Functional analytic properties and regularity of the M\"obius-invariant Willmore flow in $\mathbb{R}^n$

TL;DR

This paper investigates the analytic properties and regularity of the M"obius-invariant Willmore flow for umbilic-free tori in Euclidean space, establishing local and global analyticity results using advanced functional analysis techniques.

Contribution

It proves the local real analyticity of the evolution operator for the flow and extends the Fréchet derivative to a family of dense-range linear operators, advancing understanding of flow regularity.

Findings

Proves local real analyticity of the evolution operator.

Shows the entire flow line is real analytic for positive times.

Extends the Fréchet derivative to a family of dense-range operators.

Abstract

In this article we continue the author's investigation of the M\"obius-invariant Willmore flow moving parametrizations of umbilic-free tori in $\mathbb{R}^n$ and in the $n$-sphere $\mathbb{S}^n$. In the main theorems of this article we prove basic properties of the evolution operator of the "DeTurck modification" of the M\"obius-invariant Willmore flow and of its Fr\'echet derivative by means of a combination of the author's results about this topic with the theory of "bounded $\mathcal{H}_{\infty}$-calculus" for linear elliptic operators due to Amann, Denk, Duong, Hieber, Pr\"uss and Simonett, and with Amann's and Lunardi's work on semigroups and interpolation theory. Precisely, we prove local real analyticity of the evolution operator $[F\mapsto \mathcal{P}^*(\,\cdot\,,0,F)]$ of the "DeTurck modification" of the M\"obius-invariant Willmore flow in a small open ball in $W^{4-\frac{4}{p},p}(\Sigma,\mathbb{R}^n)$, for any $p\in (3,\infty)$, about any fixed smooth parametrization $F_0:\Sigma \longrightarrow \mathbb{R}^n$ of a compact and umbilic-free torus in $\mathbb{R}^n$. We prove moreover that the entire maximal flow line $\mathcal{P}^*(\,\cdot\,,0,F_0)$, starting to move in a smooth and umbilic-free initial immersion $F_0$, is real analytic for positive times, and that therefore the Fr\'echet derivative $D_{F}\mathcal{P}^*(\,\cdot\,,0,F_0)$ of the evolution operator in $F_0$ can be uniquely extended to a family of continuous linear operators $G^{F_0}(t_2,t_1)$ in $L^p(\Sigma,\mathbb{R}^n)$, whose ranges are dense in $L^{p}(\Sigma,\mathbb{R}^n)$, for every fixed pair of times $t_2\geq t_1$ within the interval of maximal existence $(0,T_{max}(F_0))$.