructions to the existence for (1.1). For example Kazdan and Warner proved in [33] that on the round sphere $(S^{n},g_{S^{n}})$ every solution $u$ of (1.1) must satisfy

[TABLE]

for any restriction $f$ to $S^{n}$ of an affine function on $\mathbb{R}^{n+1}$. In particular, since $u$ is positive, a necessary condition for the existence of solutions is that the function $\langle\nabla K,\nabla f\rangle_{g_{S^{n}}}$ changes sign.

One of the first answers to Nirenberg’s problem was given by J. Moser in [41] for two dimensions, where the counterpart of (1.1) has an exponential form. He proved that for $K$ being an even function on $S^{2}$ a solution always exists. A related result was given by J. Escobar and R. Schoen in [23], showing existence of solutions when $K$ is invariant under some group $G$ acting without fixed points, under suitable flatness assumptions of order $n-2$. In the same paper some results were also found for non-spherical manifolds using positivity of the mass. Other sufficient conditions for the existence in case of $G$-invariant functions were given by E. Hebey and M. Vaugon in [25], [26], allowing the possibility of fixed points.

Other existence results were obtained by A. Chang and P. Yang, see [18], [19], for the case $n=2$ without requiring any symmetry of $K$. One condition, for which they obtained existence, is the following. First they assumed, that $K$ is a positive Morse function satisfying

[TABLE]

where here and in the following $\nabla=\nabla_{g_{0}}$ and $\Delta=\Delta_{g_{0}}$, cf. (2.5) and below. Secondly, they supposed that $K$ possesses $p$ local maxima and $q$ saddle points with negative Laplacian and $p\neq q+1$. The latter condition was used to prove the result via a Leray-Schauder degree-theoretical argument. In the same papers other results were given, requiring conditions only at some prescribed levels of $K$. Typically $K$ must possess two maxima $x_{0}$ and $x_{1}$, $K(x_{1})\leq K(x_{0})$, which are connected by some path $x(t)$ for which

[TABLE]

Statements of this last kind have been obtained in [21] for $n=2$ and in [9] for $n\geq 3$. Another existence result was given by A. Bahri and J.M. Coron in [6] for $n=3$ and a Morse function $K$ satisfying (1.2) and

[TABLE]

Here $m(x,K)$ denotes the Morse index of $K$ at $x$, cf. also [12]. The result of Bahri and Coron, which relies on a topological argument, has been extended in several directions.

An extension of condition (1.3), based on Morse’s inequalities, was given by Schoen and Zhang in [45] for the case $n=3$. For a Morse function $K$ satisfying (1.2) and setting

[TABLE]

they required that either $c_{0}-c_{1}+c_{2}\neq 1$ or $c_{0}-c_{1}>1$. Note that the first condition is equivalent to (1.3) and the second one for $n=2$ corresponds to the condition $p+1>q$ in [18].

Other results of perturbative type and relying on finite-dimensional reductions were given by A. Chang and P. Yang in [20] and by A. Ambrosetti, J. Garcia-Azorero and I. Peral in [1], see also [35]. The authors considered the case in which $K$ is close to a constant and satisfies an analogue of (1.3), i.e.

[TABLE]

In [28] Y.Y. Li proved existence of solutions for every dimension, if the function $K$ near each critical point has a Morse-type structure, but with a flatness of order $\beta\in(n-2,n)$. His proof relied on a homotopy argument: considering $K_{t}=t\,K+(1-t)$, $t\in[0,1]$ the author used the degree-counting formula of [20] for $t$ small, and then a refined blow-up analysis of equation (1.1), when $t$ tends to $1$. A different degree formula under more general flatness conditions was introduced in [16]. Other results obtained by different approaches can also be found in [8], [10], [22].

A useful tool for the above results is a subcritical approximation of (1.1), namely

[TABLE]

The advantage of (1.4), compared to (1.1), is that the lower exponent makes the problem compact, so it is easier to construct solutions. However, the interesting point is passing to the limit of solutions for $\tau\longrightarrow 0$ and in general one expects some of them to diverge with zero weak limit. The approach in [12], [45], [28] was to understand in detail the behaviour of blowing-up solutions and then to use degree- or Morse-theoretical arguments to show that some solutions stay bounded.

Consider now a Morse function $K$ on the sphere satisfying (1.2). In dimension $n=3$ or under a flatness condition in higher dimensions, it turns out that blowing-up solutions to (1.4) develop a single bubble at critical points of $K$ with negative Laplacian. Bubbles correspond to solutions of (1.1) on $S^{n}$ with $K\equiv 1$ and were classified in [11], see also [2], [47], and after proper dilation represent the profiles of diverging solutions, cf. Section 2 for precise formulas.

The single-bubble phenomenon can be qualitatively explained exploiting the variational features of the problem, which admits the Euler-Lagrange energy $J=J_{K}$ given by

[TABLE]

see also (2.1) regarding (1.4). Denote by $\delta_{a,\lambda}$ a bubble centered at $a\in S^{n}$ with dilation parameter $\lambda$. Then for distinct and fixed points $a_{1},a_{2}$ and $\lambda$ large one has the expansions

[TABLE]

with constants $c_{i}>0$, where $c_{1}$ depends on $a_{1}$ and $a_{2}$. We refer to Section 5 for more accurate results. Terms similar to the above ones appear in the expression of $J_{\tau}$. By the latter formulas and for $\lambda\longrightarrow\infty$ and $n=3$ the interaction of the bubbles with $K$ is dominated by the mutual interactions among bubbles. This causes multiple bubbles to suppress each other allowing only one blow-up point at a time, which has to be close to at critical points of $K$ with negative Laplacian due to a Pohozaev identity.

This analysis was carried over in [29] also on $S^{4}$. In this case the above interactions are of the same order and multiple blow-ups occur. It was also shown there that multiple bubbles cannot accumulate at a single point. Using a terminology from [43], [44] such blow-ups are called isolated simple. In four dimensions a different constraint on multiple blow-up points replaces $\Delta K<0$, depending on the least eigenvalue of a matrix constructed out of $K$ and the location of the blow-up points, cf. (0.8) in [29]. On general four-dimensional manifolds there is an extra term due to the mass of the manifold leading to similar phenomena, but with modified formulas, see [7].

The goal of this paper is to investigate the blow-up behaviour in an opposite regime, when the dimension $n\geq 5$ and the function $K$ is Morse. In this case the second term in (1.5) dominates the first one, so it is drastically different from situation of low-dimensions or with flat curvatures. However we can still show that blow-ups are isolated simple, which is important in understanding the Morse-theoretical structure of the energy functional. Here is our main result.

Theorem 1**.**

Let $(M^{n},g_{0})$, $n\geq 5$ be a closed manifold of positive Yamabe invariant and $K:M\longrightarrow\mathbb{R}$ a smooth positive Morse function satisfying (1.2). Then positive sequences of solutions to (1.4) for $\tau_{m}\searrow 0$ with uniformly bounded $W^{1,2}$-energy and zero weak limit have only isolated simple blow-ups at critical points of $K$ with negative Laplacian.

The above theorem follows from Proposition 3.1, where a general characterization of blowing-up Palais-Smale sequences for (1.4) as $\tau\longrightarrow 0$ is given, and from Theorem 2, where a lower bound on the norm of the gradient of the Euler-Lagrange functional $J_{\tau}$ for (1.4) is proved, see (2.1).

Remark 1.1**.**

Solutions of (1.4) can be found as suitably normalized critical points of the scaling-invariant energy $J_{\tau}$ in (2.1). For a sequence of critical points $(u_{m})$ of $J_{\tau_{m}}$, with $\tau_{m}$ as in Theorem 1, there exist up to subsequences $q\in\mathbb{N}$ and distinct points $x_{1},\dots,x_{q}\in M$ with $\nabla K(x_{j})=0$ and $\Delta K(x_{j})<0$ such that

[TABLE]

for some

[TABLE]

where the multiplicative constant $\Theta$ reflects the scaling invariance of $J_{\tau_{m}}$, see (2.1), and can be fixed for instance by prescribing the conformal volume, cf. Remark 6.2. In Theorem 2 we will show much more precise estimates, that will be crucial for [36]. For example, if $n\geq 6$, we find

[TABLE]

up to errors of order $o(\lambda_{\tau_{m}}^{-3})$, where $c_{1},c_{2}$ are dimensional constants and we identify by a slight abuse of notation $a_{j,m}$ with its image in conformal normal coordinates at $x_{j}$, cf. [27]. Hence all the finite dimensional variables, i.e. $\alpha_{j,m},a_{j,m}$ and $\lambda_{j,m}$ are determined to a precision of order $o(\lambda_{\tau_{m}}^{-3})$.

Remark 1.2**.**

We next compare Theorem 1 to some existing literature and add further comments.

1. (a)

On $S^{3}$ and $S^{4}$ the isolated-simpleness of solutions was proved in **[12]**, **[28]**, **[29]**, **[45]** for arbitrary sequences of solutions by a refined blow-up analysis. The uniform $W^{1,2}$-bound is then derived a-posteriori. In dimension $n\geq 5$ the latter bound may not hold true in general - we refer the reader to **[13]**, **[14]**, **[15]**, where in some cases it is shown that blowing-up solutions for the purely critical equation (1.1) must have diverging energy and blow-ups of diverging energies and towering bubbles are also constructed, cf. also **[34]**, **[42]**, **[48]**. However, in the forthcoming paper **[37]** we will construct solutions to (1.4) via min-max or Morse theory with the purpose of finding a non-zero weak limit. These will indeed satisfy the required energy bound. This will allow us to obtain existence results under less stringent conditions compared to some others in the literature, as in **[9]** and **[17]**. 2. (b)

On manifolds not conformally equivalent to $S^{n}$ a-priori estimates were proved in **[30]** for $n=3$ in both critical and subcritical cases. Our analysis carries over for $n=4$ as well, where the matrix in Definition 6.1, introduced in **[7]**, **[29]** and also involving the mass, gives constraints on the location of multiple blow-up points. The main new aspect of our result is the isolated simple blow-up behaviour in dimension $n\geq 5$, so we chose to state Theorem 1 in a simple form only for this case. We refer to Theorem 2 for a more precise version of the result: here we derive indeed estimates on solutions with high precision as $\tau\longrightarrow 0$, as well as estimates that are uniform in this parameter. 3. (c)

In **[36]** we will show a converse statement. Given any distinct points $p_{1},\dots,p_{k}$ in $\{\nabla K=0\}\cap\{\Delta K<0\}$ and $\tau_{i}\searrow 0$ there exist solutions $(u_{i})_{i}$ to (1.4) blowing-up at $p_{1},\dots,p_{k}$ exactly as described above. Thence the characterization of Theorem 1 is optimal. We refer to **[28]**, **[29]** for the counterparts on three- and four-spheres. Proposition A2 in **[5]** regards the construction of a pseudo gradient flow for problem (1.1) ruling out multiple bubble formation at the same point for any $n$, although we believe the proof there is not complete. We refer to **[39]** for details and for the proof of a one-to-one correspondence of blowing-up sequences and critical points at infinity, cf. **[4]**. See also **[40]** for some delicate relations between $L^{2}$- and pseudo gradient flows. 4. (d)

We expect the same conclusion of Theorem 1 should hold true replacing the energy bound with a Morse index bound. It would also be interesting to understand the case of non-zero weak limits.

We discuss next some heuristics about the proof of Theorem 1. First we show a quantization result for Palais-Smale sequences of solutions to (1.4) as $\tau\longrightarrow 0$. We are inspired in this step from a result by M. Struwe in [46], where the same was proved for $\tau=0$: in our case we need extra work in the limiting process, due to a different dilation covariance of subcritical equations.

We then prove that we are in a perturbative regime and every solution to (1.4) for $\tau$ sufficiently small can be written as a finite sum of highly peaked bubbles and an error term small in $W^{1,2}$-norm, which we prove to have a minor effect in the expansions. Performing a careful analysis of the interactions of the bubbles among themselves and with $K$, it is not difficult to see that for $n\geq 5$ blow-ups should occur at critical points of $K$ with negative Laplacian only, cf. also Theorem 1.1 in [14], and we are left with excluding multiple bubbles towering at the same limit point, which is the crucial result in our paper.

We give an idea of this fact in some particular cases, that are easy to describe. Let $J_{\tau}$ be the Euler-Lagrange energy of (1.4), see (2.1). For a critical point $a$ of $K$, the following expansion holds for $J_{\tau}$ on a bubble concentrated at $a$

[TABLE]

cf. Proposition 5.1. By elementary considerations one checks that for $\Delta K(a)<0$ the function in the right-hand side has a non-degenerate minimum point at $\lambda=\lambda_{\tau}\simeq\tau^{-\frac{1}{2}}$, see also Proposition 2.1 in [45]. Since bubbles have an *attractive interaction *, cf. the first equation in (1.5), even in terms of dilations centering more bubbles at the point $a$ would make all dilation parameters collapse at $\lambda=\lambda_{\tau}$, see Figure 3. For the same reason, still by (1.6), one would get collapse with respect to the center points of multiple bubbles distributed along the unstable directions from a critical point of $K$, since points with lager values of $K$ have smaller energy, due to (1.6), see Figure 3.

We consider then the case of bubbles centered at two points $a_{1},a_{2}$ symmetrically located at distance $d$ from a critical point $\bar{p}$ such that $\Delta K(\bar{p})<0$, and along a stable direction of $K$, with the same $\lambda$’s. Here in principle the attractive force among bubbles could compensate the repulsive interaction from the critical point $\bar{p}$ of $K$, see Figure 3. For this configuration one gets an energy expansion of the form

[TABLE]

with $c_{i}>0$. From the analysis in Proposition  3.1 it turns out that $\lambda^{\tau}\simeq 1$, so imposing criticality in both $\lambda$ and $d$ one finds the relations

[TABLE]

These asymptotics imply that $\lambda^{-2}\simeq\tau+\lambda^{-\frac{2(n-2)}{n}}$, which is impossible for $\lambda$ large. The general case is rather involved to study and will be treated by a top-down cascade of estimates in Section 6.

The plan of the paper is the following. In Section 2 we introduce the variational setting of the problem and list some preliminary results. We then study some approximate solutions of (1.1), highly concentrated at arbitrary points of $M$. From these one can carry out a reduction procedure of the problem, which is done later in the paper. In Section 3 we prove a general quantization result for Palais-Smale sequences of (1.4) with uniformly bounded $W^{1,2}$-energy. In Section 4 we reduce the problem to a finite-dimensional one, while in Section 5 we derive some precise asymptotic expansions of the Euler-Lagrange energy. Section 6 is then devoted to proving suitable bounds on the gradient to exclude tower bubbles and prove our main result. We finally collect in the appendix the proofs of some useful technical estimates as well as a list of relevant constants appearing.

Acknowledgments. A.M. has been supported by the project Geometric Variational Problems and Finanziamento a supporto della ricerca di base from Scuola Normale Superiore and by MIUR Bando PRIN 2015 2015KB9WPT001. He is also member of GNAMPA as part of INdAM.

2 Variational setting and preliminaries

In this section we collect some background and preliminary material, concerning the variational properties of the problem and some estimates on highly-concentrated approximate solutions of bubble type.

We consider a smooth, closed Riemannian manifold $M=(M^{n},g_{0})$ with volume measure $\mu_{g_{0}}$ and scalar curvature $R_{g_{0}}$. Letting $\mathcal{A}=\{u\in W^{1,2}(M,g_{0})\mid u\geq 0,u\not\equiv 0\}$ the Yamabe invariant is defined as

[TABLE]

We will assume from now on that the invariant is positive. As a consequence the conformal Laplacian

[TABLE]

is a positive and self-adjoint operator. Without loss of generality we assume $R_{g_{0}}>0$ and denote by

[TABLE]

the Green’s function of $L_{g_{0}}$. Considering a conformal metric $g=g_{u}=u^{\frac{4}{n-2}}g_{0}$ there holds

[TABLE]

Note that

[TABLE]

In particular we may define and use $\|u\|^{2}=\|u\|_{L_{g_{0}}}^{2}=\int u\,L_{g_{0}}u\,d\mu_{g_{0}}$ as an equivalent norm on $W^{1,2}$. For

[TABLE]

we want to study the scaling-invariant functionals

[TABLE]

Since the conformal scalar curvature $R=R_{u}$ for $g=g_{u}=u^{\frac{4}{n-2}}g_{0}$ satisfies

[TABLE]

we have

[TABLE]

The first- and second-order derivatives of the functional are given by

[TABLE]

[TABLE]

In particular $J_{\tau}$ is of class $C^{2,\alpha}_{loc}(\mathcal{A})$ and uniformly Hölder continuous on each set of the form

[TABLE]

Indeed $u\in U_{\epsilon}$ implies

[TABLE]

Thus uniform Hölder continuity on $U_{\epsilon}$ follows from the standard pointwise estimates

[TABLE]

We consider next some approximate solutions to (1.1), highly concentrated at arbitrary points of $M$. As we will see, for suitable values of $\lambda$ these are also approximate solutions of (1.4). Let us recall the construction of conformal normal coordinates from [27]. Given $a\in M$, one chooses a special conformal metric

[TABLE]

whose volume element in $g_{a}$-geodesic normal coordinates coincides with the Euclidean one, see also [24]. In particular

[TABLE]

for the exponential maps centered at $a$, which e.g. implies

[TABLE]

and in case $\nabla K(a)=0$ also

[TABLE]

Moreover by smoothness of the exponential map $\exp_{g_{a}}=\exp^{g_{a}}_{a}$ with respect to $a$ there holds

[TABLE]

in a $g_{a}$-normal chart, as seen from the corresponding geodesic equation. We then denote by $r_{a}$ the geodesic distance from $a$ with respect to the metric $g_{a}$ just introduced. With this choice the expression of the Green’s function $G_{g_{a}}$ with pole at $a\in M$, denoted by $G_{a}=G_{g_{a}}(a,\cdot)$, for the conformal Laplacian $L_{g_{a}}$ simplifies considerably. From Section 6 in [27] one may expand

[TABLE]

where $\omega_{n}=|S^{n-1}|$. Here $H_{r,a}\in C^{2,\alpha}_{loc}$, while the singular error term satisfies

[TABLE]

Precisely the leading term in $H_{s,a}$ for $n=6$ is $-\frac{|\mathbb{W}(a)|^{2}}{288c_{n}}\ln r$, where $\mathbb{W}$ denotes the Weyl tensor. Let

[TABLE]

We notice that the constant $\gamma_{n}$ is chosen so that

[TABLE]

Evaluating the conformal Laplacian on such functions shows that they are approximate solutions.

Lemma 2.1**.**

There holds $L_{g_{0}}\varphi_{a,\lambda}=O(\varphi_{a,\lambda}^{\frac{n+2}{n-2}}).$ More precisely on a geodesic ball $B_{\alpha}(a)$ for $\alpha>0$ small

[TABLE]

where $r_{a}=d_{g_{a}}(a,\cdot)$. Since $R_{g_{a}}=O(r_{a}^{2})$ in conformal normal coordinates, cf. [27], we obtain

1. (i)

$L_{g_{0}}\varphi_{a,\lambda}=4n(n-1)[1-\frac{c_{n}}{2}r_{a}^{n-2}(H_{a}(a)+n\nabla H_{a}(a)x)]\varphi_{a,\lambda}^{\frac{n+2}{n-2}}+O\begin{pmatrix}\lambda^{-\frac{3}{2}}\varphi_{a,\lambda}^{\frac{n-1}{n-2}}&\text{ for }\,n=3\\ \frac{\ln r}{\lambda^{2}}\varphi_{a,\lambda}^{\frac{n-1}{n-2}}&\text{ for }\,n=4\\ \lambda^{-2}\varphi_{a,\lambda}&\text{ for }\,n=5\end{pmatrix};$** 2. (ii)

$L_{g_{0}}\varphi_{a,\lambda}=4n(n-1)\varphi_{a,\lambda}^{\frac{n+2}{n-2}}=4n(n-1)[1+\frac{c_{n}}{2}W(a)\ln r]\varphi_{a,\lambda}^{\frac{n+2}{n-2}}+O(\lambda^{-2}\varphi_{a,\lambda})\quad\hbox{ for }n=6$; 3. (iii)

$L_{g_{0}}\varphi_{a,\lambda}=4n(n-1)\varphi_{a,\lambda}^{\frac{n+2}{n-2}}+O(\lambda^{-2}\varphi_{a,\lambda})\quad\hbox{ for }n\geq 7.$**

The expansions stated above persist upon taking $\lambda\partial\lambda$ and $\frac{\nabla_{a}}{\lambda}$ derivatives.

Proof.

A straightforward calculation shows that

[TABLE]

which is due to $|\nabla G_{a}|^{2}_{g_{a}}G_{a}^{2\frac{n-1}{2-n}}=(n-2)^{2}|\nabla G_{a}^{\frac{1}{2-n}}|^{2}_{g_{a}}\;\text{ and }\;c_{n}\Delta_{g_{a}}G_{a}=-\delta_{a}+R_{g_{a}}G_{a}$ with $\delta_{a}$ denoting the Dirac measure at $a$. This is equivalent to

[TABLE]

Since $L_{g_{a}}=-c_{n}\Delta_{g_{a}}+R_{g_{a}}$ with $c_{n}=4\frac{n-1}{n-2}$, we obtain

[TABLE]

By conformal covariance we also get

[TABLE]

in particular $L_{g_{0}}\varphi_{a,\lambda}=O(\varphi_{a,\lambda}^{\frac{n+2}{n-2}})$. Expanding $G_{a}$ as $G_{a}=\frac{1}{4n(n-1)\omega_{n}}(r_{a}^{2-n}+H_{a}),\,r_{a}=d_{g_{a}}(a,\cdot)$ we find

[TABLE]

and conclude that

[TABLE]

Clearly these calculations transcend to the $\lambda$ and $a$ derivatives. Then the claim follows from the above expansion of the Green’s function. ∎

After introducing some notation we state a useful lemma, which will be proved in the first appendix.

Notation. Given an exponent $p\geq 1$ we will denote by $L^{p}_{g_{0}}$ the set of functions of class $L^{p}$ with respect to the measure $d\mu_{g_{0}}$. Recall also that for $u\in W^{1,2}(M,g_{0})$ we set $r_{u}=\int uL_{g_{0}}ud\mu_{g_{0}}$, while for a point $a\in M$ we denote by $r_{a}$ the geodesic distance from $a$ with respect to the metric $g_{a}$ introduced above. For a set of points $\{a_{i}\}_{i}$ of $M$ we will denote by $K_{i},\nabla K_{i}$ and $\Delta K_{i}$ for instance

[TABLE]

For $k,l=1,2,3$ and $\lambda_{i}>0,\,a_{i}\in M,\,i=1,\ldots,q$ let

1. (i)

$\varphi_{i}=\varphi_{a_{i},\lambda_{i}}$ and $(d_{1,i},d_{2,i},d_{3,i})=(1,-\lambda_{i}\partial_{\lambda_{i}},\frac{1}{\lambda_{i}}\nabla_{a_{i}});$ 2. (ii)

$\phi_{1,i}=\varphi_{i},\;\phi_{2,i}=-\lambda_{i}\partial_{\lambda_{i}}\varphi_{i},\;\phi_{3,i}=\frac{1}{\lambda_{i}}\nabla_{a_{i}}\varphi_{i}$, so $\phi_{k,i}=d_{k,i}\varphi_{i}.$

Note that with the above definitions the $\phi_{k,i}$’s are uniformly bounded in $W^{1,2}(M,g_{0})$.

Lemma 2.2**.**

Let $\theta=\frac{n-2}{2}\tau$ and $k,l=1,2,3$ and $i,j=1,\ldots,q$. Then for*

[TABLE]

there holds uniformly as $0\leq\tau\longrightarrow 0$

1. (i)

$|\phi_{k,i}|,|\lambda_{i}\partial_{\lambda_{i}}\phi_{k,i}|,|\frac{1}{\lambda_{i}}\nabla_{a_{i}}\phi_{k,i}|\leq C\varphi_{i};$** 2. (ii)

$\lambda_{i}^{\theta}\int\varphi_{i}^{\frac{4}{n-2}-\tau}\phi_{k,i}\phi_{k,i}d\mu_{g_{0}}=c_{k}\cdot id+O(\tau+\frac{1}{\lambda_{i}^{n-2+\theta}}+\frac{1}{\lambda_{i}^{2+\theta}}),\;c_{k}>0;$** 3. (iii)

for $i\neq j$ up to some error of order $O(\tau^{2}+\sum_{i\neq j}(\frac{1}{\lambda_{i}^{4}}+\frac{1}{\lambda_{i}^{2(n-2)}}+\varepsilon_{i,j}^{\frac{n+2}{n}}))$

[TABLE] 4. (iv)

$\lambda_{i}^{\theta}\int\varphi_{i}^{\frac{4}{n-2}-\tau}\phi_{k,i}\phi_{l,i}d\mu_{g_{0}}=O(\frac{1}{\lambda_{i}^{n-2}}+\frac{1}{\lambda_{i}^{2}})$* for $k\neq l$ and for $k=2,3$*

[TABLE] 5. (v)

$\lambda_{i}^{\theta}\int\varphi_{i}^{\alpha-\tau}\varphi_{j}^{\beta}d\mu_{g_{0}}=O(\varepsilon_{i,j}^{\beta})$* for $i\neq j,\;\alpha+\beta=\frac{2n}{n-2},\;\alpha-\tau>\frac{n}{n-2}>\beta\geq 1;$* 6. (vi)

$\int\varphi_{i}^{\frac{n}{n-2}}\varphi_{j}^{\frac{n}{n-2}}d\mu_{g_{0}}=O(\varepsilon^{\frac{n}{n-2}}_{i,j}\ln\varepsilon_{i,j}),\,i\neq j;$** 7. (vii)

$(1,\lambda_{i}\partial_{\lambda_{i}},\frac{1}{\lambda_{i}}\nabla_{a_{i}})\varepsilon_{i,j}=O(\varepsilon_{i,j}),\,i\neq j$.

with constants $b_{k}=\underset{\mathbb{R}^{n}}{\int}\frac{dx}{(1+r^{2})^{\frac{n+2}{2}}}$ for $k=1,2,3$ and

[TABLE]

3 Blow-up analysis

In this section we prove a result related to a well-known one in [46]. We obtain indeed similar conclusions, but allowing the exponent in the equation to vary along a sequence of approximate solutions.

Proposition 3.1**.**

Let $(u_{m})_{m}\subset W^{1,2}(M,g_{0})$ be a sequence with $u_{m}\geq 0$ and $k_{\tau_{m}}=1$ satisfying*

[TABLE]

Then up to a subsequence there exist $u_{\infty}:M\longrightarrow[0,\infty)$ smooth, $q\in\mathbb{N}_{0}$ and for $i=1,\ldots,q$ sequences

[TABLE]

such that $u_{m}=u_{\infty}+\sum_{i=1}^{q}\alpha_{i}\varphi_{a_{i,m},\lambda_{i,m}}+v_{m}$ with

[TABLE]

*and $(\varepsilon_{i,j})_{m}\longrightarrow 0$ as $m\longrightarrow\infty$ for each pair $1\leq i<j\leq q$ . *

Proof.

Setting $J=J_{\tau_{m}}$, by our assumptions we have

[TABLE]

In particular $(u_{m})\subset W^{1,2}(M,g_{0})$ is bounded, hence $u_{m}\rightharpoonup u_{\infty}$ weakly in $W^{1,2}(M,g_{0})$ and strongly in $L^{q}(M,g_{0})$, $q<\frac{2n}{n-2}$. Notice that $u_{\infty}\geq 0$ is a critical point of $J_{0}$ and therefore it is a smooth function. We may then write $u_{m}=u_{\infty}+u_{1,m}$ with $u_{1,m}\rightharpoonup 0\text{ weakly, and strongly in }L^{q}(M,g_{0}).$ Thus

[TABLE]

whence $\int u_{1,m}L_{g_{0}}u_{1,m}d\mu_{g_{0}}\longrightarrow r_{1,\infty}\geq 0$ and secondly, due to (2.4), that

[TABLE]

We may assume $r_{1,\infty}>0$, since otherwise we are done. We now claim the concentration behavior

[TABLE]

Indeed we have for a fixed cut-off function

[TABLE]

Using Hölder’s inequality and Sobolev’s embedding we obtain

[TABLE]

Thus, if $u_{1,m}$ does not concentrate in $L^{p_{m}+1}(M,g_{0})$ similarly to (3.2), then by a covering argument

[TABLE]

contradicting $r_{1,\infty}>0$. By (3.1) concentration in $L^{p_{m}+1}(M,g_{0})$ is equivalent to concentration in $L^{2}$-norm for the gradient, which had to be shown. Fixing $\varepsilon>0$ small, we measure the rate of concentration via

[TABLE]

and choose for any $\lambda_{1,m}\nearrow\infty$ with $1\leq\lim_{m\to\infty}\frac{\Lambda_{1,m}}{\lambda_{1,m}}=\delta<\infty$ up to a subsequence

[TABLE]

for some positive $c=c(\varepsilon,\delta)$ to be specified later. On a suitably small ball $B_{\rho}(a_{1,m})$ we then rescale

[TABLE]

The function $w_{1,m}$ is well defined on $B_{\rho\lambda_{1,m}}(0)$ and satisfies, with $\theta_{m}=\frac{n-2}{2}\tau_{m}$,

[TABLE]

Since $\int|\nabla u_{1,m}|^{2}d\mu_{g_{0}}$ is bounded, so it is $\int_{B_{\rho\lambda_{1,m}}(0)}|\nabla w_{1,m}|^{2}dx$ for any $\rho>0$. Hence

[TABLE]

where

[TABLE]

Given a compactly supported cut-off $\eta$, we calculate

[TABLE]

The main step here is the inequality in the above formula. Passing from $\frac{n+2}{n-2}$ to $p_{m}=\frac{n+2}{n-2}-\tau_{m}$ in the exponent is easy, as $w_{1,\infty}$ is fixed. Since $w_{1,m}\rightarrow w_{1,\infty}$ in $L^{p}(supp(\eta)),\;p<\frac{2n}{n-2}$, we have

[TABLE]

Therefore the main inequality follows from observing that

[TABLE]

Hence (3.3) is justified and we obtain as before

[TABLE]

Thus $w_{1,m}\longrightarrow w_{1,\infty}$ locally strongly, unless $w_{1,m}$ concentrates in $L^{p_{m}+1}$, but by our choice of $\Lambda_{1,m}$

[TABLE]

and $1\geq\frac{\lambda_{1,m}}{\Lambda_{1,m}}\not\longrightarrow 0$, so the $L^{2}$-gradient norm does not concentrate beyond $\varepsilon$ and, since

[TABLE]

neither the $L^{p_{m}+1}$-norm does. Thus $w_{1,m}\longrightarrow w_{1,\infty}$ locally strongly. In particular

[TABLE]

But $\sigma_{1}=0$ implies $w_{1,\infty}=0$ by harmonicity, so $\sigma_{1}\in(0,1]$, cf. (3.3), and we easily show $w_{1,\infty}>0$ and

[TABLE]

Note that $-\Delta_{\mathbb{R}^{n}}w_{1,\infty}=\sigma_{1}r_{\infty}\kappa_{1}w_{1,\infty}^{\frac{n+2}{n-2}}$ implies $\sigma_{1}r_{\infty}\kappa_{1}\alpha_{1}^{\frac{4}{n-2}}=4n(n-1).$ Moreover by construction

[TABLE]

which transfers to $w_{1,\infty}$ by locally strong convergence. This implies $a=0$ and

[TABLE]

By $\underset{m\to\infty}{\lim}\lambda_{1,m}^{-\theta_{m}}=\sigma_{1}\in(0,1]$ and $0<\varepsilon\ll 1$ we get $\tilde{\lambda}_{1}\sim\underset{m\to\infty}{\lim}\lambda_{1,m}^{\frac{2-n}{2n}\theta_{m}}$. Dilating back we may then write

[TABLE]

Moreover we know that $u_{2,m}\rightharpoonup 0$ weakly in $W^{1,2}(M,g_{0})$ and

[TABLE]

Since the initial sequence $(u_{m})$ was non-negative, it follows that $u_{\infty}\geq 0$ and the negative part of $u_{2,m}$ tends to zero as $m\longrightarrow\infty$ in $W^{1,2}$-norm. Using a dilation argument, the latter property and the above formula, it is easy to show that, if $\alpha,\beta\geq 1$ with $\alpha+\beta=\frac{2n}{n-2}$, then

[TABLE]

and that also $\int u_{2,m}L_{g_{0}}\varphi_{1,m}d\mu_{g_{0}}=o(1)$. Thence as before for $u_{1,m}$

[TABLE]

and therefore $\int u_{2,m}L_{g_{0}}u_{2,m}d\mu_{g_{0}}\longrightarrow r_{2,\infty}\geq 0$. Likewise

[TABLE]

since by expansion of the non-linear term of $\partial J_{\tau_{m}}(u_{m})$ we find

[TABLE]

The second equality follows from applying the latter formulas to any test function in $W^{1,2}(M,g_{0})$ and then applying Sobolev’s and Hölder’s inequalities together with (3.4). We may therefore iterate the afore going and find for a finite sum $u_{m}=\sum_{i}\alpha_{i}\varphi_{i,m}+v_{m},$ with energy

[TABLE]

But all $\alpha_{i}$ are uniformly lower bounded due to

[TABLE]

thence the iteration has to stop after finitely-many steps. In particular $v_{m}$ does not concentrate locally and consequently vanishes strongly as $m\longrightarrow\infty$. Now take any fixed index $j$ and recall that

[TABLE]

and that by construction $\frac{\bar{\lambda}_{k,m}}{\bar{\lambda}_{l,m}}\not\longrightarrow 0$ for $k<l$. We had seen

[TABLE]

On the other hand

[TABLE]

up to some error of order $o(1)$ locally in $W^{1,2}$, and the latter sum has to vanish, which is equivalent to

[TABLE]

Recalling (2.9), this shows that $(\varepsilon_{i,j})_{m}\longrightarrow 0$ for all $i\neq j$. We are left with proving $\bar{\lambda}_{i,m}^{\tau_{m}}\longrightarrow 1$. Ordering

[TABLE]

up to a subsequence, let

[TABLE]

Then $\frac{\bar{\lambda}_{k,m}}{\bar{\lambda}_{l,m}}\longrightarrow\infty$ for $k\leq\bar{q}<l$ and $c\leq\lim_{m\to\infty}\frac{\bar{\lambda}_{k,m}}{\bar{\lambda}_{l,m}}\leq C$ for $k,l\leq\bar{q}$. Select a half-ball $B^{+}_{\delta}(a_{k,m})$ with

[TABLE]

up to a subsequence, where for some affine function $\nu_{k,m}$ with unit gradient we have set

[TABLE]

in a local coordinate system. Then rescaling $u_{m}$ on $B_{\delta}^{a_{k,m}}\cap\{\nu_{k,m}>\frac{1}{\bar{\lambda}_{k,m}}\}$ we find

[TABLE]

On the other hand side, $w_{k,m}$ solves

[TABLE]

Recalling that $p_{m}=\frac{n+2}{n-2}-\tau_{m}$ and $\theta_{m}=\frac{n-2}{2}\tau_{m}$, this implies, that up to rotating coordinates

[TABLE]

Thus $\bar{\lambda}_{k,m}^{\theta_{m}}\longrightarrow 1$. The claim follows, since $\lim_{m\to\infty}\frac{\bar{\lambda}_{k,m}}{\bar{\lambda}_{l,m}}\geq c$ for all $l=1,\ldots,q.$ ∎

4 Reduction and v-part estimates

In this section we will consider a sequence $u_{m}$ as in Proposition 3.1, with zero weak limit. We will recall some well-known facts about finite-dimensional reductions and derive preliminary error estimates and on suitable components of the gradient of $J_{\tau}$.

For $\varepsilon>0,\;q\in\mathbb{N},\;u\in W^{1,2}(M,g_{0})$ and $(\alpha^{i},\lambda_{i},a_{i})\in(\mathbb{R}^{q}_{+},\mathbb{R}^{q}_{+},M^{q})$ we define

1. (i)

A_{u}(q,\varepsilon)=\{(\alpha^{i},\lambda_{i},a_{i})\mid\;\underset{i\neq j}{\forall\;}\;\lambda_{i}^{-1},\lambda_{j}^{-1},\varepsilon_{i,j},\bigg{|}1-\frac{r\alpha_{i}^{\frac{4}{n-2}}K(a_{i})}{4n(n-1)k_{\tau}}\bigg{|},\|u-\alpha^{i}\varphi_{a_{i},\lambda_{i}}\|<\varepsilon,\,\lambda_{i}^{\tau}<1+\varepsilon\}; 2. (ii)

$V(q,\varepsilon)=\{u\in W^{1,2}(M,g_{0})\mid A_{u}(q,\varepsilon)\neq\emptyset\},$

cf. (2.2), (2.3) and (2.8). For both conditions $\lambda_{i}>\varepsilon^{-1},\lambda_{i}^{\tau}<1+\varepsilon$ to hold, we will always assume that $\tau\ll\varepsilon$ and this is consistent with the statement of Proposition 3.1. Under the above conditions on the parameters $\alpha_{i},a_{i}$ and $\lambda_{i}$ the functions $\sum_{i=1}^{q}\alpha^{i}\varphi_{a_{i},\lambda_{i}}$ form a smooth manifold in $W^{1,2}(M,g_{0})$, which implies the following well known result, cf. [4].

Proposition 4.1**.**

For every $\varepsilon_{0}>0$ there exists $\varepsilon_{1}>0$ such that for $u\in V(q,\varepsilon)$ with $\varepsilon<\varepsilon_{1}$

[TABLE]

admits an unique minimizer $(\alpha_{i},a_{i},\lambda_{i})\in A_{u}(q,\varepsilon_{0})$ depending smoothly on $u$ and we set

[TABLE]

The term $v=u-\alpha^{i}\varphi_{i}$ is orthogonal to all $\varphi_{i},-\lambda_{i}\partial_{\lambda_{i}}\varphi_{i},\frac{1}{\lambda_{i}}\nabla_{a_{i}}\varphi_{i},$ with respect to the product

[TABLE]

For $u\in V(q,\varepsilon)$ let

[TABLE]

We next have an estimate on the projection of the gradient of $J_{\tau}$ onto $H_{u}$.

Lemma 4.1**.**

For $u\in V(q,\varepsilon)$ with $k_{\tau}=1$, cf. (2.3),and $\nu\in H_{u}(q,\varepsilon)$ there holds

[TABLE]

Proof.

Due to the fact that $k_{\tau}=1$ and $\nu\in H_{u}(q,\varepsilon)$ we have

[TABLE]

and therefore

[TABLE]

Decomposing iteratively $M$ as \big{\{}\alpha_{j}\varphi_{j}>\sum_{i>j}\alpha_{i}\varphi_{i}\big{\}}\cup\big{\{}\alpha_{j}\varphi_{j}\leq\sum_{i>j}\alpha_{i}\varphi_{i}\big{\}}, we find

[TABLE]

Using Hölder’s inequality with exponents $1=\frac{1}{p}+\frac{1}{q}=\frac{n+2}{2n}+\frac{n-2}{2n}$ and Lemma 2.2 $(v)$ applied to the latter error term, where the inequality $\varphi_{s}\lesssim\varphi_{r}$ can be used to apply it with $\beta\geq 1$, we get

[TABLE]

and by a simple expansion we also obtain

[TABLE]

Note that

[TABLE]

whence

[TABLE]

Thus up to some $O([\sum_{r}\frac{\tau}{\lambda_{r}^{\theta}}+\sum_{r}\frac{|\nabla K_{r}|}{\lambda_{r}^{1+\theta}}+\sum_{r}\frac{1}{\lambda_{r}^{2+\theta}}+\sum_{r\neq s}\frac{\varepsilon_{r,s}^{\frac{n+2}{2n}}}{\lambda_{r}^{\theta}}]\|\nu\|)$ we arrive at

[TABLE]

Finally from Lemma 2.1 and the fact that $\nu\in H_{u}(q,\varepsilon)$ (hence $\int\nu L_{g_{0}}\varphi_{i}d\mu_{g_{0}}=0$) we obtain

[TABLE]

so the claim follows. ∎

Lemma 4.2**.**

For $u\in V(q,\varepsilon)$ with $k_{\tau}=1$ and $v$ is as in (4.1) there holds

[TABLE]

Proof.

Since the Hessian of $J_{\tau}$ is uniformly Hölder continuous on bounded sets of $W^{1,2}$, we have

[TABLE]

[TABLE]

Since $v\in H_{u}(q,\varepsilon)$, by similar expansions we then find (also replacing $p$ with $\frac{n+2}{n-2}$ with an error $o(1)$)

[TABLE]

up to some $o(\|v\|^{2})$. Furthermore by definition of $V(q,\varepsilon)$ there holds $\lambda_{i}^{\theta}=1+o(1)$ and

[TABLE]

Thus

[TABLE]

This quadratic form is positive definite for $\varepsilon$ sufficiently small on the subspace $v$ belongs to, cf. [4], so

[TABLE]

Therefore the claim follows from Lemma 4.1. ∎

We now establish cancellations testing the gradient of $J_{\tau}$ orthogonally to $H_{u}(q,\varepsilon)$.

Lemma 4.3**.**

For $u\in V(q,\varepsilon)$ with $k_{\tau}=1$ the quantity $\partial J_{\tau}(u)\phi_{k,i}$ expands as

[TABLE]

Proof.

By the mean value theorem and (4.6) we have, with some $\sigma\in[0,1]$

[TABLE]

Therefore, since $v\in H_{u}(q,\varepsilon)$, up to some $O(\|v\|^{2})$ we also get

[TABLE]

Decomposing now $M$ as $\{\alpha^{j}\varphi_{j}\leq 2\|v\|\}\cup\{\alpha^{j}\varphi_{j}\geq 2\|v\|\},$ and using $|\phi_{k,i}|\leq C\alpha_{i}\varphi_{i}\leq C\alpha^{j}\varphi_{j}$, we find

[TABLE]

Now, arguing as for (4.3) and using Lemma 2.2 $(iv)$, we have

[TABLE]

[TABLE]

whence

[TABLE]

up to some O\bigg{(}\sum_{r}\frac{|\nabla K_{r}|^{2}}{\lambda_{r}^{2+2\theta}}+\sum_{r}\frac{1}{\lambda_{r}^{4+2\theta}}+\sum_{r\neq s}\frac{\varepsilon_{r,s}^{\frac{n+2}{n}}}{\lambda_{r}^{2\theta}}+\|v\|^{2}\bigg{)}. Using (4.4) and (4.5) we arrive at

[TABLE]

Yet also the first summand on the right hand side is of the same order as the second one, arguing as for (4.4) and (4.5). Combining this with Lemma 4.2, we obtain the conclusion. ∎

5 The functional and its derivatives

For $u\in V(q,\varepsilon)$ and $\varepsilon>0$ sufficiently small let

[TABLE]

Recalling the notation from the previous section we may expand the Euler-Lagrange energy as follows.

Proposition 5.1**.**

For $u=\alpha^{i}\varphi_{i}+v\in V(q,\varepsilon)$ and $\varepsilon>0$, both $J_{\tau}(u)$ and $J_{\tau}(\alpha^{i}\varphi_{i})$ can be written as

[TABLE]

with positive constants $\hat{c}_{0},\hat{c}_{1},\hat{c}_{2},\hat{b}_{1},\hat{d}_{1}$ up to errors of the form

[TABLE]

Proof.

The above expansion for $J_{\tau}(\alpha^{i}\varphi_{i})$ implies the one for $J_{\tau}(u)$ via Lemmata 4.1 and 4.2 expanding

[TABLE]

We next start analyzing $J_{\tau}(\alpha^{i}\varphi_{i})$ from the denominator. Decomposing iteratively $M$ as

[TABLE]

we may expand

[TABLE]

Recalling $\lambda_{r}^{\theta}\sim 1$ and the boundedness of $\alpha_{r}$ by the definition of $V(q,\varepsilon)$, using Lemma 2.2 and reasoning as for the proof of Lemma 4.1, the latter term is of order $O(\sum_{r\neq s}\varepsilon_{i,j}^{\frac{n+2}{n}}),$ and also

[TABLE]

Indeed we for example have

[TABLE]

with the latter norm that can be controlled by

[TABLE]

Thus Lemma 2.2, where $b_{1}$ is defined, yields

[TABLE]

and we arrive at

[TABLE]

up to an error $O(\tau^{2}+\sum_{r\neq s}\frac{|\nabla K_{r}|^{2}}{\lambda_{r}^{2}}+\frac{1}{\lambda_{r}^{4}}+\frac{1}{\lambda_{r}^{2(n-2)}}+\varepsilon_{r,s}^{\frac{n+2}{n}}).$ Finally, recalling our notation in Section 2 and denoting by $x^{i}$ a generic polynomial of degree $i$ in the $x$-variables, we expand

[TABLE]

with an extra error of order $O\left(\frac{\ln\lambda}{\lambda^{4}}\right)$ if $n=4$. For the first term on the right-hand side up to some $O(\tau^{2}+\frac{1}{\lambda_{i}^{4}})$ we may pass integrating with respect to conformal normal coordinates. Indeed

[TABLE]

and the latter term is of order $O(\frac{\tau}{\lambda_{i}^{2+\theta}})$. From (2.8) we find

[TABLE]

up to some $O(\tau^{2}+\frac{1}{\lambda_{i}^{4}}+\frac{1}{\lambda_{i}^{n}})$. Clearly

[TABLE]

letting

[TABLE]

Moreover

[TABLE]

up to some $O(\frac{1}{\lambda_{i}^{2(n-2)}}+\frac{1}{\lambda_{i}^{4}})$ and with an extra error of order $O\left(\frac{\ln\lambda}{\lambda^{4}}\right)$ if $n=4$, and

[TABLE]

whence up to some $O(\tau^{2}+\frac{1}{\lambda_{i}^{4}}+\frac{1}{\lambda_{i}^{2(n-2)}})$

[TABLE]

Likewise by radial symmetry and, since we may assume $d\mu_{g_{a_{i}}}\equiv 1$, cf. [24], we find

1. (1)

\int_{B_{c}(a_{i})}x^{3}\varphi_{i}^{p+1}d\mu_{g_{a_{i}}}=O\bigg{(}\frac{1}{\lambda_{i}^{4}}+\frac{1}{\lambda_{i}^{2(n-2)}}\bigg{)}; 2. (2)

\frac{\nabla^{2}}{2}K_{i}\int_{B_{c}(a_{i})}x^{2}\varphi_{i}^{p+1}d\mu_{g_{0}}=\frac{\Delta K_{i}}{2n\lambda_{i}^{2+\theta}}\underset{\mathbb{R}^{n}}{\int}\frac{r^{2}dx}{(1+r^{2})^{n}}+O\bigg{(}\tau^{2}+\frac{1}{\lambda_{i}^{4}}+\frac{1}{\lambda_{i}^{2(n-2)}}\bigg{)}; 3. (3)

\int_{B_{c}(a_{i})}x\varphi_{i}^{p+1}d\mu_{g_{0}}=O\bigg{(}\frac{1}{\lambda_{i}^{4}}+\frac{1}{\lambda_{i}^{2(n-2)}}\bigg{)}

with an extra error of order $O\left(\frac{\ln\lambda}{\lambda^{4}}\right)$ if $n=4$. Collecting all terms we arrive at

[TABLE]

up to an error $O(\tau^{2}+\frac{1}{\lambda_{i}^{4}}+\frac{1}{\lambda_{i}^{2(n-2)}})$, and thus obtain

[TABLE]

up to some $O(\tau^{2}+\sum_{r\neq s}\frac{|\nabla K_{r}|^{2}}{\lambda_{r}^{2}}+\frac{1}{\lambda_{r}^{4}}+\frac{1}{\lambda_{r}^{2(n-2)}}+\varepsilon_{r,s}^{\frac{n+2}{n}}).$ Consequently up to the same error

[TABLE]

Next for $i\neq j$ using Lemma 2.1 we get

[TABLE]

For example to check the error term, we may estimate

[TABLE]

which is of order $O(\frac{\varepsilon_{i,j}^{\frac{n+2}{2n}}}{\lambda_{i}^{n-2}})$ thanks to Lemma 2.2, and likewise for e.g. $n\geq 7$

[TABLE]

whence $\lambda_{i}^{-2}\int\varphi_{i}\varphi_{j}d\mu_{g_{0}}=o(\frac{\varepsilon_{i,j}^{\frac{n+2}{2n}}}{\lambda_{i}^{2}})$. Thus Lemma 2.2 shows that

[TABLE]

Finally from (2.8) and Lemma 2.1 we find

[TABLE]

up to some error terms of order $O(\lambda_{i}^{-3},\lambda_{i}^{-4}\ln\lambda_{i},\lambda_{i}^{-4},\lambda_{i}^{-4},\lambda_{i}^{-4}),$ whence

[TABLE]

up to $O(\frac{1}{\lambda_{i}^{4}}+\frac{1}{\lambda_{i}^{2(n-2)}}).$ Recalling (5.7), we obtain

[TABLE]

up to some $O(\tau^{2}+\frac{1}{\lambda_{i}^{4}}+\frac{1}{\lambda_{i}^{2(n-2)}})$. As $\bar{d}_{1}=\tilde{d}_{1}$, cf. (5.6), we simply get

[TABLE]

up to an error of order O\bigg{(}\tau^{2}+\sum_{r}\frac{1}{\lambda_{r}^{4}}+\frac{1}{\lambda_{r}^{2(n-2)}}+\sum_{r\neq s}\varepsilon_{r,s}^{\frac{n+2}{n}}\bigg{)}. Plugging this into (5.9), we obtain

[TABLE]

up to some $O(\tau^{2}+\sum_{r}\frac{|\nabla K_{r}|^{2}}{\lambda_{r}^{2}}+\frac{1}{\lambda_{r}^{4}}+\frac{1}{\lambda_{r}^{2(n-2)}}+\sum_{r\neq s}\varepsilon_{r,s}^{\frac{n+2}{n}})$. Recalling

[TABLE]

and setting

[TABLE]

we may rewrite this as

[TABLE]

Then the claim follows from Lemma 5.1. ∎

We next state three lemmas with some expansions for the derivatives of the functionals with respect to the parameters involved (recall our notation from Section 2). The proofs are given in appendix B.

Lemma 5.1**.**

For $u\in V(q,\varepsilon)$ and $\varepsilon>0$ sufficiently small the three quantities $\partial J_{\tau}(u)\phi_{1,j}$, $\partial J_{\tau}(\alpha^{i}\varphi_{i})\phi_{1,j}$, $\partial_{\alpha_{j}}J_{\tau}(\alpha^{i}\varphi_{i})$ can be written as

[TABLE]

with positive constants $\grave{c}_{0},\grave{c}_{2},\grave{b}_{1},\grave{d}_{1}$ up to an error of order

[TABLE]

In particular for all $j$

[TABLE]

Lemma 5.2**.**

For $u\in V(q,\varepsilon)$ and $\varepsilon>0$ sufficiently small the three quantities $\partial J_{\tau}(u)\phi_{2,j}$, $\partial J_{\tau}(\alpha^{i}\varphi_{i})\phi_{2,j}$ and $\frac{\lambda_{j}}{\alpha_{j}}\partial_{\lambda_{j}}J_{\tau}(\alpha^{i}\varphi_{i})$ can be written as

[TABLE]

with positive constants $\tilde{c}_{1},\tilde{c}_{2},\tilde{d}_{1},\tilde{b}_{2}$ up to some error of the form

[TABLE]

Lemma 5.3**.**

For $u\in V(q,\varepsilon)$ and $\varepsilon>0$ sufficiently small the three quantities $\partial J_{\tau}(u)\phi_{3,j}$, $\partial J_{\tau}(\alpha^{i}\varphi_{i})\phi_{3,j}$ and $\frac{\nabla_{a_{j}}}{\alpha_{j}\lambda_{j}}J_{\tau}(\alpha^{i}\varphi_{i})$ can be written as

[TABLE]

with positive constants $\check{c}_{3},\check{c}_{4},\check{b}_{3}$ up to some error of the form

[TABLE]

6 Gradient bounds

Theorem 2 will give suitable lower norm-bounds on the gradient of $J_{\tau}$, yielding Theorem 1 as a corollary. We recall that on $S^{3}$ and $S^{4}$ the result was proved in [12], [28], [29], [45] in more generality.

Definition 6.1**.**

Let $H$ be as in (2.7). We call a positive Morse function $K$ on $M$ non-degenerate

1. (i)

of degree* $q\in\mathbb{N}$ in case $n=4$, if $\{\nabla K=0\}\cap\{\tilde{c}_{2}\frac{\Delta K}{K}+\tilde{c}_{3}H=0\}=\emptyset$ and if for every $1\leq k\leq q$ and every subset $\{x_{1},\ldots,x_{k}\}\subseteq\{\nabla K=0\}\cap\{\tilde{c}_{2}\frac{\Delta K}{K}+\tilde{c}_{3}H<0\}$ the matrices *

[TABLE]

have non-vanishing least eigenvalues, where $\tilde{c}_{2}=\sqrt{3\omega_{4}},\;\tilde{c}_{3}=24\sqrt{3\omega_{4}}=\tilde{c}_{4}$. We say that $K$ is non-degenerate, if it is non-degenerate of all degrees. 2. (ii)

in case $n\geq 5$, if $\{\nabla K=0\}\cap\{\Delta K=0\}=\emptyset$, i.e. (1.2) holds.

Remark 6.1**.**

Non-degeneracy in case $n=4$ implies the existence of a least eigenvalue

[TABLE]

and such that $\lambda_{x_{1,}\ldots,x_{k}}$ is simple and has a positive eigenvector, i.e.

[TABLE]

Theorem 2**.**

Let $\mathcal{M}_{x_{1},\ldots,x_{k}}$ be as in Definition 6.1, and suppose that

[TABLE]

Then for $\varepsilon>0$ sufficiently small there exists $c>0$ such that for any $u\in V(q,\varepsilon)$ with $\;k_{\tau}=1$ there holds

[TABLE]

cf. (5.1), unless there is a violation of at least one of the four conditions

1. (i)

$\tau>0$; 2. (ii)

there exists $x_{i}\neq x_{j}\in\left\{\begin{matrix}[l]\{\nabla K=0\}\cap\{\tilde{c}_{2}\frac{\Delta K}{K}+\tilde{c}_{3}H<0\}&\text{for }\;n=4,\\ \{\nabla K=0\}\cap\{\Delta K<0\}&\text{for }\;n\geq 5\end{matrix}\right\}$ and $d(a_{i},x_{i})=O(\frac{1}{\lambda_{i}})$; 3. (iii)

\left\{\begin{matrix}[l]\alpha_{j}=\Theta\,\cdot\left(\frac{\lambda_{j}^{\theta}}{K_{j}}\biggr{(}1+\frac{1}{8}\big{(}\frac{\Delta K_{j}}{K_{j}\lambda_{j}^{2}}-60\frac{H_{j}}{\lambda_{j}^{2}}-\frac{\sum_{k}(\frac{\Delta K_{k}}{K_{k}^{2}\lambda_{k}^{2}}-60\frac{H_{k}}{K_{k}\lambda_{k}^{2}})}{\sum_{k}\frac{1}{K_{k}}}\big{)}\biggr{)}\right)^{\frac{1}{p-1}}+o(\frac{1}{\lambda_{j}^{2}})&\text{for }\;n=4,\\ \alpha_{j}=\Theta\,\cdot(\frac{\lambda_{j}^{\theta}}{K_{j}})^{\frac{1}{p-1}}+o(\frac{1}{\lambda_{j}^{2}})&\text{for }\;n\geq 5\end{matrix}\right\}; 4. (iv)

$\left\{\begin{matrix}[l]\mathcal{M}_{x_{1},\ldots,x_{q}}>0\;\text{ and }\;\lambda_{j}=\frac{\sigma_{j}+o(1)}{\sqrt{\tau}}&\text{for }\;n=4,\\ \tilde{c}_{1}\tau=-\tilde{c}_{2}\frac{\Delta K_{k}}{K_{k}\lambda_{k}^{2}}+o(\frac{1}{\lambda_{k}^{2}})&\text{for }\;n\geq 5\end{matrix}\right\}$**

for all $j\neq i,j=1,\ldots,q$, where $\sigma=(\sigma_{1},\ldots,\sigma_{q})$ in case $n=4$ is the unique solution of

[TABLE]

while $\Theta$ is given in Remark 6.2. In the latter case there holds  $\lambda_{1}\simeq\ldots\lambda_{q}\simeq\lambda=\frac{1}{\sqrt{\tau}}$  and setting

[TABLE]

we still have up to an error $o(\frac{1}{\lambda^{3}})$ the lower bound

[TABLE]

in case $n=4$ and

[TABLE]

in case $n=5$ and

[TABLE]

in case $n\geq 6$. The constants appearing above are defined by $\bar{c}_{0}=\int_{\mathbb{R}^{n}}\frac{dx}{(1+r^{2})^{n}}$,

[TABLE]

and

[TABLE]

The differences in the above expressions for $n=5$ and $n\geq 6$ is caused by a different decay of bubble functions causing stronger mutual interactions in lower dimension.

Remark 6.2**.**

Under non-degeneracy conditions, Theorem 2 has the following immediate implications.

In case $\tau=0$ there are no solutions of $\partial J(u)=\partial J_{0}(u)=0$ in $V(q,\varepsilon)$, cf. Theorem 1.4 in **[14]**. 2. 2.

In case $\tau>0$ every solution $\partial J_{\tau}(u)=0$ in $V(q,\varepsilon)$ satisfies

[TABLE]

and has isolated simple blow-ups occurring close to

[TABLE] 3. 3.

The $\alpha_{j},\lambda_{j}$ and $a_{j}$’s are determined to a precision $o(\tau^{\frac{3}{2}})+O(|\partial J_{\tau}(u)|)$. Indeed, for e.g. $n=6$

[TABLE]

determines $\lambda_{j}$ up to the latter error from $\tau$ and $x_{j}$, whence $a_{j}$ is determined as well by

[TABLE]

from $\lambda_{j}$ and $x_{j}$, and finally up to the multiplicative constant $\Theta$ also $\alpha_{j}$ is determined by

[TABLE]

from $\lambda_{j},a_{j}$ and $\tau$, recalling $\theta=\frac{n-2}{2}\tau$ and $p=\frac{n+2}{n-2}-\tau$. As for the multiplicative constant we have

[TABLE]

up to some $o(\tau^{\frac{3}{2}})$, cf. (4.5), Lemma 4.2, Lemma 2.2 and (5.8), whence

[TABLE]

up to the same error and so the multiplicative constant $\Theta$ is determined as well.

Proof of Theorem 2.

First we note that $k_{\tau}=1$ implies, that all the $\alpha_{i}$ do not tend to infinity and least one of them does not approach zero. Hence by definition of $V(q,\varepsilon)$ all the $\alpha_{i}$ are uniformly bounded away from zero and infinity. Secondly, if for some index $j=1,\ldots,q$ we have

[TABLE]

then the claim follows from Lemma 5.1, whence we may henceforth assume that for all $j=1,\dots,q$

[TABLE]

Thus we have to show

[TABLE]

and arguing by contradiction we may assume that

[TABLE]

Then by Lemmata 5.2 and 5.3 we have

[TABLE]

[TABLE]

up to some errors of the form $O(\frac{1}{\lambda_{j}^{3}})+O(\tau^{2}+\sum_{r\neq s}\frac{|\nabla K_{r}|^{2}}{\lambda_{r}^{2}}+\frac{1}{\lambda_{r}^{4}}+\varepsilon_{r,s}^{\frac{n+2}{n}}),$ where we have to add for ($\lambda$) $\tilde{d}_{1}\frac{H_{i}}{\lambda_{i}^{2}}$ to $\tilde{c}_{2}\frac{\Delta K_{i}}{K_{i}\lambda_{i^{2}}}$ in case $n=4$. Ordering indices so that $\lambda_{1}\geq\ldots\geq\lambda_{q}\Longleftrightarrow\frac{1}{\lambda_{1}}\leq\ldots\leq\frac{1}{\lambda_{q}}$ and recalling (2.9), we have

[TABLE]

and therefore

[TABLE]

From ($a$) and ($\lambda$) above we find uniformly bounded vector fields $A_{1},\Lambda_{1}$ on $V(q,\varepsilon)$ such that

[TABLE]

[TABLE]

with $\tilde{c}_{4}=\frac{n-2}{2}\tilde{b}_{2}$, and combining $X_{1}=\Lambda_{1}+\epsilon A_{1}$ with some $\epsilon>0$ small and fixed such that we keep a positive coefficient in front of $\varepsilon_{1,i}$, we get

[TABLE]

Likewise from ($a$) and ($\lambda$) we find uniformly bounded vector fields $A_{2},\Lambda_{2}$ defined on $V(q,\varepsilon)$ such that

[TABLE]

[TABLE]

and combining them as $X_{2}=\Lambda_{2}+\epsilon A_{2}$ with $\epsilon>0$ small we obtain

[TABLE]

Therefore combining $B_{1}$ and $B_{2}$ so that the coefficient of $\varepsilon_{i,j}$ is positive

[TABLE]

Iteratively, for all $k=1,\ldots,q$ we can find uniformly bounded vector fields $A_{k},\Lambda_{k}$ such that

[TABLE]

[TABLE]

[TABLE]

where we have to add $\tilde{c}_{3}\frac{H_{j}}{\lambda_{j}^{2}}$ to $\tilde{c}_{2}\frac{\Delta K_{j}}{K_{j}\lambda_{j}^{2}}$ in case $n=4$, where

[TABLE]

In particular

[TABLE]

Then, if either

[TABLE]

we obviously have (6.2) from $(\mathbf{C_{q}})$. Thus we may assume

[TABLE]

whence we may simplify the above formulas to

[TABLE]

[TABLE]

[TABLE]

adding $\tilde{c}_{3}\frac{H_{j}}{\lambda_{j}^{2}}$ to $\tilde{c}_{2}\frac{\Delta K_{j}}{K_{j}\lambda_{j}^{2}}$ for $n=4$. We first consider the pair $(q-1,q)$. Suppose

[TABLE]

To prove (6.2) we then may assume from $(\mathbf{C_{q-1}})$ and (6.5) that also $\tau+\sum_{r\neq s}\varepsilon_{r,s}=o(\frac{1}{\lambda_{q}^{2}}),$ since

[TABLE]

As the coefficient of $\lambda_{q}^{-2}$ in $(\mathbf{\Lambda_{q}})$ is non zero by non-degeneracy, (6.2) follows. So we may assume

[TABLE]

and therefore, still by (6.5),

[TABLE]

So, if $a_{q-1}$ is close to $a_{q}$, these points are close to the same critical point of $K$, which, as $K$ is Morse, implies $d(a_{q-1},a_{q}))\lesssim\frac{1}{\lambda_{q}}\simeq\frac{1}{\lambda_{q-1}}$. This however contradicts the fact that by Proposition 3.1

[TABLE]

Therefore for the pair $(q-1,q)$ we may assume

[TABLE]

In particular in case $n\geq 5$ we have $\varepsilon_{q-1,q}\simeq\frac{1}{\lambda_{q}^{n-2}}=o(\frac{1}{\lambda_{q}^{2}}),$ whereas in case $n=4$

[TABLE]

We turn to consider the triple $(q-2,q-1,q)$. Suppose that $\frac{1}{\lambda_{q-2}^{2}}=o(\frac{1}{\lambda_{q-1}^{2}}).$ To get (6.2) we then may assume from $(\mathbf{C_{q-2}})$ and (6.5) that

[TABLE]

as well. But then clearly in case $n\geq 5$ we obtain (6.2) from $(\mathbf{\Lambda_{q-1}})$ or $(\mathbf{\Lambda_{q}})$, since $\varepsilon_{q-1,q}=o(\lambda_{q}^{-2})$ is already known. In case $n=4$ we have to argue more subtly. From ($\lambda$) we find

[TABLE]

and

[TABLE]

up to an error of order $o(\frac{1}{\lambda_{q}^{2}})$, cf. (6.3). Obviously (6.2) then follows if either

[TABLE]

We may thus assume both summands to be negative. Recalling (6.1), we then obtain

[TABLE]

up to an error $o(\frac{1}{\lambda_{q}^{2}})$ letting

[TABLE]

and thus $|\partial J_{\tau}(u)|\gtrsim\lambda_{q}^{-2},$ since otherwise $a_{q-1},a_{q}$ close to $x_{q-1},x_{q}\in\{\nabla K=0\}\cap\{\tilde{c}_{2}\frac{\Delta K}{K}+\tilde{c}_{3}H<0\}$ and

[TABLE]

would have after a blow-up for $\tau\longrightarrow 0$ a vanishing eigenvalue with strictly positive eigenvector, which by Remark 6.1 is impossible. So (6.2) again follows. We may thus assume

[TABLE]

and therefore by (6.5)

[TABLE]

So, if $a_{q-2}$ is close to either $a_{q-1}$ or $a_{q}$, these points are close to the same critical point of $K$, whence

[TABLE]

as before, contradicting Proposition 3.1. Thus for $(q-2,q-1,q)$ we may assume

[TABLE]

and

[TABLE]

analogously to the previous case of the pair $(q-1,1)$. In particular in case $n\geq 5$

[TABLE]

whereas in case $n=4$ up to an error $O(\frac{1}{\lambda_{q}^{4}})$

[TABLE]

Iteratively, we then may assume for all $k\neq l=1,\ldots,q$

[TABLE]

In particular $\varepsilon_{k,l}=o(\frac{1}{\lambda_{q}^{2}})$ for $n\geq 5$ and $\varepsilon_{k,l}=\frac{G_{g_{0}}(a_{k},a_{l})}{\lambda_{k}\lambda_{l}}$ for $n=4$. But then

[TABLE]

in case $n\geq 5$ and thus

[TABLE]

up to some $o(\frac{1}{\lambda_{q}^{2}})$. Therefore (6.2) holds unless $\tilde{c}_{1}\tau+\tilde{c}_{2}\frac{\Delta K_{k}}{K_{k}\lambda_{k}^{2}}=o(\frac{1}{\lambda_{q}^{2}})$, while now for $n=4$

[TABLE]

up to some $o(\frac{1}{\lambda_{q}^{2}})$, cf. (6.3), for all $j=1,\ldots,q$. Obviously (6.2) then follows, if for some $j=1,\ldots,q$

[TABLE]

whence we may assume all these summands to be negative, proving (ii). From ($\lambda$) and (6.1) we then have

[TABLE]

up to some $o(\frac{1}{\lambda_{q}^{2}})$ letting as before $\beta_{j}=\frac{(\alpha_{K,\tau}^{\frac{2n}{n-2}})^{\frac{n-2}{n}}}{K_{j}\alpha_{j}}$. Therefore

[TABLE]

up to the same error. This implies that (6.2) holds true, unless we can solve

[TABLE]

and we may already assume, by (ii), that $a_{j}$ is close to

[TABLE]

In particular (6.2) follows in case $\tau=0$ by the non-degeneracy condition on $K$, proving (i). In case $\tau>0$, writing $\sigma_{j}=\sqrt{\tau}\lambda_{j}$, we find passing to the limit $\tau\longrightarrow 0$, that there has to exist a solution to

[TABLE]

In particular, testing the above relation with ${\bf{x}}={\bf{x}}_{x_{1},\ldots,x_{q}}$, cf. Remark 6.1, we find

[TABLE]

where $\lambda=\lambda_{x_{1},\ldots,x_{q}}$ is the least eigenvalue of $\mathcal{M}_{x_{1},\ldots,x_{q}}$. Thus necessarily $\mathcal{M}_{x_{1},\ldots,x_{q}}>0$. Since

[TABLE]

is a sum of convex functions, there exists a unique critical point of $F$ satisfying (6.7). Hence we have comparability $\lambda_{1}\simeq\ldots\simeq\lambda_{1}\simeq\nicefrac{{1}}{{\sqrt{\tau}}}\simeq\lambda$ like in case $n\geq 5$. Thus (iv) follows upon checking constants for $n=4$, i.e. $\bar{c}_{0}=\underset{\mathbb{R}^{n}}{\int}(\frac{1}{1+r^{2}})^{n}=\frac{\omega_{4}}{12}$ and

[TABLE]

cf. (7.14) from the corresponding Lemma 5.2. We turn next to (iii). In case $n\geq 5$ we may now assume

[TABLE]

which by Lemma 5.1 implies

[TABLE]

Note that $\alpha_{j}^{p-1}=\Theta^{p-1}\cdot\frac{\lambda_{j}^{\theta}}{K_{j}}$ is modulo scaling the unique and non-degenerate maximum of

[TABLE]

Now (6.2) follows, unless $\alpha_{j}^{p-1}=\Theta^{p-1}\cdot\frac{\lambda_{j}^{\theta}}{K_{j}}+o(\frac{1}{\lambda^{2}})$ and there holds

[TABLE]

In case $n=4$ we may rewrite Lemma 5.1 up to some $o(\frac{1}{\lambda^{2}})$ with constant given below as

[TABLE]

using (6.1) and $\lambda_{j}^{\theta}\simeq(\frac{1}{\sqrt{\tau}})^{\frac{n-2}{2}\tau}=1+O(\frac{\ln\lambda}{\lambda^{2}})$. Moreover, up to an error $o(1)$ there holds

[TABLE]

and due to (6.6)

[TABLE]

and

[TABLE]

up to some $o(\frac{1}{\lambda^{2}})$. We may therefore cancel out the interaction terms in (6.9) and obtain

[TABLE]

Checking constants for $n=4$, i.e. with $\bar{c}_{0}=\int_{\mathbb{R}^{n}}\frac{dx}{(1+r^{2})^{n}}=\frac{\omega_{4}}{12}$

[TABLE]

cf. (7.9) from the corresponding Lemma 5.1, we then find

[TABLE]

Note that setting

[TABLE]

there holds $E_{j}=O(\frac{1}{\lambda^{2}})$, $\sum_{j}\frac{E_{j}}{K_{j}}=0$, and $\alpha_{j}^{p-1}=\Theta^{p-1}\frac{\lambda_{j}^{\theta}}{K_{j}}(1+E_{j})$ is modulo scaling the unique and non-degenerate maximum of

[TABLE]

and satisfies

[TABLE]

due to $(\frac{\lambda_{i}^{\theta}}{K_{i}})^{\frac{2}{p-1}}=\frac{1}{K_{i}}+O(\frac{\ln\lambda}{\lambda^{2}})$. Thus (6.2) follows unless, up to some $o(\frac{1}{\lambda^{2}})$,

[TABLE]

We have therefore proved (i)-(iv), which will be used for showing the second statement of the proposition. In this case the error terms in Lemmata 5.1, 5.2 and 5.3 are of type $o(\lambda^{-3})+O(|\partial J_{\tau}(u)|^{2}).$ This follows immediately in case $n\geq 5$, while the terms $\varepsilon_{r,s}^{\frac{n+2}{n}}\simeq\lambda^{-3}$ in case $n=4$, for which however the underlying estimates can be improved to derive a quadratic error in $\varepsilon_{r,s}$, cf. [38]. Let us first treat the lower bounds arising from Lemma 5.3. In case $n\geq 5$ we find from the latter lemma

[TABLE]

up to some $o(\lambda^{-3})$ and therefore, writing $a_{j}=\exp_{g_{x_{j}}}(\bar{a}_{j})$, that

[TABLE]

Similarly in case $n=4$ we find up to some $o(\lambda^{-3})$

[TABLE]

From (iii) we have $\alpha_{i}=\Theta(\frac{\lambda_{i}^{\theta}}{K_{i}})^{\frac{1}{p-1}}+O(\frac{1}{\lambda^{2}})$, which by $\theta=\frac{n-2}{2}\tau$ and $\lambda_{i}\simeq\tau^{-\frac{1}{2}}$ due to (iv) becomes $\alpha_{i}=\frac{\Theta}{\sqrt{K_{i}}}+O(\frac{\ln\lambda}{\lambda^{2}}).$ Thus, still up to some $o(\lambda^{-3})$

[TABLE]

and checking constants from Lemma 5.3, cf. (7.20), we have

[TABLE]

We conclude that, up to some $o(\frac{1}{\lambda^{3}})$

[TABLE]

By this, i.e. $\bar{a}_{j}=O(\frac{1}{\lambda^{2}})$, and $\alpha_{i}=\frac{\Theta}{\sqrt{K_{i}}}+O(\frac{\ln\lambda}{\lambda^{2}})$ we then infer from Lemma 5.2 that up to some $o(\frac{1}{\lambda^{3}})$

[TABLE]

with constants, cf. above, given for $n=4,5$ respectively by

[TABLE]

we conclude

[TABLE]

By similar reasoning, using $\bar{a}_{j}=O(\frac{1}{\lambda^{2}})\;\text{ and }\;\alpha_{i}=\frac{\Theta}{\sqrt{K_{i}}}+O(\frac{\ln\lambda}{\lambda^{2}})$ we finally have, up to some $o(\frac{1}{\lambda^{3}})$

[TABLE]

This follows in case $n\geq 6$ immediately from Lemma 5.1 and for $n=4$ by repeating the arguments leading to (6.9) and (6.10), while the case $n=5$ follows by arguing as in case $n=4$ using (6.13) to cancel out the interaction terms when passing from (6.9) to (6.10). Then arguing as for the passage from (6.10) to (6.11) we finally obtain that up to some $o(\frac{1}{\lambda^{3}})$

[TABLE]

Thus the second statement of the theorem follows from combining (6.12), (6.13) and (6.14). ∎

In [36] the next result will be needed.

Lemma 6.1**.**

For every $u\in V(q,\varepsilon)$ there holds

[TABLE]

Proof.

Recalling (4.2) we can find $|\beta_{k.i}|,|\beta|=O(1)$ and $\nu\in H_{u}(p,\varepsilon),\|\nu\|=1$ such that

[TABLE]

From Lemmata 5.1, 5.2 and 5.3 we then find

[TABLE]

whereas from Lemma 4.1 we have

[TABLE]

From this the claim follows. ∎

7 Appendix

7.1 Interactions

Proof of Lemma 2.2.

1. (i)

follows using straightforwardly the expression of $\phi_{k,i}$. 2. (ii)

1. $(\alpha)$

Case $k=1$. We have $\phi_{k,i}=\varphi_{i}$ for $k=1$, and thus for $c>0$ small

[TABLE]

On $B_{c}(a_{i})$ one has $u_{a_{i}}^{-\tau}=1+O(\tau|x-a_{i}|^{2})$, and by (2.8)

[TABLE]

whence passing to normal coordinates at $a_{i}$

[TABLE]

up to some error $O(\frac{\tau}{\lambda_{i}^{2+\theta}})$, whence the claim follows with $c_{1}=\underset{\mathbb{R}^{n}}{\int}\frac{dx}{(1+r^{2})^{n}}$. 2. $(\beta)$

Case $k=2$. The proof works analogously to the one of case $k=1$ above. 3. $(\gamma)$

Case $k=3$. We have $\phi_{k,i}=\frac{2-n}{2}u_{a_{i}}\frac{\lambda_{i}\gamma_{n}\nabla_{a_{i}}G^{\frac{2}{2-n}}_{a_{i}}}{1+\lambda_{i}^{2}\gamma_{n}G^{\frac{2}{2-n}}_{a_{i}}}\varphi_{i}+\frac{\nabla_{a_{i}}u_{a_{i}}}{\lambda_{{i}}}\varphi_{i},$ whence

[TABLE]

Moreover $u_{a_{i}}=1+O(r_{a_{i}}^{2})$, implies $\nabla_{a_{i}}u_{a_{i}}=O(r_{a_{i}})$. Thus

[TABLE]

From this the claim follows. 3. (iii)

We just prove the case $k=2$ and start showing that

[TABLE]

up to some O\big{(}\tau^{2}+\sum_{i\neq j}\big{(}\frac{1}{\lambda_{i}^{4}}+\frac{1}{\lambda_{i}^{2(n-2)}}+\varepsilon_{i,j}^{\frac{n+2}{n}}\big{)}\big{)}, so we may evaluate either of these integrals. Clearly

[TABLE]

up to an error $O(\frac{1}{\lambda_{i}^{\frac{n+2}{2}}}\frac{1}{\lambda_{j}^{\frac{n-2}{2}}})$, whence using Lemma 2.1 we find

[TABLE]

up to $O(\lambda_{i}^{-2}+\lambda_{i}^{-(n-2)})\varepsilon_{i,j}^{\frac{n+2}{2n}})$. Indeed we clearly have \lambda_{i}^{-\frac{n+2}{2}}\lambda_{j}^{-\frac{n-2}{2}}=O\big{(}\lambda_{i}^{-1}\varepsilon_{i,j}^{\frac{n+2}{2n}}\big{)}, and the difference from $L_{g_{0}}\varphi_{i}$ to $4n(n-1)\varphi_{i}^{\frac{n+2}{n-2}}$ can be estimated by Lemma 2.1 via quantities of the type

[TABLE]

thanks to case (v). Passing back to integrating on the whole manifold $M$ we find , estimating also mixed products of gradients of $\varphi_{i}$ and $\varphi_{j}$,

[TABLE]

By direct calculation $\Delta_{g_{0}}\varphi_{i}^{-\tau}=O(\tau\varphi_{i}^{\frac{4}{n-2}-\tau})$, whence

[TABLE]

Now applying Lemma 2.1 as before, but in differentiated form, (7.1) follows. Let

[TABLE]

denote a quantity such order. We now assume the non-exclusive alternative

[TABLE]

For $c>0$ small and fixed we have by the expression in (2.8)

[TABLE]

whence passing to $g_{a_{i}}$-normal coordinates and recalling (2.8) we find

[TABLE]

up to the error $R_{i,j}$. Indeed for e.g. $n\geq 7$ (2.8) tells us that on $B_{c}(0)$

[TABLE]

in conformal normal coordinates, whence by Hölder’s inequality and Lemma 2.2

[TABLE]

Due to (7.2) we have that either

[TABLE]

and for $\epsilon>0$ sufficiently small may expand on

[TABLE]

the integrand in (7.3) as

[TABLE]

Using radial symmetry we then get, with $\tilde{b}_{2}=\frac{n-2}{2}\underset{\mathbb{R}^{n}}{\int}\frac{dx}{(1+r^{2})^{\frac{n+2}{2}}}=\frac{n-2}{2}b_{1}$,

[TABLE]

up to errors of the form $R_{i,j}$ and $I_{\mathcal{A}^{c}}$, where

[TABLE]

In case $\varepsilon_{i,j}^{\frac{2}{2-n}}\sim\frac{\lambda_{i}}{\lambda_{j}}$, we obviously have

[TABLE]

Otherwise we may assume $\mathcal{A}^{c}\neq\emptyset$, thus $d(a_{i},a_{j})\ll 1$, and write $\mathcal{A}^{c}\subseteq\,\mathcal{B}_{1}\cup\mathcal{B}_{2},$ where

[TABLE]

for a sufficiently large constant $E>0$. We then may estimate

[TABLE]

Changing coordinates via $d_{i,j}=\exp_{g_{a_{i}}}^{-1}\exp_{g_{a_{j}}}$, we get

[TABLE]

and thus $I_{\mathcal{B}_{1}}=O(\varepsilon_{i,j}^{\frac{n}{n-2}-\tau})=o(\varepsilon_{i,j}^{\frac{n+2}{n}})$ using, (7.2). Moreover

[TABLE]

This shows $I_{\mathcal{A}^{c}}\lesssim I_{\mathcal{B}_{1}}+I_{\mathcal{B}_{2}}=o(\varepsilon_{i,j}^{\frac{n+2}{n}})$, and we arrive at

[TABLE]

up to some error of the form $R_{i,j}$. Due to conformal covariance, there holds

[TABLE]

and we therefore conclude

[TABLE]

We turn to the case left by (7.2), i.e.

[TABLE]

and, recalling (7.1), estimate for $c>0$ small

[TABLE]

up to some error $R_{i,j}$, whence up to the same error

[TABLE]

On \mathcal{A}=\left\{\big{|}\frac{x}{\lambda_{j}}\big{|}\leq\varepsilon\sqrt{\gamma_{n}G^{\frac{2}{2-n}}_{a_{i}}(a_{j})}\right\}\cup\left\{\big{|}\frac{x}{\lambda_{j}}\big{|}\leq\epsilon\frac{1}{\lambda_{i}}\right\} we may expand for $\epsilon>0$ sufficiently small

[TABLE]

With analogous estimates as in the previous case we derive

[TABLE]

with

[TABLE]

and indeed $\bar{b}_{2}=\tilde{b}_{2}=\frac{n-2}{2n}\omega_{n}$ whence, using conformal covariance, as before (7.5) implies

[TABLE]

Now the claim follows comparing (7.4) under (7.2) and (7.7) under (7.5). 4. (iv)

The first claim, i.e. that for $k\neq l$

[TABLE]

follows like in case (ii), just with vanishing leading terms. The second one is proved analogously to (ii), cf. case ($\alpha$) in the proof. 5. (v)

The case $\tau=0$ is known, cf. e.g. [38], Lemma 3.4. By Lemma 2.2 we therefore have

[TABLE]

To estimate the integral in the above right-hand side, we write

[TABLE]

From the case $\tau=0$ and $\alpha+\beta=\frac{2n}{n-2}$ we then get

[TABLE]

By direct evaluation the latter norm is of order $\lambda_{i}^{-\theta}$ and the claim follows. 6. (vi)

also follows from the same above reference in [38], while (vii) is a straightforward computation.

∎

7.2 Derivatives

In this appendix we give the remaining proofs from Section 5.

Proof of Lemma 5.1.

First note that the equalities up to the error in (5.14)

[TABLE]

follow from Lemma 4.3 and the chain rule of differentiation. So we evaluate

[TABLE]

and start expanding

[TABLE]

The above error term is of order O\big{(}\sum_{r\neq s}\varepsilon_{r,s}^{\frac{n+2}{n}}\big{)} by Lemma 2.2, whence

[TABLE]

up to an error of order $O(\sum_{r\neq s}\varepsilon_{r,s}^{\frac{n+2}{n}})$. Similarly

[TABLE]

up to an error $O(\sum_{r\neq s}\varepsilon_{r,s}^{\frac{n+2}{n}})$, and thus

[TABLE]

Iteratively we obtain $\underset{\{\alpha_{j}\varphi_{j}\leq\sum_{j\neq i}\alpha_{i}\varphi_{i}\}}{\int}K(\sum_{j\neq i}\alpha_{i}\varphi_{i})^{p}\varphi_{j}d\mu_{g_{0}}=\sum_{j\neq i}\int K\alpha_{i}^{p}\varphi_{i}^{p}\varphi_{j}d\mu_{g_{0}}$ and conclude

[TABLE]

up to an error of order $O(\sum_{r\neq s}\varepsilon_{r,s}^{\frac{n+2}{n}}).$ From this, we obviously have

[TABLE]

up to some $O(\sum_{r\neq s}\varepsilon_{r,s}^{\frac{n+2}{n}}).$ Then (5.8) and (5.11) applied to the second and third summands above show

[TABLE]

up to an $O(\tau^{2}+\sum_{r\neq s}\frac{|\nabla K_{r}|^{2}}{\lambda_{r}^{2}}+\frac{1}{\lambda_{r}^{4}}+\frac{1}{\lambda_{r}^{2(n-2)}}+\varepsilon_{r,s}^{\frac{n+2}{n}}).$ Then applying (5.11) as well as (5.10) and Lemma 2.2 to the first summand above we find

[TABLE]

Using (5.8) for the first term in the right-hand side, we then get

[TABLE]

up to an error of order

[TABLE]

Applying now (5.8) to the first coefficient above we find

[TABLE]

and obviously the second summand is of order of the previous error term. Thus

[TABLE]

up to the same error, and applying finally (5.7) and (5.8) we arrive at

[TABLE]

again up to the same error term. Recalling that $\bar{b}_{1}=\frac{2n}{n-2}b_{1}$, we can rewrite this as

[TABLE]

up to an error of the form

[TABLE]

Note that by (5.1) the coefficient of $\bar{c}_{1}$ in the above term vanishes. This then tells us in a first step, that

[TABLE]

and therefore

[TABLE]

Using this we derive up to an error of the form O\big{(}\tau^{2}+\sum_{r\neq s}\frac{|\nabla K_{r}|^{2}}{\lambda_{r}^{2}}+\frac{1}{\lambda_{r}^{4}}+\frac{1}{\lambda_{r}^{2(n-2)}}+\varepsilon_{r,s}^{\frac{n+2}{n}}+|\partial J_{\tau}(u)|^{2}\big{)}

[TABLE]

Finally note that the last summand can be simplified to

[TABLE]

From this the lemma follows setting

[TABLE]

cf. (5.6), (5.7) and Lemma 2.2. ∎

Proof of Lemma 5.2.

From Lemma 4.3 and the chain rule of differentiation we obtain

[TABLE]

up to the error in (5.15), and evaluate $\partial J_{\tau}(\alpha^{i}\varphi_{i})\phi_{2,j}=\frac{2\Lambda}{(\int K(\alpha^{i}\varphi_{i})^{p+1}d\mu_{g_{0}})^{\frac{2}{p+1}}}$ with

[TABLE]

Arguing as for (7.8), we find

[TABLE]

and arguing as for (5.2) (5.10), (5.11) we see that

[TABLE]

and

[TABLE]

as well as \int\varphi_{j}L_{g_{0}}\lambda_{j}\partial_{\lambda_{j}}\varphi_{j}d\mu_{g_{0}}=O\big{(}\tau^{2}+\frac{1}{\lambda_{j}^{4}}+\frac{1}{\lambda_{j}^{2(n-2)}}\big{)}. Using these, we arrive at

[TABLE]

Moreover, still arguing as for (5.10) and using Lemma 2.2, we have up to the same error as above

[TABLE]

Combining this with (5.8), (5.10) and (5.11) we get with the same precision

[TABLE]

Using Lemma 5.1 we find by cancellation

[TABLE]

up to some O\big{(}\tau^{2}+\sum_{r\neq s}\frac{|\nabla K_{r}|^{2}}{\lambda_{r}^{2}}+\frac{1}{\lambda_{r}^{4}}+\frac{1}{\lambda_{r}^{2(n-2)}}+\varepsilon_{r,s}^{\frac{n+2}{n}}+|\partial J_{\tau}(u)|^{2}\big{)}. Moreover from Lemma 2.2 we have

[TABLE]

whence recalling (5.8) we get

[TABLE]

up to some O\big{(}\tau^{2}+\sum_{r\neq s}\frac{|\nabla K_{r}|^{2}}{\lambda_{r}^{2}}+\frac{1}{\lambda_{r}^{4}}+\frac{1}{\lambda_{r}^{2(n-2)}}+\varepsilon_{r,s}^{\frac{n+2}{n}}+|\partial J_{\tau}(u)|^{2}\big{)}. Therefore

[TABLE]

up to the same error. Thus we are left with analysing

[TABLE]

Expanding the bubble $\varphi_{j}$ and its derivative $\lambda_{j}\partial_{\lambda_{j}}\varphi_{j}$ in conformal normal coordinates, i.e.

[TABLE]

and arguing as for (5.4) we find using radial symmetry

1. (1)

\int_{B_{c}(a_{j})}x\varphi_{j}^{p}\partial_{\lambda_{j}}\varphi_{j}d\mu_{g_{0}}\;,\;\int_{B_{c}(a_{j})}x^{3}\varphi_{j}^{p}\partial_{\lambda_{j}}\varphi_{j}d\mu_{g_{0}}=O\big{(}\tau^{2}+\frac{1}{\lambda_{j}^{4}}+\frac{1}{\lambda_{j}^{2(n-2)}}\big{)}; 2. (2)

$\frac{\nabla^{2}}{2}K_{j}\int_{B_{c}(a_{j})}x^{2}\varphi_{j}^{p}\partial_{\lambda_{j}}\varphi_{j}d\mu_{g_{0}}=\frac{n-2}{4n}\frac{\Delta K_{j}}{\lambda_{j}^{2+\theta}}\underset{\mathbb{R}^{n}}{\int}\frac{r^{2}(1-r^{2})}{(1+r^{2})^{n+1}}dx+O(\tau^{2}+\frac{1}{\lambda_{j}^{4}}+\frac{1}{\lambda_{j}^{2(n-2)}})$,

Finally we have

[TABLE]

up to some $O(\tau^{2}+\frac{1}{\lambda_{j}^{4}}+\frac{1}{\lambda_{j}^{2(n-2)}})$, and see that for the first summand above there holds

[TABLE]

up to the same error. Defining

[TABLE]

it can be shown, that

[TABLE]

so we arrive at

[TABLE]

up to some $O(\tau^{2}+\frac{1}{\lambda_{j}^{4}}+\frac{1}{\lambda_{j}^{2(n-2)}})$ and arguing as for (5.6) we find

[TABLE]

We conclude that

[TABLE]

Plugging this into (7.11), we then have

[TABLE]

Now the claim follows from Lemma 5.1 by replacing the constants as follows

[TABLE]

cf. (7.10), (7.12) and (7.13) as well as Lemma 2.2. ∎

Proof of Lemma 5.3.

From Lemma 4.3 and the chain rule we obtain up to the error in (5.16)

[TABLE]

and write

[TABLE]

with

[TABLE]

Arguing as for (7.8), we find

[TABLE]

and arguing as for (5.2) and (5.10), in particular using Lemma 2.2, we obtain

[TABLE]

up to some

[TABLE]

using Lemma 5.1 for the last step. Consider a cut-off function $\eta$ such that

[TABLE]

with $c>0$ sufficiently small and some $a\in M$ sufficiently close to $a_{j}$. Then

[TABLE]

and passing to conformal normal coordinates around $a_{j}$ we have

[TABLE]

where

[TABLE]

up to some $O(\frac{1}{\lambda_{j}^{n-\theta}})$. From (2.6) anda(2.8) and using radial symmetry we obtain

[TABLE]

up to some O\big{(}\tau^{2}+\frac{|\nabla K_{j}|^{2}}{\lambda_{j}^{2}}+\frac{1}{\lambda_{j}^{4}}+\frac{1}{\lambda_{j}^{2(n-2)}}\big{)} . By (2.8) we have $\nabla H_{a_{j}}H_{a_{j}}=O(1)$ for $n=3,4,5$ and

[TABLE]

whence up to some O\big{(}\frac{1}{\lambda_{j}^{4}}+\frac{1}{\lambda_{j}^{2(n-2)}}\big{)}

[TABLE]

and we obtain

[TABLE]

up to some $O(\frac{1}{\lambda_{j}^{4}}+\frac{1}{\lambda_{j}^{2(n-2)}})$. Collecting terms we arrive at

[TABLE]

up to some O\big{(}\tau^{2}+\frac{|\nabla K_{j}|^{2}}{\lambda_{j}^{2}}+\frac{1}{\lambda_{j}^{4}}+\frac{1}{\lambda_{j}^{2(n-2)}}\big{)} and conclude

[TABLE]

up to some O\big{(}\tau^{2}+\sum_{r\neq s}\frac{|\nabla K_{r}|^{2}}{\lambda_{r}^{2}}+\frac{1}{\lambda_{r}^{4}}+\frac{1}{\lambda_{r}^{2(n-2)}}+\varepsilon_{r,s}^{\frac{n+2}{n}}+|\partial J_{\tau}(u)|^{2}\big{)}. Applying Lemma 5.1 we find

[TABLE]

up to the same error. We are left with estimating

[TABLE]

Then from Lemma 2.1 we see that in case $n=4,5$

[TABLE]

up to some $O(\frac{1}{\lambda_{j}^{4}}+\frac{1}{\lambda_{j}^{2(n-2)}})$, and thus due to (7.17)

[TABLE]

up to some O\big{(}\tau^{2}+\frac{|\nabla K_{j}|^{2}}{\lambda_{j}^{2}}+\frac{1}{\lambda_{j}^{4}}+\frac{1}{\lambda_{j}^{2(n-2)}}\big{)}. Finally we observe that

[TABLE]

and using the smoothness of conformal normal coordinates with respect to $a_{j}$ we find

[TABLE]

This gives

[TABLE]

up to some O\big{(}\tau^{2}+\frac{|\nabla K_{j}|^{2}}{\lambda_{j}^{2}}+\frac{1}{\lambda_{j}^{4}}+\frac{1}{\lambda_{j}^{2(n-2)}}\big{)}. Passing to conformal normal coordinates around $a_{j}$, we find

[TABLE]

up to some $O(\frac{1}{\lambda_{j}^{4}}+\frac{1}{\lambda_{j}^{2(n-2)}}))$. We therefore conclude

[TABLE]

up to some O\big{(}\tau^{2}+\frac{|\nabla K_{j}|^{2}}{\lambda_{j}^{2}}+\frac{1}{\lambda_{j}^{4}}+\frac{1}{\lambda_{j}^{2(n-2)}}\big{)}. Plugging into (7.18) we arrive at

[TABLE]

up to some

[TABLE]

Recalling (7.15) the claim follows by setting or replacing

[TABLE]

cf. 7.16 and Lemma 2.2. ∎

7.3 List of constants

We give here a list of constants, referring to where they can be found.

[TABLE]

For instance, $c_{2}$ is found in Lemma 2.2, $\bar{c}_{2}$ in equation (5.7) and $\hat{d}_{1}$ in equation (5.13). For the empty fields the corresponding combination of accent and symbol is non-existent. As a caveat please note that we have within some proofs redefined constants for the sake for normalization, hence we point to the final definition, from which upwards mentioned constants can be easily recovered. Finally we recall that $c_{n}$ is the normalizing constants in the definition of the conformal laplacian

[TABLE]