Explicit conformally constrained Willmore minimizers in arbitrary codimension

Abstract

In our previous work (Ndiaye and Schätzle, 2014), we proved that the flat constant mean curvature tori

$$\begin{aligned} T_r := r S^1 \times \sqrt{1 - r^2} S^1 \subseteq S^3 \quad \hbox {for } 0 < r \le 1/\sqrt{2} \end{aligned}$$

minimize the Willmore energy in their conformal class in codimension one when $r \approx 1 / \sqrt{2}$, that is $T_r$ is close to the Clifford torus $T_{Cliff} = T_{1/\sqrt{2}}$. In this article, we extend this to arbitrary codimension. Moreover we prove that the Clifford torus minimizes the Willmore energy in an open neighbourhood of its conformal class, again in arbitrary codimension, but the neighbourhood may depend on the codimension.

Appendix

1.1 Convergence in Teichmüller space

In this appendix, we establish convergence in Teichmüller space under weak convergence in $W^{2,2}$, as it was used in Theorem 1.3.

Proposition 5.1

Let $f_m, f: {\Sigma }\rightarrow {\mathbb R}^n$ be smooth immersions of a closed, orientable surface with

$$\begin{aligned} \begin{array}{l} f_m \rightarrow f \hbox { weakly in } W^{2,2}({\Sigma }), \\ \limsup \limits _{m \rightarrow \infty } {{\mathcal {W}}}(f_m) < \min (8 \pi , {{\mathcal {W}}}(f) + { e_{n} }), \\ \end{array} \end{aligned}$$

(5.1)

where

$$\begin{aligned} { e_{n} } := \left\{ \begin{array}{l@{\quad }l} 4 \pi &{} \hbox {for } n = 3, \\ 8 \pi / 3 &{} \hbox {for } n = 4, \\ 2 \pi &{} \hbox {for } n \ge 5. \\ \end{array} \right. \end{aligned}$$

Then the pull-back metrics $g_m := f_m^* { g_{{\mathbb R}^n} }$ differ from unit volume constant curvature metrics ${ g_{{ poin },m} }:= e^{-2 u_m} g_m$ by a bounded conformal factor for $m$ large, more precisely

$$\begin{aligned} \limsup \limits _{m \rightarrow \infty } \parallel u_m \parallel _{L^\infty (\Sigma )} < \infty \end{aligned}$$

(5.2)

and

$$\begin{aligned} { \pi }(f_m^* { g_{{\mathbb R}^n} }) \rightarrow { \pi }(f^* { g_{{\mathbb R}^n} }). \end{aligned}$$

(5.3)

Proof

By the weak convergence in $W^{2,2}({\Sigma },{\mathbb R}^n)$ we get in local charts

$$\begin{aligned} \begin{array}{l} \nabla f_m \rightarrow \nabla f, \\ g_{m,ij} \rightarrow g_{ij}, \sqrt{g_m} \rightarrow \sqrt{g}, \\ \end{array} \left\{ \begin{array}{l} \hbox {strongly in } L^p({\Sigma }), p < \infty , \\ \hbox {pointwise almost everywhere } \\ \end{array} \right. \end{aligned}$$

after passing to an appropriate subsequence. By Egoroff’s theorem, we can select for any $\varepsilon > 0$ a Borel subset $B_\varepsilon \subseteq \Sigma$ with

$$\begin{aligned} \begin{array}{l} \nabla f_m \rightarrow \nabla f, g_{m,ij} \rightarrow g_{ij}, g_m^{ij} \rightarrow g^{ij}, \sqrt{g_m} \rightarrow \sqrt{g} \quad \hbox {uniformly on } B_\varepsilon , \\ \mu _g(\Sigma - B_\varepsilon ) < \varepsilon . \\ \end{array} \end{aligned}$$

Then

$$\begin{aligned} A_{f_m,ij} = \partial _{ij} f_m - g_m^{kl} \langle \partial _{ij} f_m , \partial _l f_m \rangle \partial _k f_m \rightarrow A_{f,ij} \quad \hbox {weakly in } L^2(B_\varepsilon ) \end{aligned}$$

and

$$\begin{aligned} {\vec {{\mathbf H}}}_{f_m} = g_m^{ij} A_{f_m,ij} \rightarrow g^{ij} A_{f,ij} = {\vec {{\mathbf H}}}_f \quad \hbox {weakly in } L^2(B_\varepsilon ), \end{aligned}$$

hence

$$\begin{aligned} \int \limits _{B_\varepsilon } |{\vec {{\mathbf H}}}_f|^2 { \ \mathrm{d} }\mu _g \le \liminf \limits _{m \rightarrow \infty } \int \limits _{B_\varepsilon } |{\vec {{\mathbf H}}}_{f_m}|^2 { \ \mathrm{d} }\mu _{g_m} \end{aligned}$$

and by (5.1)

$$\begin{aligned} {{\mathcal {W}}}(f) \le \liminf \limits _{m \rightarrow \infty } {{\mathcal {W}}}(f_m) < 8 \pi . \end{aligned}$$

By the Li–Yau inequality, see [7, Theorem 6], we get that $f_m \hbox { and } f$ are embeddings for large $m$. Again by the weak convergence in $W^{2,2}({\Sigma },{\mathbb R}^n)$

$$\begin{aligned} f_m \rightarrow f \hbox { uniformly}, \end{aligned}$$

(5.4)

and hence we may assume

$$\begin{aligned} f_m({\Sigma }), f({\Sigma }) \subseteq B_R(0) \end{aligned}$$

for some $R < \infty$. Clearly $f$ is not constant, hence $0 < osc\ f \leftarrow osc\ f_m \le 2 R$ and by [14, Lemma 1.1]

$$\begin{aligned} c_0 \le { \mathcal{H}^2 }(f_m(\Sigma )) = \mu _{g_m}({\Sigma }) \le C(n) \end{aligned}$$

for some $0 < c_0 \le C(n) < \infty$. After passing to a subsequence, we get

$$\begin{aligned} { \mathcal{H}^2 }\lfloor f_m(\Sigma ) \rightarrow \mu \ne 0 \quad \hbox {weakly as Radon measures} \end{aligned}$$

and again by monotonicity formula, we get as in [14, Theorem 3.1]

$$\begin{aligned} f_m(\Sigma ) \rightarrow spt\ \mu \quad \hbox {in Hausdorff-distance}, \end{aligned}$$

hence by (5.4)

$$\begin{aligned} f(\Sigma ) = spt\ \mu . \end{aligned}$$

(5.5)

Next by Allard’s integral compactness theorem, see [1, Theorem 6.4] or [13, Remark 42.8], and (5.1), we get that $\mu$ is an integral varifold and has weak mean curvature in $L^2(\mu )$, more precisely by lower semicontinuity

$$\begin{aligned} {{\mathcal {W}}}(\mu ) \le \liminf \limits _{m \rightarrow \infty } {{\mathcal {W}}}(f_m) < 8 \pi . \end{aligned}$$

Then by the Li–Yau inequality in [7] or [5, (A.17)]

$$\begin{aligned} \theta ^2(\mu ) \le {{\mathcal {W}}}(\mu ) / (4 \pi ) < 2, \end{aligned}$$

hence $\theta ^2(\mu ) = 1 \hbox { alomost everywhere with respect to } \mu$. Then by (5.5) we get ${ \mathcal{H}^2 }\lfloor f(\Sigma ) = \mu$ and

$$\begin{aligned} {{\mathcal {W}}}(f) = {{\mathcal {W}}}(\mu ). \end{aligned}$$

(5.6)

Combining (5.1), (5.5) and (5.6), we see

$$\begin{aligned} \begin{array}{l} genus(spt\ \mu ) = genus(\Sigma ) \ge 1, \\ \limsup \limits _{m \rightarrow \infty } {{\mathcal {W}}}(f_m) < {{\mathcal {W}}}(\mu ) + { e_{n} }, \\ \end{array} \end{aligned}$$

and obtain from [11, Proposition 2.4] that the smooth unit volume constant curvature metrics ${ g_{{ poin },m} }= e^{-2 u_m} g_m$ conformal to $g_m$ satisfy

$$\begin{aligned} \parallel u_m \parallel _{L^\infty ({\Sigma })} \le C(n, 8 \pi , K , genus({\Sigma }) , \delta ) \end{aligned}$$

(5.7)

for $m$ large, which establishes (5.2).

Then by standard weak compactness, see [11, Proposition 6.1], there exist diffeomorphisms $\phi _m: {\Sigma }\mathop {\longrightarrow }\limits ^{\approx } {\Sigma }$, such that for $\tilde{f}_m := f_m \circ \phi _m, \tilde{g}_m := \tilde{f}_m^* { g_{{\mathbb R}^n} }= e^{2 \tilde{u}_m} { \tilde{g}_{{ poin },m} }, { \tilde{g}_{{ poin },m} }$ a unit volume constant curvature metric,

$$\begin{aligned}&\tilde{f}_m \rightarrow \tilde{f} \hbox { weakly in } W^{2,2}({\Sigma }), \hbox {weakly}^* \hbox { in } W^{1,\infty }({\Sigma }),&\\&\tilde{u}_m \rightarrow \tilde{u} \hbox { weakly in } W^{1,2}({\Sigma }), \hbox {weakly}^* \hbox { in } L^\infty ({\Sigma }),&\\&{ \tilde{g}_{{ poin },m} }:= \phi _m^* { g_{{ poin },m} }\rightarrow { \tilde{g}_{{ poin }} }\quad \hbox {smoothly},&\\&\tilde{f}^* { g_{{\mathbb R}^n} }= e^{2 \tilde{u}} { \tilde{g}_{{ poin }} },&\end{aligned}$$

to some smooth unit volume constant curvature metric ${ \tilde{g}_{{ poin }} }$, and in particular

$$\begin{aligned} \tilde{f}_m \rightarrow \tilde{f} \hbox { uniformly}. \end{aligned}$$

For $\tilde{p}_m \rightarrow \tilde{p} \in {\Sigma }$, we get by compactness of ${\Sigma }$ that $\phi _m(\tilde{p}_m) \rightarrow p \in {\Sigma }$ for a subsequence, hence with (5.4),

$$\begin{aligned} \tilde{f}(\tilde{p}) \leftarrow \tilde{f}_m(\tilde{p}_m) = f_m(\phi _m(\tilde{p}_m))) \rightarrow f(p). \end{aligned}$$

(5.8)

Firstly for $\tilde{p}_m = \tilde{p}$, we get $\tilde{f}({\Sigma }) \subseteq f({\Sigma })$. Secondly for $\tilde{p}_m := \phi _m^{-1}(p) \hbox { and } \tilde{p}_m \rightarrow \tilde{p}$ for a subsequence, we get $\tilde{f}({\Sigma }) \supseteq f({\Sigma })$, and together

$$\begin{aligned} \tilde{f}({\Sigma }) = f({\Sigma }). \end{aligned}$$

(5.9)

We have already seen that $f$ is an embedding, in particular it is injective and $f^{-1}: f({\Sigma }) \rightarrow {\Sigma }$ is smooth. Then $\phi := f^{-1} \circ \tilde{f}: {\Sigma }\rightarrow {\Sigma }\in W^{2,2}({\Sigma }) \cap W^{1,\infty }({\Sigma })$ and $p = (f^{-1} \circ \tilde{f})(\tilde{p}) = \phi (\tilde{p})$ is uniquely determined in (5.8), hence

$$\begin{aligned} \phi _m \rightarrow \phi \quad \hbox {uniformly on } {\Sigma }\end{aligned}$$

(5.10)

and $\tilde{f} = f \circ \phi$. For the smooth unit volume constant curvature metric ${ g_{{ poin }} }= e^{-2u} g = e^{-2u} f^* { g_{{\mathbb R}^n} }$ for some smooth $u$, we calculate by above

$$\begin{aligned} e^{2 \tilde{u}} { \tilde{g}_{{ poin }} }= \tilde{f}^* { g_{{\mathbb R}^n} }= \phi ^* f^* { g_{{\mathbb R}^n} }= \phi ^*(e^{2u} { g_{{ poin }} }) = e^{2 (u \circ \phi )} \phi ^* { g_{{ poin }} }. \end{aligned}$$

(5.11)

Then $\phi : ({\Sigma },{ \tilde{g}_{{ poin }} }) \rightarrow ({\Sigma },{ g_{{ poin }} })$ is conformal, in particular satisfies the Cauchy–Riemann equations in local conformal charts with respect to ${ g_{{ poin }} }\hbox { and } { \tilde{g}_{{ poin }} }$, hence is harmonic and smooth. Then $\tilde{f} = f \circ \phi$ is smooth as well, hence an immersion. As ${{\mathcal {W}}}(\tilde{f}) = {{\mathcal {W}}}(f) < 8 \pi$, we see that $\tilde{f}$ is an embedding by an inequality of Li and Yau [7], in particular $\tilde{f} \hbox { and hence } \phi$ are injective. Clearly by (5.9), we see

$$\begin{aligned} \phi ({\Sigma }) = f^{-1}(\tilde{f}({\Sigma })) = f^{-1}(f({\Sigma })) = {\Sigma }, \end{aligned}$$

and $\phi$ is also surjective, hence $\phi$ is bijective. As $u, \tilde{u} \in L^\infty ({\Sigma })$ and $D \phi$ is continuous, we conclude further that $D \phi$ is of full rank everywhere on ${\Sigma }$, hence $\phi$ is a diffeomorphism. By (5.11),

$$\begin{aligned} { \tilde{g}_{{ poin }} }= \phi ^* { g_{{ poin }} }. \end{aligned}$$

(5.12)

Then we can replace $\phi _m \hbox { by } \phi _m \circ \phi ^{-1}$ and get by (5.10) that $\phi _m \rightarrow id_{\Sigma }\hbox { uniformly on } {\Sigma }$, in particular $\phi _m \simeq id_{{\Sigma }} \hbox { for } m$ large and ${ \tilde{g}_{{ poin }} }= { g_{{ poin }} }$ by (5.12). This yields

$$\begin{aligned}&{ \pi }(f_m^* { g_{{\mathbb R}^n} }) = { \pi }({ g_{{ poin },m} }) = { \pi }(\phi _m^* { g_{{ poin },m} }) = { \pi }({ \tilde{g}_{{ poin },m} }) \rightarrow&\\&\rightarrow { \pi }({ \tilde{g}_{{ poin }} }) = \pi ({ g_{{ poin }} }) = { \pi }(e^{2u} { g_{{ poin }} }) = \pi (f^* { g_{{\mathbb R}^n} }),&\end{aligned}$$

which establishes (5.3). $\square$