Interaction energy between vortices of vector fields on Riemannian surfaces

TL;DR

This paper analyzes the interaction energy of vortices in tangent vector fields on Riemannian surfaces using a variational Ginzburg-Landau model, deriving a precise limit as the core size shrinks.

Contribution

It provides a detailed second-order $\Gamma$-limit characterization of vortex interaction energy on Riemannian surfaces, extending previous models to vector fields with singularities.

Findings

Interaction energy characterized as $\Gamma$-limit

Vortices correspond to singular points with non-zero index

Results applicable to tangent vector fields on 2D surfaces

Abstract

We study a variational Ginzburg-Landau type model depending on a small parameter $\epsilon>0$ for (tangent) vector fields on a $2$-dimensional Riemannian surface. As $\epsilon\to 0$, the vector fields tend to be of unit length and will have singular points of a (non-zero) index, called vortices. Our main result determines the interaction energy between these vortices as a $\Gamma$-limit (at the second order) as $\epsilon\to 0$.

Equations

$$u:S\to TS,\text{ i.e., }u(x)\in T_{x}S\text{ for every }x\in S$$

$$E_{\epsilon }(u)={\int }_S{e}_{\epsilon }(u)\text{mbox}{vol}_g,e_{\epsilon }(u):=\frac{1}{2}|Du{|}_g^2+\frac{1}{4{\epsilon }^2}F(|u{|}_g^2),$$

$$i^{2}w=-w,(iw,v{)}_g=-(w,iv{)}_g=\text{mbox}{vol}_g(w,v),$$

$$A(v):=(D_v{\tau }_2,{\tau }_1{)}_g=-(D_v{\tau }_1,{\tau }_2{)}_g\text{ in }O.$$

$$D_v(\varphi {\tau }_1)=(d\varphi (v)-iA(v)\varphi ){\tau }_1\text{ in }O.$$

$$dA=\kappa \text{mbox}{vol}_g,$$

$${\int }_S\kappa \text{mbox}{vol}_g=2\pi \chi (S),$$

$$j(u)=(Du,iu{)}_g.$$

$$\mathrm{deg}(u;\partial O):=\frac{1}{2\pi }\left({\int }_{\partial O}\frac{j(u)}{|u{|}_g^2}+{\int }_O\kappa \text{mbox}{vol}_g\right)$$

$$\omega (u):=dj(u)+\kappa \text{mbox}{vol}_g.$$

$$\sum _{k=1}^n{d}_k=\chi (S).$$

$$\omega (u^{\ast })=2\pi \sum _{k=1}^n{d}_k{\delta }_{a_k}\text{ in }S,$$

$$E_{\varepsilon}(u_{\varepsilon})=n\pi\log\frac{1}{\varepsilon}+\lim_{r\to 0}\bigg{(}\int_{S\setminus\cup_{k=1}^{n}B_{r}(a_{k})}\frac{1}{2}|Du^{*}|_{g}^{2}\,\,\mbox{vol}_{g}+n\pi\log r\bigg{)}+n\gamma_{F}+o(1),\textrm{ as }\,\varepsilon\to 0,$$

$$d^{\ast }j(u^{\ast })=0\text{ in }S.$$

$$\int_{S}\big{(}d^{*}j(u^{*}),\zeta)_{g}\,\mbox{vol}_{g}=\int_{S}\big{(}j(u^{*}),d\zeta)_{g}\,\mbox{vol}_{g}\quad\textrm{for every smooth $0$-form $\zeta$}.$$

$$-\Delta \psi =-\kappa \text{mbox}{vol}_g+2\pi \sum _{k=1}^n{d}_k{\delta }_{a_k}\text{ in }S,{\int }_S\psi =0,$$

$$Harm^{1}(S)=\{\mbox{integrable $1$-forms $\eta$ on $S$}\ :\ d\eta=d^{*}\eta=0\mbox{ as distributions}\}.$$

$${\int }_S({\eta }_k,{\eta }_l{)}_g\text{mbox}{vol}_g={\delta }_{kl}\text{ for k,l=1,…,2g}.$$

$$j(u^{\ast })=d^{\ast }\psi +\sum _{k=1}^{2\mathfrak{g}}{\Phi }_k{\eta }_k\text{ in }S$$

$${\Phi }_k={\int }_S(j(u^{\ast }),{\eta }_k{)}_g\text{mbox}{vol}_g,\text{ for }k=1,\dots ,2\mathfrak{g}.$$

$$\gamma$$

$${\int }_{\gamma }\zeta -\sum _{\ell =1}^{2\mathfrak{g}}{c}_{\ell }{\int }_{{\gamma }_{\ell }}\zeta ={\int }_{S}fd\zeta \text{ for all smooth 1-forms ζ}.$$

$${\alpha }_{\ell k}:={\int }_{{\gamma }_{\ell }}{\eta }_k,k,\ell =1,\dots ,2\mathfrak{g}.$$

$${\zeta }_{\ell }={\zeta }_{\ell }(a;d)\in \mathbb{R}/2\pi \mathbb{Z},\ell =1,\dots ,2\mathfrak{g}$$

$$\sum _{k=1}^{2\mathfrak{g}}{\alpha }_{\ell k}{\Phi }_k+{\zeta }_{\ell }(a,d)\in 2\pi \mathbb{Z},\ell =1,\dots ,2\mathfrak{g},$$

$$\Vert \mu {\Vert }_{W^{-1,p}}:=\sup \left\{{\int }_{S}f\mu :f\in W^{1,q}(S;\mathbb{R}),\Vert f{\Vert }_{W^{1,q}}:=\Vert f{\Vert }_{L^q}+\Vert df{\Vert }_{L^q}\le 1\right\}.$$

$${\mu }^t:=2\pi \sum _{l=1}^{n_t}{d}_l^t{\delta }_{a_l^t}\to {\mu }^0:=2\pi \sum _{l=1}^{n_0}{d}_l^0{\delta }_{a_l^0}\text{in W−1,1}\text{ as }t\downarrow 0,$$

$${\zeta }_{\ell }(a,d):={\int }_{{\lambda }_{\ell }}(d^{\ast }\psi +A)\mathrm{mod}2\pi ,\ell =1,\dots ,2\mathfrak{g},$$

$$\mathcal{L}(a,d):=\{\Phi =({\Phi }_1,\dots ,{\Phi }_{2\mathfrak{g}})\in {\mathbb{R}}^{2\mathfrak{g}}:\sum _{k=1}^{2\mathfrak{g}}{\alpha }_{\ell k}{\Phi }_k+{\zeta }_{\ell }(a,d)\in 2\pi \mathbb{Z},\ell =1,\dots ,2\mathfrak{g}\}.$$

$$W(a,d,\Phi):=\lim_{r\to 0}\bigg{(}\int_{S\setminus\cup_{k=1}^{n}B_{r}(a_{k})}\frac{1}{2}|Du^{*}|^{2}_{g}\,\,\mbox{vol}_{g}+\pi\log r\sum_{k=1}^{n}d_{k}^{2}\bigg{)},$$

Full text

1 Introduction

Let $(S,g)$ be a closed (i.e., compact, connected without boundary) $2$-dimensional Riemannian manifold of genus $\mathfrak{g}$. We will focus on (tangent) vector fields

[TABLE]

where $TS=\cup_{x\in S}T_{x}S$ is the tangent bundle of $S$. It is well known that there are no smooth vector fields ${\mathcal{X}}(S)$ (or more generally, of Sobolev regularity ${\mathcal{X}}^{1,2}(S)$) of unit length $|u|_{g}=1$ on $S$ (unless $\mathfrak{g}=1$). In fact, vector fields of unit length have in general singular points with a (non-zero) index. Our aim is to determine the interaction energy between these singular points in a variational model of Ginzburg-Landau type depending on a small parameter $\varepsilon>0$ where the penalty $|u|_{g}=1$ in $S$ is relaxed.

Model. For vector fields $u:S\to TS$, we define the energy functional

[TABLE]

where $|Du|_{g}^{2}:=|D_{\tau_{1}}u|_{g}^{2}+|D_{\tau_{2}}u|_{g}^{2}$ in $S$, $\,\mbox{vol}_{g}$ is the volume $2$-form on $(S,g)$ and $D_{v}$ denotes covariant differentiation (with respect to the Levi-Civita connection) of $u$ in direction $v$ and $\{\tau_{1},\tau_{2}\}$ is any local orthonormal basis of $TS$. The potential $F:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ is a continuous function with $F(1)=0$ and there exists some $c>0$ such that $F(s^{2})\geq c(1-s)^{2}$ for every $s\geq 0$; in particular, $1$ is the unique zero of $F$. The parameter $\varepsilon>0$ is small penalizing $|u|_{g}\neq 1$ in $S$; the goal is to analyse the asymptotic behaviour of $E_{\varepsilon}$ in the framework of $\Gamma$-convergence (at first and second order) in the limit $\varepsilon\to 0$. This is a “toy” problem for some physical models arising for thin shells in micromagnetics or nematic liquid crystals (see e.g., [4, 5]).

Connection $1$-form. On an open subset $O\subset S$, a moving frame is a pair of smooth, properly oriented, orthonormal vector fields $\tau_{k}\in{\mathcal{X}}(O)$, $k=1,2$, i.e., $(\tau_{k},\tau_{l})_{g}=\delta_{k\ell}$, $k,l=1,2,$ and $\,\mbox{vol}_{g}(\tau_{1},\tau_{2})=1$ in $O$, where $(\cdot,\cdot)_{g}$ is the scalar product on $TS$. (We will use the same notation $(\cdot,\cdot)_{g}$ for the inner product associated to $k$-forms, $k=0,1,2$.) Defining $i:TS\to TS$ such that $i$ is an isometry of $T_{x}S$ to itself for every $x\in S$ satisfying

[TABLE]

then every smooth vector field $\tau\in{\mathcal{X}}(O)$ of unit length provides a moving frame $\{\tau_{1},\tau_{2}\}:=\{\tau,i\tau\}$ on $O$. Moreover, if $\{\tau_{1},\tau_{2}\}$ is any moving frame in $O$, then $\tau_{2}=i\tau_{1}$. 333In general a moving frame exists only locally on $S$. Given a moving frame $\{\tau_{1},\tau_{2}\}$ on an open subset $O\subset S$, the connection 1-form $A$ associated to $\{\tau_{1},\tau_{2}\}$ is defined for every smooth vector field $v\in{\mathcal{X}}(O)$:

[TABLE]

In particular, $D_{v}\tau_{1}=-A(v)\tau_{2}$ and $D_{v}\tau_{2}=A(v)\tau_{1}$ in $O$. In complex notation, it yields for any smooth complex-valued function $\phi$ on $O$:

[TABLE]

The definition of $A$ depends on the choice of the moving frame. However, the exterior derivative $dA$ of the connection $1$-form is independent of the moving frame, in particular, the following identity holds

[TABLE]

where $\kappa$ is the Gaussian curvature of $S$ (see [7] Proposition 2, Chapter 5.3). We recall the Gauss-Bonnet theorem that states

[TABLE]

where $\chi(S)$ is the Euler characteristic, related to the genus $\mathfrak{g}$ of $S$ by $\chi(S)=2-2\mathfrak{g}.$

Vortices. We will identify vortices of a vector field $u$ with small geodesic balls centered at some points around which $u$ has a (non-zero) index. To be more precise, we introduce the Sobolev space ${\mathcal{X}}^{1,p}(S)$ of vector fields $u:S\to TS$ such that $|u|_{g}$ and $|Du|_{g}$ belong to $L^{p}(S)$ (with respect to the volume $2$-form), $p\geq 1$. Given $u\in{\mathcal{X}}^{1,p}(S)\cap L^{q}(S)$ such that $\frac{1}{p}+\frac{1}{q}=1$, $p,q\in[1,\infty]$, we define the $1$-form $j(u)$ by 444Note that if $\{\tau_{1},\tau_{2}\}$ is a moving frame on an open set $O\subset S$, then the connection $1$-form $A$ associated to the moving frame is given by $A=-j(\tau_{1})$ on $O$. In particular, $dj(u)=-k\,\mbox{vol}_{g}$ in $O$ for every smooth $u\in{\mathcal{X}}(O)$ of unit length.

[TABLE]

In particular, $j(u)$ is a well-defined $1$-form in $L^{1}(S)$ if $u\in{\mathcal{X}}^{1,1}(S)$ with $|u|_{g}=1$ almost everywhere in $S$; the same is true if $u\in{\mathcal{X}}^{1,p}(S)$ for $p\geq\frac{4}{3}$. To introduce the notion of index, we assume that $O$ is a simply connected open subset of $S$ and $u\in{\mathcal{X}}^{1,2}(N)$ is a vector field in a neighborhood $N$ of $\partial O$ such that $|u|_{g}\geq\frac{1}{2}$ a.e. in $N$; then the index (or winding number) of $u$ along $\partial O$ is defined by

[TABLE]

(see [7] Chapter 6.1). In particular, if $u$ is defined in $O$ and has unit length on $\partial O$, then one has $\displaystyle\int_{O}\omega(u)=2\pi\deg(u;\partial O)$ where $\omega(u)$ is the vorticity associated to the vector field $u$:

[TABLE]

Sometimes we can identify the index of $u$ at a point $P\in S$ with the index of $u$ along a curve around $P$. Note that every smooth vector field $u\in{\mathcal{X}}(O)$ (or more generally, $u\in{\mathcal{X}}^{1,2}(O)$) of unit length in $O$ has $\deg(u;\partial O)=0$; moreover, a vortex with non-zero index will carry infinite energy $E_{\varepsilon}$ as $\varepsilon\to 0$.

We will prove a $\Gamma$-convergence result (at the second order) of $E_{\varepsilon}$ as $\varepsilon\to 0$. In particular, at the level of minimizers $u_{\varepsilon}$ of $E_{\varepsilon}$, we show that $u_{\varepsilon}$ converges in ${\mathcal{X}}^{1,1}(S)$ (for a subsequence) to a canonical harmonic vector field $u^{*}$ of unit length that is smooth 555In the case of a surface $(S,g)$ with genus $1$ (i.e., homeomorphic with the flat torus), then $n=0$ and $u^{*}$ is smooth in $S$. away from $n=|\chi(S)|$ distinct singular points $a_{1},\dots,a_{n}$, each singular point $a_{k}$ carrying the same index $d_{k}=\operatorname{sign}\chi(S)$ so that 666In fact, $\deg(u^{*};\gamma)=d_{k}$ for every closed simple curve $\gamma$ around $a_{k}$ and lying near $a_{k}$.

[TABLE]

The vorticity $\omega(u^{*})$ detects the singular points $\{a_{k}\}_{k=1}^{n}$ of $u^{*}$:

[TABLE]

where $\delta_{a_{k}}$ is the Dirac measure (as a $2$-form) at $a_{k}$. The expansion of the minimal energy $E_{\varepsilon}$ at the second order is given by

[TABLE]

where $\gamma_{F}>0$ is a constant depending only on the potential $F$ and $B_{r}(a_{k})$ is the geodesic ball centered at $a_{k}$ of radius $r$. The second term in the above RHS is called the renormalized energy between the vortices $a_{1},\dots,a_{n}$ and governs the optimal location of these singular points as in the Euclidian case (see the seminal book [3]). In particular, if $S$ is the unit sphere in ${\mathbb{R}}^{3}$ endowed with the standard metric $g$, then $n=2$ and $a_{1}$ and $a_{2}$ are two diametrically opposed points on $S$.

Outline of the note. The note is divided as follows. Section 2 is devoted to characterize canonical harmonic vector fields of unit length. In Section 3, we determine the renormalized energy between singular points of canonical harmonic vector fields. The main $\Gamma$-convergence result is stated in the last section. The proofs of these results are part of our forthcoming article [9].

2 Canonical harmonic vector fields of unit length

We will say that a canonical harmonic vector field of unit length having the singular points $a_{1},\ldots,a_{n}\in S$ of index $d_{1},\ldots,d_{n}\in{\mathbb{Z}}$ for some $n\geq 1$, is a vector field $u^{*}\in{\mathcal{X}}^{1,1}(S)$ such that $|u^{*}|_{g}=1$ in $S$, (3) holds and

[TABLE]

Here, $d^{*}$ is the adjoint of the exterior derivative $d$, i.e., $d^{*}j(u^{*})$ is the unique [math]-form on $S$ such that

[TABLE]

If $u^{*}$ satisfies (3), then the Gauss-Bonnet theorem combined with (1) imply that necessarily (2) holds.

We will see that condition (2) is also sufficient. Indeed, if (2) holds, we will construct solutions of (3) and (4), as follows: let $\psi=\psi(a,d)$ be the unique $2$-form on $S$ solving:

[TABLE]

with the sign convention that $-\Delta=dd^{*}+d^{*}d$. The idea is to find $u^{*}$ such that $j(u^{*})-d^{*}\psi$ is an harmonic $1$-form, i.e.,

[TABLE]

The dimension of the space $Harm^{1}(S)$ is twice the genus (i.e., $2\mathfrak{g}$) of $(S,g)$ and we fix an orthonormal basis $\eta_{1},\ldots,\eta_{2\mathfrak{g}}$ of $Harm^{1}(S)$ such that

[TABLE]

Therefore, it is expected that

[TABLE]

for some constant vector $\Phi=(\Phi_{1},\ldots,\Phi_{2\mathfrak{g}})\in{\mathbb{R}}^{2\mathfrak{g}}$. These constants are called flux integrals as they can be recovered by

[TABLE]

Note that (6) combined with (5) automatically yield (3) and (4). One important point is to characterize for which values of $\Phi$ the RHS of (6) arises as $j(u^{*})$ for some vector field $u^{*}$ of unit length in $S$. For that condition, we need to recall the following theorem of Federer-Fleming [8]: there exist $2\mathfrak{g}$ simple closed geodesics $\gamma_{\ell}$ on $S$, $\ell=1,\ldots,2\mathfrak{g}$, such that for any closed Lipschitz curve $\gamma$ on $S$, one can find integers $c_{1}\ldots,c_{2\mathfrak{g}}$ such that

[TABLE]

i.e., there exists an integrable function $f:S\to{\mathbb{Z}}$ such that

[TABLE]

Having chosen the geodesic curves $\{\gamma_{\ell}\}_{\ell=1}^{2\mathfrak{g}}$ and the harmonic $1$-forms $\{\eta_{k}\}_{k=1}^{2\mathfrak{g}}$, we fix the notation

[TABLE]

Theorem 1

Let $n\geq 1$ and $d=(d_{1},\ldots,d_{n})\in{\mathbb{Z}}^{n}$ satisfy (2). Then for every $a=(a_{1},\ldots,a_{n})\in S^{n}$, there exists

[TABLE]

such that if a vector field $u^{*}\in{\mathcal{X}}^{1,1}(S)$ of unit length solves (3) and (4), then $j(u^{*})$ has the form (6) for constants $\Phi_{1},\ldots,\Phi_{2\mathfrak{g}}$ such that

[TABLE]

where $(\alpha_{\ell k})$ were defined in (7). Conversely, given any $\Phi_{1},\ldots,\Phi_{2\mathfrak{g}}$ satisfying (8), there exists a vector field $u^{*}\in{\mathcal{X}}^{1,1}(S)$ of unit length solving (3) and (4) and such that $j(u^{*})$ satisfies (6). In addition, the following hold:

$\zeta_{\ell}(\cdot;d)$* depends continuously on $a\in S^{n}$ for every $\ell=1,\ldots,2\mathfrak{g}$. More generally, if *777If $\mu$ is a $2$-form (possibly measure-valued) then we write for $p,q\in[1,\infty]$ with $\frac{1}{p}+\frac{1}{q}=1$:

$\|\mu\|_{W^{-1,p}}:=\sup\left\{\int_{S}f\mu\ :\ f\in W^{1,q}(S;{\mathbb{R}}),\,\|f\|_{W^{1,q}}:=\|f\|_{L^{q}}+\|df\|_{L^{q}}\leq 1\right\}.$

**

[TABLE]

$\{d^{t}_{l}\}_{l}$* are integers with (2) and $\sum_{l=1}^{n_{t}}|d^{t}_{l}|$ is uniformly bounded in $t$, then $\zeta_{\ell}(a^{t},d^{t})\rightarrow\zeta_{\ell}(a^{0},d^{0})$ as $t\downarrow 0$.*

• 2)

any $u^{*}$ solving (3) and (4) belongs to ${\mathcal{X}}^{1,p}(S)$ for all $1\leq p<2$, and is smooth away from $\{a_{k}\}_{k=1}^{n}$.

• 3)

If $u^{*},\tilde{u}^{*}$ both satisfy (6) for the same $(a,d)$ and the same $\{\Phi_{k}\}_{k=1}^{2\mathfrak{g}}$, then $\tilde{u}^{*}=e^{i\beta}u^{*}$ for some $\beta\in{\mathbb{R}}$.

The constants $\{\zeta_{\ell}(a;d)\}_{\ell=1}^{2\mathfrak{g}}$ are determined as follows. For every $\ell=1,\ldots,2\mathfrak{g}$, we let $\lambda_{\ell}$ be some smooth simple closed curve such that $\lambda_{\ell}$ is homologous to $\gamma_{\ell}$ (the geodesics fixed in (7)) so that $\{a_{k}\}_{k=1}^{n}$ is disjoint from $\lambda_{\ell}$; for example, $\lambda_{\ell}$ is either $\gamma_{\ell}$ or, if $\gamma_{\ell}$ intersects some $a_{k}$, a small perturbation thereof. We now define $\zeta_{\ell}(a,d)$ to be the element of ${\mathbb{R}}/2\pi{\mathbb{Z}}$ such that

[TABLE]

where $\psi=\psi(a,d)$ is the $2$-form given by (5) and $A$ is the connection $1$-form associated to any moving frame defined in a neighborhood of $\lambda_{\ell}$. The integral in (9) is independent, modulo $2\pi{\mathbb{Z}}$, of the choice of moving frame and of the curve $\lambda_{\ell}$ homologous to $\gamma_{\ell}$. In examples in which it can be explicitly computed, in general $\zeta_{\ell}(a,d)\neq 0\mod 2\pi$ for $\ell=1,\ldots,2\mathfrak{g}$.

3 Renormalized energy

For any $n\geq 1$, we consider $n$ distinct points $a=(a_{1},\ldots,a_{n})\in S^{n}$. Let $d=(d_{1},\ldots,d_{n})\in{\mathbb{Z}}^{n}$ satisfying (2), $\{\zeta_{\ell}(a;d)\}_{\ell=1}^{2\mathfrak{g}}$ be given in Theorem 1 and $\Phi\in{\mathbb{R}}^{2\mathfrak{g}}$ be a constant vector inside the set:

[TABLE]

We define the renormalized energy between the vortices $a$ of indices $d$ by

[TABLE]

where $u^{*}=u^{*}(a,d,\Phi)$ is the unique (up to a multiplicative complex number) canonical harmonic vector field given in Theorem 1 and $B_{r}(a_{k})$ is the geodesic ball centered at $a_{k}$ of radius $r$. Our arguments show that the above limit indeed exists. As in the euclidian case (see [3]), we can compute the renormalized energy by using the Green’s function. For that, let $G(x,y)$ be the unique function on $S\times S$ such that

[TABLE]

with $\mbox{Vol}_{g}(S):=\int_{S}\,\mbox{vol}_{g}$. Then $G$ may be represented in the form (see [2] Chapter 4.2):

[TABLE]

where $G_{0}$ is smooth away from the diagonal, with

[TABLE]

The $2$-form $\psi=\psi(a,d)$ defined at (5) can be written as:

[TABLE]

where $\psi_{0}\in C^{\infty}(S)$ has zero average on $S$ and solves

[TABLE]

In other words, the $2$-form $x\mapsto\psi(x)+d_{k}\log\operatorname{dist}(x,a_{k})\,\mbox{vol}_{g}$ is $C^{1}$ in a neighborhood of $a_{k}$ for every $1\leq k\leq n$. We have the following expression of the renormalized energy:

Proposition 2

Given $n\geq 1$ distinct points $a_{1},\ldots,a_{n}\in S$, integers $d_{1},\ldots,d_{n}$ with (2) and $\Phi\in\mathcal{L}(a,d)$, then

[TABLE]

where $\psi_{0}$ is defined in (10).

In the case of the unit sphere $S$ in ${\mathbb{R}}^{3}$ endowed with the standard metric (in particular, $\psi_{0}$ vanishes in $S$), if $n=2$ and $d_{1}=d_{2}=1$, then the second term in the RHS of (11) is independent of $a_{k}$ (as $x\mapsto H(x,x)$ is constant, see [14]); moreover, $\Phi=0$ and so, minimizing $W$ is equivalent by minimizing the Green’s function $G(a_{1},a_{2})$ over the set of pairs $(a_{1},a_{2})$ in $S\times S$, namely, the minimizing pairs are diametrically opposed.

4 $\Gamma$-convergence

Given the potential $F$ in Section 1, we compute the energy $E_{\varepsilon}$ of the radial profile of a vortex of index $1$ inside a geodesic ball of radius $R>0$:

[TABLE]

Then $I_{F}(R,\varepsilon)=I_{F}(\lambda R,\lambda\varepsilon)=I_{F}(1,\frac{\varepsilon}{R})=:I_{F}(\frac{\varepsilon}{R})$ for every $\lambda>0$, and the following limit exists (see [3]):

[TABLE]

We state our main result:

Theorem 3

The following $\Gamma$-convergence result holds.

(Compactness) Let $(u_{\varepsilon})_{\varepsilon\downarrow 0}$ be a family of vector fields on $S$ satisfying $E_{\varepsilon}(u_{\varepsilon})\leq N\pi|\log\varepsilon|+C$ for some integer $N\geq 0$ and a constant $C>0$. Denoting by

[TABLE]

then there exists a sequence $\varepsilon\downarrow 0$ such that

[TABLE]

where $\{a_{k}\}_{k=1}^{n}$ are distinct points in $S$ and $\{d_{k}\}_{k=1}^{n}$ are nonzero integers satisfying (2) and $\sum_{k=1}^{n}|d_{k}|\leq N$ and $\Phi\in{\mathcal{L}}(a,d)$. Moreover, if $\sum_{k=1}^{n}|d_{k}|=N$, then $n=N$ and $|d_{k}|=1$ for every $k=1,\dots,n$ (in particular, $n=\chi(S)$ modulo $2$).

• 2)

($\Gamma$-liminf inequality) Assume that the vector fields $u_{\varepsilon}\in{\mathcal{X}}^{1,2}(S)$ satisfy (12) for $n$ distinct points $\{a_{k}\}_{k=1}^{n}\in S^{n}$ and $|d_{k}|=1$, $k=1,\dots n$ that satisfy (2) and $\Phi\in{\mathcal{L}}(a,d)$. Then

[TABLE]

• 3)

($\Gamma$-limsup inequality) For every $n$ distinct points $a_{1},\ldots,a_{n}\in S$ and $d_{1},\ldots,d_{n}\in\{\pm 1\}$ satisfying (2) and every $\Phi\in{\mathcal{L}}(a,d)$ there exists a sequence of vector fields $u_{\varepsilon}$ on $S$ such that (12) holds and

[TABLE]

This theorem is the generalization of the $\Gamma$-convergence result for $E_{\varepsilon}$ in the euclidian case (see [6, 11, 13, 1]) and it is based on topological methods for energy concentration (vortex ball construction, vorticity estimates etc.) as introduced in [10, 12].

Acknowledgements

R.I. acknowledges partial support by the ANR project ANR-14-CE25-0009-01. The research of R.J. was partially supported by the National Science and Engineering Research Council of Canada under operating grant 261955.