函数分析性质和Möbius不变Willmore流的正则性
Functional analytic properties and regularity of the Möbius-invariant Willmore flow in $\mathbb{R}^n$
AI总结：
本文研究Möbius不变Willmore流的函数分析性质和正则性，证明了其演算算子的局部实解析性和Fréchet导数的扩展性。
摘要：
在本文中，我们继续研究Möbius不变Willmore流在R^n和n球面S^n中移动参数化无脐环面的性质。在本文的主要定理中，我们证明了
英文摘要：
In this article we continue the author's investigation of the Möbius-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 Möbius-invariant Willmore flow and of its Fréchet 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, Prüss 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 Möbius-invariant Willmore flow in a small open ball in $W^{4-\frac{4}{p},p}(Σ,\mathbb{R}^n)$, for any $p\in (3,\infty)$, about any fixed smooth parametrization $F_0:Σ\longrightarrow \mathbb{R}^n$ 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 Fréchet 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 $L^p(Σ,\mathbb{R}^n)$, whose ranges are dense in $L^{p}(Σ,\mathbb{R}^n)$, for every fixed pair of times $t_2\geq t_1$ within the interval of maximal existence $(0,T_{max}(F_0))$.
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