Compactness of Surfaces in $\pmb {\mathbb {R}}^n$ with Small Total Curvature

Abstract

In this paper, we study the compactness and local structure of immersed surfaces in the unit ball of $\mathbb {R}^n$ with uniformly bounded area and small total curvature. A key ingredient is a new quantity we call isothermal radius. By estimating the lower bound of the isothermal radius, we establish a compactness theorem of such surfaces in intrinsic $L^p$-topology and extrinsic $W^{2,2}$-weak topology. As applications, we get bi-Lipschitz parametrization of such surfaces, explain Leon Simon’s decomposition theorem (Simon in Commun Anal Geom 1(2):281–326, 1993) in the viewpoint of convergence and prove a non-collapsing version of Hélein’s convergence theorem (Hélein in Harmonic maps, conservation laws and moving frames, Cambridge University Press, Cambridge, 2002; Kuwert and Li in Commun Anal Geom 20(2):313–340, 2012).

Appendices

Appendix A: The Proof of Intrinsic Compactness

In this appendix, we give the proof of the intrinsic compactness Theorem 4.5.

Proof of Theorem 4.5

Step 1. (Gromov–Hausdorff convergence)

In the first paragraph we omit the footprint k. Let $f_0:D_1(0)\rightarrow U(p)$ be an conformal parametrization such that $U(p)\supset B_{i_0}^{\Sigma }(p)$ and define u by $f_0^*g=e^{2u}g_0$. By Lemma 3.1, we know $\Vert u\Vert _{L^{\infty }(D_r)}\le C(r)=C(n,\varepsilon , i_0, A_0,V,r)$. Thus for any curve $\gamma :[0,1]\rightarrow D_r$ with $\gamma (0)=x,\gamma (1)=y$, we have

$$\begin{aligned} e^{-C(r)}l_{g_0}(\gamma )\le l_g(f_0(\gamma ))\le e^{C(r)}l_{g_0}(\gamma ), \end{aligned}$$

which means

• (a) $d(f_0(x),f_0(y))\le e^{C(r)}|x-y|, \forall x, y \in D_r(0)$,

• (b) $d(f_0(x),f_0(y))\ge e^{-C(r)}\mathop {\mathrm {min}}\{|x-y|,2r-|x|-|y|\}, \forall x,y \in D_r(0).$

The same argument as in [24, Sect. 10.3.4] implies the capacity estimate, i.e., for any $R>0$ and $((\Sigma ,g,p)\rightarrow \mathbb {R}^n)\in \mathcal {E}(n,\varepsilon , i_0,A_0,V)$, there exists $N(\alpha )=N(\alpha ,R,r,C(r))$ and $\delta =\frac{1}{10}e^{-C(\frac{r+1}{2})}r$ such that $\mathop {\mathrm {Cap}_{B^{\Sigma }_R(p)}(\alpha )}\le N(\alpha )$ for any $\alpha \le {\alpha }_0=\delta$, where the capacity $\mathop {\mathrm {Cap}_{B^{\Sigma }_R(p)}(\alpha )}$ is defined by

$$\begin{aligned} \mathop {\mathrm {Cap}_X(\alpha )}=\mathrm { maximum\ number\ of\ disjoint}\ \frac{\alpha }{2}-\mathrm {balls\ in }\ X \end{aligned}$$

for compact metric space X. For a sequence $\{(\Sigma _k,g_k,p_k)\rightarrow \mathbb {R}^n\}_{k=1}^{\infty }\subset \mathcal {E}(n,\varepsilon , i_0, A_0, V)$, choose conformal coordinates covering $\{f_{ks}:D_1(0)\rightarrow U_{ks}(p^k_s)\subset \Sigma _k\}_{k,s=1}^{\infty }$ s.t. $f_{ks}(0)=p_s^k$, $p^k_1=p_k$, $U_{ks}\supset B^{\Sigma _k}_{i_0}(p^k_s)$, $\Sigma _k=\cup _s f_{ks}(D_r(0))$ and $\bar{B}^{\Sigma _k}_{l\cdot \frac{\delta }{2}}(p_k)\subset \cup _s^{N^l} f_{ks}(D_r(0))$. By Gromov’s compactness theorem [12] (see also [24, Sect. 10.1.4], the sequence $\{(\bar{B}_{l\cdot \frac{\delta }{2}(p_k)},g_k,p_k)\}_{k=1}^{\infty }$ converges(after passing to a subsequence) to a metric space $X_l,d_l,p)$ in pointed Gromov–Hausdorff topology. W.L.O.G., we can assume $(X_l,d_l,p)\subset (X_{l+1},d_{l+1},p)$. After taking direct limit, we have

$$\begin{aligned} (\Sigma _k,g_k,p_k)=\lim _{\longrightarrow }\bar{B}_{l\cdot \frac{\delta }{2}}(p_k)\xrightarrow []{p-GH} \lim _{\longrightarrow }X_l=:(X,d,p). \end{aligned}$$

From now on, we assume all $(\Sigma _k,g_k,p_k)$ and $(\Sigma ,d,p)$ are in a same metric space Y locally since Gromov-Hausdorff convergence is equal to Hausdorff convergence after passing to a subsequence.

Step 2.(Convergence of complex structure)

Assume $f_{ks}:D_1(0)\rightarrow U_{ks}(p^k_s)(\hookrightarrow Y)$ are the conformal parametrizations taken above and $e^{2u_{ks}}g_0=f_{ks}^*g_k=\langle df_{ks},df_{ks}\rangle =({|\frac{\partial f_{ks}}{\partial x}|}^2+{|\frac{\partial f_{ks}}{\partial y}|}^2)g_0$. Then, Lemma 3.1 implies $\Vert \nabla f_{ks}\Vert _{L^{\infty }(D_r)}\le e^{\Vert u_{ks}\Vert _{L^{\infty }(D_r)}}\le e^{C(r)}$, i.e., $\{f_{ks}\}_{k=1}^{\infty }$ are all local Lipschitz with uniform Lipschitz constant. So Arzela-Ascoli’s lemma implies (after passing to a subsequence) $f_{ks}\xrightarrow []{C^{0,\beta }(D_r)}f_s:D_r(0)\rightarrow Y$ for $\beta \in (0,1)$. Furthermore, we have

$$\begin{aligned} d(f_s(x),f_s(y))=\lim _{k\rightarrow \infty }d(f_{ks}(x),f_{ks}(y))\le e^{C(r)}|x-y|, \end{aligned}$$

and

$$\begin{aligned} d(f_s(x),f_s(y))\ge e^{-C(r)}\mathop {\mathrm {min}}\{|x-y|,2r-|x|-|y|\}\ge e^{-C(r)}\frac{r-\sigma }{\sigma }|x-y|, \end{aligned}$$

for $x,y\in D_{\sigma }(0)\subset \subset D_r(0)$, i.e., $f_s$ is local bilipschitz with $\mathrm {Lip}_{D_r}f_s\le e^{C(r)}$ and hence also injective. If we define $p_s=\lim _{k\rightarrow \infty }p_{ks}$, then when noticing that all $X_k$ are length spaces, one know $U_{ks}(p_s^k)$ converges to some $U_s(p_s)\supset B^X_{i_0}(p_s)\subset X$ in Hausdorff topology as subsets in Y. So $f_{ks}\xrightarrow []{C^0}f_s:\bar{D}_r(0)\rightarrow Y$ implies $\mathrm {Im}(f_s)\subset X$, i.e., $f_s:(D_1(0),0)\rightarrow (U_s(p_s))\subset X$ is an injective map from a locally compact space to a Hausdorff space, hence is embedding when restricted to any $\bar{D}_r(0)$.

Moreover, we claim $f_s:D_1(0)\rightarrow U_s(p_s)$ is surjective (hence homeomorphic). In fact, for any $x\in U_s(p_s)=\cup _r\lim _{k\rightarrow \infty }f_{ks}(D_r)$ (Hausdorff convergence as subsets in Y), there exists $r\in (0,1)$ and $x_k=:f_{ks}(a_k)\in f_{ks}(D_r(0))$ s.t. $x_k\rightarrow x$. W.L.O.G., we can assume $a_k\rightarrow a\in \bar{D}_r(0)$. Then

$$\begin{aligned} 0\le d(f_s(a),x)&\le d(f_s(a_k),f_s(a))+d(f_s(a_k),x_k)+d(x_k,x)\\&\le e^{C(r)}|a_k-a|+d(f_s(a_k),f_{ks}(a_k))+d(x_k,x)\rightarrow 0, \end{aligned}$$

since $f_{ks}\rightarrow f_s$ uniformly on compact subsets of $D_1(0)$. So, $x=f_s(a)$ and $f_s$ is surjective. This means $\Sigma :=X$ is a topological manifold.

To construct a complex structure on X, we consider the transport function $f_{ks}^{-1}\circ f_{kt}$ with domain $\mathrm {Dom}(f_{ks}^{-1}\circ f_{kt})\rightarrow \mathrm {Dom}(f_{s}^{-1}\circ f_{t})$ in Hausdorff topology as subsets in $\mathbb {C}$. We have $d(f_{ks}^{-1}\circ f_{kt}(z),f_{s}^{-1}\circ f_{t}(z))\le d(f_{ks}^{-1}\circ f_{kt}(z),f_{ks}^{-1}\circ f_{t}(z))+d(f_{ks}^{-1}\circ f_{t}(z),f_{s}^{-1}\circ f_{t}(z))\rightarrow 0$ uniformly on compact subsets of $\mathrm {Dom}(f_{s}^{-1}\circ f_{t})$ since $f_{kt}\xrightarrow []{C_c^0}f_t$ and $f_{ks}^{-1}$ are locally uniformly bilipschitz. This means

$$\begin{aligned} f_{ks}^{-1}\circ f_{kt}\xrightarrow []{C_c^0} f_s^{-1}\circ f_t\ \mathrm {on} \ \mathrm {Dom}(f_s^{-1}\circ f_t). \end{aligned}$$

But we know $f_{ks}^{-1}\circ f_{kt}$ is analytic since $f_{ks}$ and $f_{kt}$ are both conformal coordinates in their intersection domain, so Montel’s theorem implies $f_s^{-1}\circ f_t$ is also analytic on its domain. This means $\mathcal {O}=\{f_s:D_1(0)\}_{s=1}^{\infty }$ is a complex structure on $\Sigma$ and

$$\begin{aligned} (\Sigma _k,\mathcal {O}_k,p_k)\xrightarrow []{C^{\omega }}(\Sigma ,\mathcal {O},p), \end{aligned}$$

as pointed Riemann surfaces, where $\mathcal {O}_k=\{f_{ks}:D_r(0)\rightarrow U_{ks}(p^k_s)\subset \Sigma _k\}_{s=1}^{\infty }$.

Step 3. (Riemannian metric on $\Sigma$)

For the conformal parametrization $f_{ks}:D_1\rightarrow U_{ks}\subset \Sigma _k$ with pull back metric represented as $f_{ks}^*g_k=e^{2u_{ks}}g_0$. Recall $-\triangle u_{ks}=K_{k}e^{2u_{ks}}=w_{ks}$ in $D_1$ for some $w_{ks}\in \mathcal {H}^1(\mathbb {C})$. As before, we define $v_{ks}=-{\triangle }^{-1}w_{ks}$ and $h_{ks}=u_{ks}-v_{ks}$. Then, $(\Sigma _k\rightarrow \mathbb {R}^n)\in \mathcal {E}(n,\varepsilon , i_0, A_0, V)$ and Lemma 2.3 implies

$$\begin{aligned} \Vert v_{ks}\Vert _{L^{\infty }(\mathbb {C})}+\Vert v_{ks}\Vert _{W^{1,2}(\mathbb {C})}\le C\Vert w_{ks}\Vert _{\mathcal {H}^1(\mathbb {C})}\le C(n,\varepsilon ,A_0,i_0). \end{aligned}$$

Now, Rellich’s lemma and the weak compactness of $\mathcal {H}^1(\mathbb {C})$ (see [28, Chap. 3.5.1]) imply there exist $w_s\in \mathcal {H}^1(\mathbb {C})$ and $v_s\in W^{1,2}(\mathbb {C})$ s.t.

$$\begin{aligned} w_{ks}\rightharpoonup w_s \ \mathrm {as\ distribution}, \ and\ v_{ks}\xrightarrow []{L^q}v_s,\ \forall q\in (1,+\infty ). \end{aligned}$$

Moreover, for any $\varphi \in C_c^{\infty }(\mathbb {C})$, we have

$$\begin{aligned} \int _{\mathbb {C}}\nabla v_s \nabla \varphi =\lim _{k\rightarrow \infty }\int _{\mathbb {C}}\nabla v_{ks} \nabla \varphi =\lim _{k\rightarrow \infty }\int _{\mathbb {C}}w_{ks}\varphi =\int _{\mathbb {C}}w_s\varphi , \end{aligned}$$

i.e., $v_s\in W_0^{1,2}(\mathbb {C})$ satisfies the weak equation $-\triangle v_s=w_s$ in $\mathbb {C}$. And weak lower semi-continuity of the norm of the Banach space $\mathcal {H}^1=(VMO)^*$ imply $\Vert w_s\Vert _{\mathcal {H}^1(\mathbb {C})}\le \liminf _{k\rightarrow \infty }\Vert w_{ks}\Vert _{\mathcal {H}^1(\mathbb {C})}\le C(n,\varepsilon , A_0,i_0)$. So, by Lemma 2.3 again, we get $v_s\in C^0(\mathbb {C})$ and

$$\begin{aligned} \Vert v_s\Vert _{L^{\infty }(\mathbb {C})}+\Vert v_s\Vert _{W^{1,2}(\mathbb {C})}\le C \Vert w_s\Vert _{\mathcal {H}^1(\mathbb {C})}\le C(n,\varepsilon , A_0,i_0). \end{aligned}$$

And by (2), there exists $h_s$ harmonic on $D_1$ such that $h_{ks}\xrightarrow []{C_c^{\infty }(D_1)}h_s$. Denote $u_s:=h_s+v_s$. Then we get $u_{ks}\xrightarrow []{L^q_{loc}(D_1)}u_s$ and $u_s\in C^0(D_1)$ with $\Vert u_s\Vert _{L^{\infty }(D_r)}\le C(n,\varepsilon ,A_0,i_0,V, r), \forall r \in (0,1)$.

With this $u_s$, we can construct a continuous local metric g on $\Sigma$ by defining

$$\begin{aligned} g={(f_s^{-1})}^*(e^{2u_s}g_0), \end{aligned}$$

where $f_s:D_1\rightarrow U_s(p_s)$ is a coordinate in the complex structure $\mathcal {O}$ constructed in Step 2. In fact, Step 2. also implies $\mathcal {O}_k\rightarrow \mathcal {O}$, i.e., $f_{ks}^{-1}\circ f_{kt}\xrightarrow []{C_c^{\infty }}f_s^{-1}\circ f_t$, from which we can get ${(f_s^{-1})}^*(e^{2u_s}g_0)={(f_t^{-1})}^*(e^{2u_t}g_0)$ a.e.(hence everywhere since both are continuous) on their common domain, i.e., the metric g is globally well defined. Moreover, $u_{ks}\rightarrow u_s\ a.e.$ on $D_1$ and $\Vert u_s\Vert _{L^{\infty }(D_r)}\le C(n,\varepsilon ,A_0,i_0,V, r)$ imply $\lim _{k\rightarrow \infty }l_{g_k}(\gamma )=l_g(\gamma ), \forall \gamma \in C^1([0,1],\Sigma )$ by dominate convergence theorem. Thus the induced metric $d_g$ is compatible with the limit metric d. More precisely, we have

$$\begin{aligned} d_g(x,y)=\inf _{\gamma } l_g(\gamma )=\inf _{\gamma }\lim _{k\rightarrow \infty }l_{g_k}(\gamma )\ge \lim d_{g_k}(x,y)=d(x,y), \forall x,y \in \Sigma , \end{aligned}$$

where we use the Gromov–Hausdorff convergence result from Step 1. in the last equation. On the other hand, since $\Vert u_{ks}\Vert _{L^{\infty }(D_r)}+\Vert u_s\Vert _{L^{\infty }(D_r)}\le C(n,\varepsilon ,A_0,i_0,V, r)$, one know all the metrics are uniformly equivalent to the Euclidean metric in every coordinate and

$$\begin{aligned} d_g\le C d \end{aligned}$$

for some $C=C(n, \varepsilon , A_0, i_0,V)$.

Moreover, in the case $\int _{\Sigma _k}|A_k|^2d\mu _{g_k}\rightarrow 0$, by Corollary 2.5, we know $u_{ks}$ converges to $u_s$ in $C^0_{loc}(D_1)$.So for any $\delta >0$, $u(x)-\delta \le u_k(x)\le u(x)+\delta$ for large k independent on $x\in D_{r}(0)$. For any $\gamma :[0,1]\rightarrow D_r(0)$,

$$\begin{aligned} \int _0^1 e^{-\delta }e^{u\circ \gamma }|\gamma '|\le \int _0^1 e^{u_k\circ \gamma }|\gamma '|\le \int _0^1 e^{\delta }e^{u\circ \gamma }|\gamma '|. \end{aligned}$$

Since the last estimate is uniform for all $\gamma$, we can take infimum for $\gamma$ joining x and y and then let $k\rightarrow \infty$ and $\delta \rightarrow 0$ to get

$$\begin{aligned} d_{g}(x,y)=\lim _{k\rightarrow \infty }d_{g_k}(x,y)=d(x,y). \end{aligned}$$

Step 4: ($L^q$-convergence of the metric structure)

By gluing the local diffeomorphisms $\phi _{ks}=f_{ks}\circ f_s^{-1}:U_s\rightarrow U_{ks}$( which converge to identity) together by partition of unity, we get the following global description of pointed convergence of differential structure.

Lemma A.1

If $(\Sigma _k, \mathcal {O}_k,p_k)$ converges to $(\Sigma , \mathcal {O},p)$ as pointed differential surfaces, then for any fixed l, there exist differential maps $\Phi _{kl}:{\Omega }_l=\cup _{s=1}^{l}U_s\rightarrow {\Omega }_{kl}=\cup _{s=1}^l U_{ks}$ for k large enough, such that $\Phi _{kl}$ are embeddings when restricted to compact subsets of $\Omega _l$.

Proof

(It can be found in [24, Sect. 10.3.4], we write it here for the convenience of readers.) Assume $\mathcal {O}_k=\{f_{ks}:D_1\rightarrow U_{ks}\}_{s=1}^{\infty }$ and $\mathcal {O}=\{f_s:D_1\rightarrow U_s\}_{s=1}^{\infty }$. Define $\phi _{ks}:f_{ks}\circ f_s^{-1}:U_s\rightarrow U_{ks}$. Then for $t\ne s$, if $U_s \cap U_t\ne \varnothing$, when putting $\phi _{kt}:U_t\rightarrow U_{kt}$ in local coordinates $f_s:D_1\rightarrow U_s$ and $f_{ks}:D_1\rightarrow U_{ks}$ of $\Sigma$ and $\Sigma _k$, we have

$$\begin{aligned} \tilde{\phi }_{kt}=f_{ks}^{-1}\circ \phi _{kt}\circ f_s=f_{ks}^{-1}\circ f_{kt}\circ f_t^{-1}\circ f_s\xrightarrow []{C_c^{\infty }}f_s^{-1}\circ f_t \circ f_t^{-1} \circ f_s=id. \end{aligned}$$

That is, the local map $\phi _{kt}$ between $\Sigma$ and $\Sigma _k$ converges smoothly to identity w.r.t. the differential structures $\mathcal {O}$ and $\mathcal {O}_k$. That is, if we denote $\hat{\phi }_{ks}=f_{ks}^{-1}\circ \phi _{ks}$, then for any compact subset $K\subset \mathrm {Dom}(\phi _{ks})\cap \mathrm {Dom}(\phi _{kt})=U_s\cap U_t$ and integer m,

$$\begin{aligned} \Vert \hat{\phi }_{kt}-\hat{\phi }_{ks}\Vert _{C^{m}(K)}\le \Vert \hat{\phi }_{kt}-id\Vert _{C^{m}(K)} +\Vert id-\hat{\phi }_{ks}\Vert _{C^{m}(K)} \rightarrow 0, \text { as }k\rightarrow +\infty . \end{aligned}$$

Now, choose a partition of unity $\{\lambda _1, \lambda _2\}$ for $\{U_s, U_t\}$, i.e., smooth functions $\lambda _1,\ \lambda _2$ on $\Sigma$ with $\mathrm {supp}\lambda _1\subset U_s$, $\mathrm {supp}\lambda _2\subset U_t$ and $\lambda _1+\lambda _2=1$ on $U_s\cup U_t$. Then $\lambda _1=1$ on $U_s\backslash U_t$ and $\lambda _2=1$ on $U_t \backslash U_s$. Let $\hat{\Phi }_k=\lambda _1\hat{\phi }_{ks}+\lambda _2\hat{\phi }_{kt}$. Then $\hat{\Phi }_k-\hat{\phi }_{ks}=(\lambda _1-1)\hat{\phi }_{ks}+\lambda _2\hat{\phi }_{kt}=\lambda _2(\hat{\phi }_{kt}-\hat{\phi }_{ks}) \rightarrow 0$ in $C_c^{\infty }(U_s)$. For the same reason, $\hat{\Phi }_k-\hat{\phi }_{kt}\rightarrow 0$ in $C_c^{\infty }(U_t)$. But we know $\hat{\phi }_{ks}\rightarrow id$ in $C_c^{\infty }(U_s)$ and $\hat{\phi }_{kt}\rightarrow id$ in $C_c^{\infty }(U_t)$, so if we define

$$\begin{aligned} \Phi _k= {\left\{ \begin{array}{ll} f_{ks}\circ \hat{\phi }_{ks}\ \mathrm {on}\ U_s,\\ f_{kt}\circ \hat{\Phi }_k\ \mathrm {on}\ U_t\backslash U_s,\\ \end{array}\right. } \end{aligned}$$

then $\Phi _k$ is well defined and for any compact subset $K\subset \subset U_s\cup U_t$, $\Phi _k:K\rightarrow \Phi _k(K)\subset U_{ks}\cup U_{kt}$ is a diffeomorphism for k large enough.

The above argument show that we can glue two sequences of diffeomorphisms together if they are arbitrary close for k large enough. So, by induction, for the finite sequences of local coordinates $\{\phi _{ks}\}_{s=1}^l:U_s\rightarrow U_{ks}$, they are all close to the identity(hence to each other) for k large enough. When denoting $\Omega _l=\cup _{s=1}^l U_s$ and $\Omega _{kl}=\cup _{s=1}^l U_{ks}$, we can glue them together to be a local diffeomorphism $\Phi _{kl}:\Omega _l\rightarrow \Omega _{kl}$ such that $\Phi _{kl}^{-1}\circ \phi _{ks}$ converges smoothly to the identity on its domain. $\square$

By this lemma, we have ${(\Phi _{kl})}^*g_k\xrightarrow []{L^q_{loc}}g$ since $(\phi _{ks})^*g_k\xrightarrow []{L^q}g$ by the construction of g and $\Phi _{kl}$ is arbitrary close to $\phi _{ks}$ in $U_s$ for k large enough. That is,

$$\begin{aligned} (\Sigma _k,g_k,p_k)\rightarrow (\Sigma ,g,p)\sim (X,d,p)\ \ \mathrm {in\ pointed\ } L^q\ \mathrm {topology}. \end{aligned}$$

In the case $\int _{\Sigma _k}|A_k|^2d\mu _{g_k}\rightarrow 0$, by gluing the local uniform convergence of $u_{ks}\rightarrow u_{s}$, we know $(\Sigma _k,g_k,p_k)$ converges to $(\Sigma ,g,p)$ in $C^0_{loc}$ topology. $\square$

Appendix B: Weak Lower Semi-continuity

In this appendix, we write a proof of the weak lower semi-continuity for reader’s convenience.

Proof of Corollary 4.7

The key observation is, as a function of $(\xi ,\zeta ,\eta )=(F,DF,D^2F)$, $J(\xi ,\zeta ,\eta ):=|A_g|_g^2d\mu _g$ depends on $\eta =D^2F$ convexly for each fixed $\xi$ and $\zeta$, since it is an nonnegative quadratic form of $D^2F$ in any fixed coordinate. For any compact domain $D\subset \subset \Sigma$, we suppose $\int _{D}|A|^2_gd\mu _g<\infty$. Since $F_k$ converges to F weakly in $W^{2,2}_{loc}(\Sigma ,\mathbb {R}^n)$ and strongly in $W^{1,q}_{loc}(\Sigma ,\mathbb {R}^n)$, by Lusin’s theorem, Egorov’s theorem and the absolute continuity of integral, there exists a compact set $S\subset \subset D$ such that $F, DF, D^2F$ are continuous on S, $(F_k,DF_k)$ converges to (F, DF) uniformly on S and

$$\begin{aligned} \int _{S}|A|^2_gd\mu _g\ge \int _{D}|A|^2_gd\mu _g-\varepsilon . \end{aligned}$$

In the case $\int _{D}|A|^2_gd\mu _g=\infty$, we can take $\int _{S}|A|^2_gd\mu _g\ge M$ for any $M>0$. By the convexity of J we know

$$\begin{aligned} J(F_k,DF_k, D^2F_k)&\ge J(F_k,DF_k,D^2F)+D_{\eta }J(F,DF,D^2F)\cdot (D^2F_k-D^2F)\\&\quad +[D_{\eta }J(F_k,DF_k,D^2F)-D_{\eta }J(F,DF,D^2F)]\cdot (D^2F_k-D^2F). \end{aligned}$$

Since $F,DF,D^2F\in C(S)$ and $dF\otimes dF$ is a metric, we know $D_{\eta }J(F,DF,D^2F)\in C(S)$. By the weak $L^2$ convergence of $D^2F_k$ to $D^2F$, we get

$$\begin{aligned} \int _{S}D_{\eta }J(F,DF,D^2F)\cdot (D^2F_k-D^2F)\rightarrow 0. \end{aligned}$$

Moreover, the uniform convergence of $(F_k,DF_k)$ to (F, DF) on S and uniform boundness of $D^2F_k$ and $D^2F$ in $L^2$ imply that $D_{\eta }J(F_k,DF_k,D^2F)$ converges to $D_{\eta }J(F,DF,D^2F)$ uniformly on S and

$$\begin{aligned} \int _{S}[D_{\eta }J(F_k,DF_k,D^2F)-D_{\eta }J(F,DF,D^2F)]\cdot (D^2F_k-D^2F)\rightarrow 0. \end{aligned}$$

As a result, we get

$$\begin{aligned} \liminf _{k\rightarrow \infty }\int _SJ(F_k,DF_k, D^2F_k)&\ge \liminf _{k\rightarrow \infty }\int _SJ(F_k,DF_k, D^2F)\\&=\int _SJ(F,DF, D^2F)\\&\ge \int _DJ(F,DF, D^2F)-\varepsilon . \end{aligned}$$

Letting $\varepsilon \rightarrow 0$ and $D\rightarrow \Sigma$, we get

$$\begin{aligned} \int _{\Sigma }{|A|}^2\le \liminf _{k\rightarrow \infty }\int _{\Sigma _k}{|A_k|}^2. \end{aligned}$$

$\square$