Short-Time Existence of the Möbius-Invariant Willmore Flow

Abstract

In this article the author proves existence and uniqueness of a smooth short-time solution of the “Möbius-invariant Willmore flow” Eq. (9) starting in a $C^{\infty }$-smooth immersion $F_0$ of a fixed smooth compact torus ${\varSigma }$ into $\mathbb {R}^n$ without umbilic points. Hence, for some sufficiently small $T^*>0$ there exists a unique smooth family $\{f_t\}$ of smooth immersions of the torus ${\varSigma }$ into $\mathbb {R}^n$, with $f_0=F_0$, which solve the evolution Eq. (9) for $t \in [0,T^*]$ and whose tracefree parts $A^{0}_{f_t}(x)$ of their second fundamental forms do not vanish in any $(x,t) \in {\varSigma }\times [0,T^*]$. The right-hand side of Eq. (9) has the specific property that any family $\{f_t\}$ of umbilic-free $C^4$-immersions $f_t:{\varSigma }\longrightarrow \mathbb {R}^n$ solves Eq. (9) if and only if its composition ${\varPhi }(f_t)$ with any applicable Möbius-transformation ${\varPhi }$ of $\mathbb {R}^n$ solves Eq. (9) as well.

Appendix

The aim of this section is to show that the right-hand side of (9) is the only modification of the usual Willmore gradient $\delta \mathcal {W}(f)$ by means of some scalar factor which only depends on f, Df and $D^2f$, in order to obtain a Möbius-invariant flow. Let ${\varSigma }$ be a surface without boundary and let $Imm _{ uf }({\varSigma },\mathbb {R}^n)$ denote the open subset of $C^{\infty }({\varSigma },\mathbb {R}^n)$ which consists of all umbilic-free $C^{\infty }$-immersions of ${\varSigma }$ into $\mathbb {R}^n$. Firstly, we state

Proposition 4

Let $B: \mathbb {R}^n \times \mathbb {R}^{2n} \times \mathbb {R}^{4n} \rightarrow \mathbb {R}_+$ denote some arbitrary real-analytic, positive function. The flow

$$\begin{aligned} \partial _t f_t = - B(f_t,Df_t,D^2f_t) \, \big ( \triangle ^{\perp }_{f_t} H_{f_t} + Q(A^{0}_{f_t}) (H_{f_t}) \big ) \qquad on \quad {\varSigma }\times [0,T) \end{aligned}$$

meets the property of Möbius-invariance—in the sense of Part 2 of Corollary 1—if and only if the composition $\varphi (f):=B(f,Df,D^2f)$ satisfies the following structure conditions:

• (1)
  $$\begin{aligned} \varphi (\lambda \, f) = \lambda ^4\, \varphi (f) \nonumber \\ \varphi (M(f)) = \varphi (f) \end{aligned}$$

  (58)

  $\forall \, \lambda > 0$ and for all rigid motions $M(y)=O(y) + c$, $\, O \in O(n)$,  $c \in \mathbb {R}^n$.

• (2) There has to hold

  $$\begin{aligned} \varphi (I(f)) = |f|^{-8} \, \varphi (f) \qquad on \, {\varSigma }\end{aligned}$$

  for every immersion $f \in Imm _{ uf }({\varSigma },\mathbb {R}^n\setminus \{0\})$, where $I(y):=\frac{y}{|y|^2}$ denotes inversion at the unit sphere $\mathbb {S}^{n-1}$.

• (3) The map $f \mapsto \varphi (f) \, \big ( \triangle ^{\perp }_{f} H_{f} + Q(A^{0}_{f}) (H_{f}) \big )$ has to be a differential operator on $Imm _{ uf }({\varSigma },\mathbb {R}^n)$.

This proposition follows immediately from the statements of Lemma 1. We note that the function B does not depend on its 1st variable $y \in \mathbb {R}^n$ due to property (58) of $\varphi$. We now restrict our attention to the special case $n=3$, because in this case we can use the precise knowledge of local, conformally invariant operators on $Imm _{ uf }({\varSigma },\mathbb {R}^3)$ due to Cairns et al. [2, 3].

Theorem 4

Let $\varphi : Imm _{ uf }({\varSigma },\mathbb {R}^3) \rightarrow C^{\infty }({\varSigma },\mathbb {R}_+)$ be a real-analytic map of the form $\varphi (f)=B(Df,D^2f)$ which satisfies the structure conditions (1)–(3) of Proposition 4. Then there holds

$$\begin{aligned} \varphi (f)= c \, | A^0_f |^{-4} \end{aligned}$$

for some $c>0$ and for any $f \in Imm _{ uf }({\varSigma },\mathbb {R}^3)$.

Proof

By (4) and (5) one can easily compute that the map $f \mapsto | A^0_f |^{-4}$ satisfies the requirements (1)–(3) of Proposition 4 on $Imm _{ uf }({\varSigma },\mathbb {R}^3)$ and is ”local of second order”, which means that its value in any point $p \in {\varSigma }$ only depends on the first and second partial derivatives of f. Now, let $\varphi : Imm _{ uf }({\varSigma },\mathbb {R}^3) \rightarrow C^{\infty }({\varSigma },\mathbb {R}_+)$ be an arbitrary map which is local of second order, i.e. of the form $\varphi (f)=B(Df,D^2f)$, and which satisfies the conditions (1)–(3) of Proposition 4. We consider the quotient $Q(f):= \frac{\varphi (f)}{| A^0_f |^{-4}}$ and see that $f \mapsto Q(f)$ is a map from $Imm _{ uf }({\varSigma },\mathbb {R}^3)$ to $C^{\infty }({\varSigma },\mathbb {R}_+)$ which is again local of second order and satisfies $Q({\varPhi }(f))=Q(f)$ for any Möbius-transformation ${\varPhi }$ of $\mathbb {R}^3$ which is applicable to f. This means that Q is a local, conformally invariant operator from $Imm _{ uf }({\varSigma },\mathbb {R}^3)$ to $C^{\infty }({\varSigma },\mathbb {R}_+)$ of second order. Now, by Theorem 5.6 in [3] any non-constant, local and conformally invariant operator from $Imm _{ uf }({\varSigma },\mathbb {R}^3)$ to $C^{\infty }({\varSigma },\mathbb {R}_+)$ has to be at least of third order. Hence, we conclude $Q(f) \equiv const. >0$, i.e. $\varphi (f) = c\, | A^0_f |^{-4}$, for some $c>0$. $\square$