\documentclass[reqno]{amsart} \usepackage{hyperref} %\usepackage{cite} \AtBeginDocument{{\noindent\small Ninth MSU-UAB Conference on Differential Equations and Computational Simulations. \emph{Electronic Journal of Differential Equations}, Conference 20 (2013), pp. 151--164.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \setcounter{page}{151} \title[\hfilneg EJDE-2013/Conf/20/ \hfil Real analytic solutions] {Real analytic solutions for the Willmore flow} \author[Y. Shao \hfil EJDE-2013/conf/20 \hfilneg] {Yuanzhen Shao} % in alphabetical order \address{Yuanzhen Shao \newline Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA} \email{yuanzhen.shao@vanderbilt.edu} \thanks{Published October 31, 2013.} \subjclass[2000]{35B65, 35K55, 53A05, 53C44, 58J99} \keywords{Real analytic solution; Willmore flow; mean curvature; \hfill\break\indent Gaussian curvature, geometric evolution equation; implicit function theorem; \hfill\break\indent maximal regularity} \begin{abstract} In this article, we present a regularity result for the Willmore flow. This is obtained by using a truncated translation technique in conjunction with the implicit function theorem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} The Willmore flow consists in looking for an oriented, closed, compact moving hypersurface $\Gamma(t)$ immersed in $\mathbb{R}^3$ evolving subject to the law \begin{equation} \label{original eq 1.1} \begin{cases} V(t)=-\Delta_{\Gamma(t)}H_{\Gamma(t)}-2H_{\Gamma(t)} (H^2_{\Gamma(t)}-K_{\Gamma(t)}),\\ \Gamma(0)=\Gamma_0. \end{cases} \end{equation} Here $V(t)$ denotes the velocity in the normal direction of $\Gamma(t)$ at time $t$. $\Delta_{\Gamma(t)}$ and $H_{\Gamma(t)}$ stand for the Laplace-Beltrami operator and the normalized mean curvature of $\Gamma(t)$, respectively. Finally, $K_{\Gamma(t)}$ denotes the Gaussian curvature. The equilibria of \eqref{original eq 1.1} appear as the critical points of the Willmore functional, or sometimes called the Willmore energy. For a smooth immersion $f:{\Gamma}\to \mathbb{R}^3$ of a closed oriented two-dimensional manifold $\Gamma$, the Willmore functional is defined as \begin{equation} \label{Willmore functional} W(f)=\int_{f(\Gamma)} H_{f(\Gamma)}^{2}\, d\sigma, \end{equation} where $d\sigma$ is the area element on $f(\Gamma)$ with respect to the Euclidean metric in $\mathbb{R}^3$. The critical surfaces of this functional, called the Willmore surfaces, satisfy the equation \begin{equation} \label{Euler-Lagrange equation} \Delta_{f(\Gamma)}H_{f(\Gamma)}+2H_{f(\Gamma)}^{3} -2H_{f(\Gamma)}K_{f(\Gamma)}=0. \end{equation} The reader may consult \cite[Section~7.4]{TJWR} for a brief historical account and a proof of this variational formula. The proof therein is derived by computing the critical points of all normal variations of the hypersurface ${f(\Gamma)}$. A generalization of the Willmore functional \eqref{Willmore functional} in higher dimensions is studied by Chen \cite{BYCV}. He extends \eqref{Willmore functional} for smooth immersions $f:{\Gamma}\to \mathbb{R}^{m+1}$ of the $m$-dimensional closed oriented manifold $\Gamma$ into $\mathbb{R}^{m+1}$: \begin{align*} W(f)=\int_{f(\Gamma)} H_{f(\Gamma)}^{m}\, d\sigma \end{align*} with $d\sigma$ standing for the volume element with respect to the Euclidean metric in $\mathbb{R}^{m+1}$. The critical points of this functional are now of the form \begin{align*} \Delta_{f(\Gamma)}H_{f(\Gamma)}^{m-1}+m(m-1)H_{f(\Gamma)}^{m+1} -H_{f(\Gamma)}^{m-1}R_{f(\Gamma)}=0\,. \end{align*} Here $R_{f(\Gamma)}$ denotes the scalar curvature. We may observe that $R_{f(\Gamma)}=2K_{f(\Gamma)}$ when $m=2$, so this Euler-Lagrange equation agrees with \eqref{Euler-Lagrange equation} in the two-dimensional case. However, this generalization has the drawback that the corresponding Willmore functional is no longer conformally invariant except when $m=2$. The Willmore problem has been studied by many authors, among them Thomsen, Blaschke, Willmore, Chen, Weiner, Li, Yau, Bryant, Kusner, Simon, Mayer, Simonett, Bauer, Kuwert, Sch\"atzle, Pinkall, Sterling, Schmidt, Marques, and Neves; see \cite{MBEK,WBDGIII,RBDW,BYCV,FAWC,RKCW,EKSE,KSGF,EKRS,LYCW,MSSI,MSWF,UPHT, PSWS,MSWC,LSMW,GSWF,GTKGI,JWPC,TJWR}. It is well-known that the Willmore functional is bounded below by $4\pi$ with equality only for the round sphere. Then the famous Willmore conjecture due to Willmore asserts that for any immersed $2$-dimensional torus into $\mathbb{R}^3$ we have $W(f)\geq{2\pi^2}$, and it suggests that the $2$-dimensional Clifford torus achieves the minimum of the Willmore functional amongst all immersed tori in $\mathbb{R}^3$. In 1982, Li and Yau \cite{LYCW} showed that any immersion with $W(f)<8\pi$ must in fact be an embedding. In other words, it will suffice to estimate $W(f)$ for embeddings. A classification of all Willmore immersions $f:\mathbb{S}^2\to \mathbb{R}^3$ is obtained by Bryant \cite{RBDW}. The possible values of \begin{align*} W(f)=\int_{f(\mathbb{S}^2)} H_{f(\mathbb{S}^2)}^{2}\, d\sigma \end{align*} are $4n\pi$ with $n=1$, or $n\geq{4}$ and $n$ even, or $n\geq{9}$ and $n$ odd. Existence and regularity for embedded tori in the Willmore conjecture has been proven by Simon \cite{LSMW}, and later this result is generalized by Bauer, Kuwert \cite{MBEK} for an extension of the conjecture by Kusner \cite{RKCW} to higher genus cases. An existence, uniqueness and regularity result on the Willmore flow is presented by Simonett \cite{GSWF}. It is proven therein that the Willmore flow admits a unique smooth solution. Moreover, this solution exists globally when it is initially close enough to spheres in the $C^{2+\alpha}$-topology and is exponentially attracted by spheres. In \cite{MSSI}, Mayer and Simonett proved that the Willmore flow can drive embedded surfaces to a self-intersection in a finite time interval. Moreover, numerical simulations in \cite{MSWF} indicate that the Willmore flow can develop true singularities (topological changes) in finite time. Kuwert and Sch\"atzle \cite{EKSE} show that the smooth solutions are global as long as the initial Willmore energy is sufficiently small. Later, the same authors improve this result in \cite{EKRS} by finding an explicit optimal bound for the restriction on the initial energy; that is, if the smooth immersion $f_0:\Gamma\to \mathbb{R}^3$ satisfies $W(f_0)\leq{8\pi}$, then the solution with initial data $f_0$ exists smoothly for all time and converges to a round sphere. Recently, in a breakthrough paper, Marques and Neves \cite{FAWC} prove the Willmore conjecture for surfaces of arbitrary genus $g\geq 1$; i.e., $W(f)\geq 2\pi^2$ for all embedded $\Gamma$ with genus $g\geq 1$, and the equality holds if and only if $\Gamma$ is conformal to the Clifford torus. \subsection*{Assumptions} Throughout this paper, we assume that $({\sf{M}},g)$ is a compact, closed, embedded, oriented, real analytic hypersurface in $\mathbb{R}^3$ endowed with the Euclidean metric $g$ with the exception of Section~3, wherein we remove the restriction on the dimension of ${\sf{M}}$. The notation $(\cdot|\cdot)$ always stands for the standard inner product in $\mathbb{R}^3$. We may find for ${\sf{M}}$ a \emph{normalized atlas} $\mathfrak{K}:=({\sf{O}_{\kappa}},\varphi_{\kappa})_{{\kappa}\in \mathfrak{K}}$. An atlas $\mathfrak{K}$ is called \emph{normalized} if $\varphi_{\kappa}({\sf{O}_{\kappa}})= {\mathbb{B}^2}$ for all $\kappa\in\mathfrak{K}$. Here ${\mathbb{B}^2}$ is the open unit ball centered at the origin in $\mathbb{R}^2$. Put $\psi_{\kappa}=\varphi_{\kappa}^{-1}$. A family $({\pi_\kappa})_{{\kappa}\in\mathfrak{K}}$ is called a \emph{localization system subordinate to} $\mathfrak{K}$ if: \begin{itemize} \item[(L1)] ${\pi_\kappa}\in\mathcal{D}({\sf{O}_{\kappa}},[0,1])$ and $(\pi_{\kappa}^{2})_{\kappa\in{\mathfrak{K}}}$ is a partition of unity subordinate to $\mathfrak{K}$. \item[(L2)] Any ${\pi_\kappa}$ and $\pi_{\eta}$ satisfying $\operatorname{supp}({\pi_\kappa})\cap \operatorname{supp}({\pi_\eta})\neq \emptyset$ have their supports located within the same local chart. \end{itemize} For any manifold satisfying the above assumptions, there exists a localization system. See \cite[Lemma~3.2]{FSM} for a proof. Condition (L2) is not an additional assumption, because of the compactness of $\sf{M}$. \subsection*{Notation} Throughout this paper, ${\mathbb{N}}_0$ stands for the set of natural numbers including $0$. For any interval $I$, $\mathring{I}$ denotes the interior of $I$, and $\dot{I}:=I\setminus\{0\}$. For a fix $0<\alpha<1$. Put $E_0:=h^{\alpha}({\sf{M}})$, $E_1:=h^{4+\alpha}({\sf{M}})$. Please refer to the remark below Theorem~\ref{main theorem} for the precise definition of the spaces $h^{s}(\sf{M})$. For notational brevity, we simply write $\mathfrak{F}(\mathcal{O},\mathbb{R})$ and $\mathfrak{F}({\sf{M}},\mathbb{R})$ as $\mathfrak{F}(\mathcal{O})$ and $\mathfrak{F}({\sf{M}})$, where $\mathcal{O}$ is any open subset of $\mathbb{R}^2$ and $\mathfrak{F}$ stands for any of the function spaces in this paper. Let $\gamma\in(0,1]$. In the sequel, we denote $(E_0,E_1)_{\gamma}$ by $E_{\gamma}$, where $(\cdot,\cdot)_{\gamma}$ is the continuous interpolation method. See \cite[Definition~1.2.2]{LSOR} for a definition. In particular, we set $(E_0,E_1)_1:=E_1$. For some fixed interval $I=[0,T]$ and some Banach space $E$, we define \begin{gather*} BU\!C_{1-\gamma}(I,E):=\{u\in{C(\dot{I},E)};[t\mapsto{t^{1-\gamma}}u] \in{BU\!C(\dot{I},E)},\lim_{t\to{0^+}}{t^{1-\gamma}}\|u\|=0\}, \\ \|u\|_{C_{1-\gamma}}:=\sup_{t\in{\dot{I}}}{t^{1-\gamma}}\|u(t)\|_{E}, \\ BU\!C_{1-\gamma}^1(I,E):=\{u\in{C^1(\dot{I},E)}: u,\dot{u}\in{BU\!C_{1-\gamma}(I,E)}\}. \end{gather*} In particular, we put \[ BU\!C_0(I,E):=BU\!C(I,E), \quad BU\!C^1_0(I,E):=BU\!C^1(I,E). \] In addition, if $I=[0,T)$ is a half open interval, then \begin{gather*} C_{1-\gamma}(I,E):=\{v\in{C(\dot{I},E)}:v\in{BU\!C_{1-\gamma}([0,t],E)},\; t0$. Moreover, \[ \mathcal{M}:=\cup_{t\in(0,T)}(\{t\}\times\Gamma(t)) \] is a real analytic submanifold in $\mathbb{R}^{4}$. In particular, each manifold $\Gamma(t)$ is real analytic for $t\in(0,T)$. \end{theorem} For any open subset $\mathcal{O}\subset\mathbb{R}^{2}$, the little H\"older space $h^{s}(\mathcal{O})$ of order $s>0$ with $s\notin{\mathbb{N}}$ is the closure of $BU\!C^{\infty}(\mathcal{O})$ in $BU\!C^{s}(\mathcal{O})$. Here $BU\!C^{s}(\mathcal{O})$ is the Banach space of all bounded and uniformly H\"older continuous functions. The little H\"older space $h^{s}(\sf{M})$ on $\sf{M}$ is defined in terms of the atlas $\mathfrak{K}$; that is, a function $u$ belongs to $h^{s}({\sf{M}})$ if and only if $\psi_{\kappa}^{\ast}{\pi_\kappa}u\in h^{s}(\mathbb{R}^{2})$, for each $\kappa\in\mathfrak{K}$. \section{Parameterization over a reference manifold} In equation \eqref{original eq 1.1}, if we fix an embedded initial hypersurface $\Gamma_0$ belonging to the class $h^{2+\alpha}$, then by the discussion in \cite[Section~4]{MCH} we can find a real analytic compact closed embedded oriented hypersurface ${\sf{M}}$, a function $\rho_0 \in h^{2+\alpha}(\sf{M}) $ and a parameterization \[ \Psi_{\rho_0}:{\sf{M}}\to \mathbb{R}^3,\quad \Psi_{\rho_0}(p):=p+\rho_0(p){\nu}_{\sf{M}}(p) \] such that $\Gamma_0=\operatorname{im}(\Psi_{\rho_0})$. Here ${\nu}_{\sf{M}}(p)$ denotes the unit normal with respect to a chosen orientation of $\sf{M}$ at $p$, and $\rho_0:{\sf{M}}\to (-a,a)$ is a real-valued function on ${\sf{M}}$, where $a$ is a sufficiently small positive number depending on the inner and outer ball condition of ${\sf{M}}$. The reader may consult \cite[Section~4.1]{MCH} for the precise bound of $a$. Thus $\Gamma_0$ lies in the $a$-tubular neighborhood of ${\sf{M}}$. In fact, it will suffice to assume $\Gamma_0$ to be a $C^2$-manifold for the existence of such a parameterization and a real analytic reference manifold. See \cite[Section~4]{MCH} for a detailed proof. Analogously, if $\Gamma(t)$ is $C^1$-close enough to ${\sf{M}}$, then we can find a function $\rho:[0,T)\times{\sf{M}}\to (-a,a)$ for some $T>0$ and a parameterization \[ \Psi_{\rho}:{[0,T)}\times{\sf{M}}\to \mathbb{R}^3,\quad \Psi_{\rho}(t,p):=p+\rho(t,p){\nu}_{\sf{M}}(p) \] such that $\Gamma(t)=\operatorname{im}(\Psi_{\rho}(t,\cdot))$ for every $t\in [0,T)$. It is worthwhile to mention that $\Psi_{\rho}$ admits an extension on $\mathbb{R}^{3}$, called Hanzawa transform, which is first introduced by Hanzawa in \cite{HCSS}. For any fixed $t$, I do not distinguish between $\rho(t,\cdot)$ and $\rho(t,\psi_{\kappa}(\cdot))$ in each local coordinate $({\sf{O}_{\kappa}},\varphi_{\kappa})$ and abbreviate $\Psi_{\rho}(t,\cdot)$ to be $\Psi_{\rho}:=\Psi_{\rho}(t,\cdot)$. In addition, the hypersurface $\Gamma(t)$ will be simply written as $\Gamma_{\rho}$ as long as the choice of $t$ is of no importance in the context, or $\rho$ is independent of $t$. We put \[ \mho:=\{\rho\in{h^{2+\alpha}({\sf{M}})}: \|\rho\|_{\infty}^{\sf{M}}0$, the operator $(I-\rho{L_{\sf{M}}^{\mathcal{E}}})$ is invertible. One can check that \[ I-\rho{L_{\sf{M}}^{\mathcal{E}}}=(\delta_{i}^{j}-\rho{l}_{i}^{j}) \tau^{i}\otimes\tau_{j}+\nu_{\sf{M}}\otimes\nu_{\sf{M}}. \] Thus \begin{equation} \label{Inverse formula} (I-\rho{L_{\sf{M}}^{\mathcal{E}}})^{-1}=r_i^j(\rho)\tau^{i} \otimes\tau_{j}+\nu_{\sf{M}}\otimes\nu_{\sf{M}}, \end{equation} where ${R}_{\rho}=(r_i^j(\rho))_{ij}=[(\delta_{i}^{j}-\rho{l}_{i}^{j})_{ij}]^{-1}$. By Cramer's rule, all the entries of ${R}_{\rho}$ possess the expression \[ r_i^j(\rho)=\frac{P_{i}^{j}(\rho)}{Q_{i}^{j}(\rho)} \] in every local chart, where $P_{i}^{j}$ and $Q_{i}^{j}$ are polynomials in $\rho$ with real analytic coefficients and $Q_{i}^{j}\neq{0}$. Substituting $(I-\rho{L_{\sf{M}}^{\mathcal{E}}})^{-1}$ by \eqref{Inverse formula}, we obtain \[ |a(\rho)|^2=( r^j_i(\rho) \partial_j\rho\tau^i| r^l_k(\rho) \partial_l\rho\tau^k)=g^{ik} r_i^j(\rho) r_k^l(\rho) \partial_{j} \rho \partial_{l}\rho. \] Then \[ \beta(\rho)=[1+|a(\rho)|^2]^{-1/2} =[1+g^{ik} r_i^j(\rho) r_k^l(\rho) \partial_{j}\rho \partial_{l}\rho]^{-1/2}. \] Note that in every local chart \[ \beta^2(\rho)=\frac{P^{\beta}(\rho)}{Q^{\beta}(\rho,\partial_{j}\rho)}, \] where $P^{\beta}(\rho)$ is a polynomial in $\rho$ with real analytic coefficients and $Q^{\beta}(\rho,\partial_{j}\rho)\neq{0}$ is a polynomial in $\rho$ and its first order derivatives with real analytic coefficients. The normal velocity can be expressed as \[ V(t)=(\partial_{t}\Psi_{\rho}|\nu_{\Gamma}) =(\rho_{t}\nu_{\sf{M}}|\nu_{\Gamma})=\beta(\rho)\rho_{t}. \] Therefore, the first line of \eqref{original eq 1.1} is equivalent to \[ \rho_{t}=-\frac{1}{\beta(\rho)}[\Psi^{\ast}_{\rho} \Delta_{\Gamma_{\rho}}H_{\Gamma_{\rho}}+2\Psi^{\ast}_{\rho} H_{\Gamma_{\rho}}(H^2_{\Gamma_{\rho}}-K_{\Gamma_{\rho}})]. \] Next we shall calculate the Gaussian curvature $K_{\Gamma_{\rho}}$ in terms of $\rho$. For simplicity, we write $K_{\rho}$ instead of $\Psi^{\ast}_{\rho}K_{\Gamma_{\rho}}$. Using that \[ \partial_j \tau_i=\Gamma^{k}_{ij}\tau_k+l_{ij}\nu_{\sf{M}},\quad \partial_j \tau^{i}=-\Gamma^i_{jk}\tau^k+l^i_j\nu_{\sf{M}}, \] one may readily obtain \begin{equation} \label{derivative of WT} \partial_{j}L_{\sf{M}}^{\mathcal{E}} =\partial_{j}l^{k}_{i}\tau^{i}\otimes\tau_{k} -\Gamma^{i}_{jl}l_{i}^{k}\tau^{l}\otimes\tau_{k} +\Gamma^{l}_{jk}l^{k}_{i}\tau^{i}\otimes\tau_{l} +l_{j}^{i}l^{k}_{i}\nu_{\sf{M}}\otimes\tau_{k} +l_{jk}l^{k}_{i}\tau^{i}\otimes\nu_{\sf{M}}. \end{equation} Denote by $L^{\Gamma}=(l_{ij}^{\Gamma})_{ij}$ the second fundamental form of $\Gamma_{\rho}$ with respect to $g_{\Gamma}$. Then by \eqref{nu} and \eqref{standard basis}, we can compute its components $l_{ij}^{\Gamma}$ as follows: \begin{align*} l_{ij}^{\Gamma} &=-(\tau^{\Gamma}_{i}|\partial_{j}\nu_{\Gamma})\\ &=-((I-\rho{L_{\sf{M}}^{\mathcal{E}}})\tau_{i}+\nu_{\sf{M}}\partial_{i}\rho|\beta({\partial_{j}\nu_{\sf{M}}-\partial_{j}a(\rho)}))-(\tau^{\Gamma}_{i}|\frac{\partial_{j}\beta}{\beta}\nu_{\Gamma})\\ &=\beta\{{l_{ij}}+\rho(L_{\sf{M}}^{\mathcal{E}}\tau_{i}|\partial_{j}\nu_{\sf{M}})+(\tau_{i}|\partial_{j}(\nabla_{\sf{M}}\rho))+((I-\rho{L_{\sf{M}}^{\mathcal{E}}})\tau_{i}|\partial_{j}[(I-\rho{L_{\sf{M}}^{\mathcal{E}}})^{-1}]\nabla_{\sf{M}}\rho)\\ &\quad +\partial_{i}\rho(\nu_{\sf{M}}|\partial_{j}[(I-\rho{L_{\sf{M}}^{\mathcal{E}}})^{-1}]\nabla_{\sf{M}}\rho)+\partial_{i}\rho(\nu_{\sf{M}}|(I-\rho{L_{\sf{M}}^{\mathcal{E}}})^{-1}[\partial_{j}(\nabla_{\sf{M}}\rho)])\}\\ &=\beta\{{l_{ij}}+\rho{l_{ik}(\tau^{k}|\partial_{j}\nu_{\sf{M}})}+(\tau_{i}|\partial_{j}(\nabla_{\sf{M}}\rho))+(\tau_{i}|\partial_{j}(\rho{L_{\sf{M}}^{\mathcal{E}}})(I-\rho{L_{\sf{M}}^{\mathcal{E}}})^{-1}\nabla_{\sf{M}}\rho)\\ &\quad +\partial_{i}\rho(\nu_{\sf{M}}|\partial_{j}(\rho{L_{\sf{M}}^{\mathcal{E}}})(I-\rho{L_{\sf{M}}^{\mathcal{E}}})^{-1}\nabla_{\sf{M}}\rho)+\partial_{i}\rho(\nu_{\sf{M}}|\partial_{j}(\nabla_{\sf{M}}\rho))\}\\ &=\beta[{l_{ij}}-{l_{ik}}{l^{k}_{j}}\rho+{\partial_{ij}\rho}-\Gamma^{k}_{ij}\partial_{k}\rho +r_k^l(\rho)(\partial_{j}l_{i}^{k}+\Gamma^{k}_{jh}l_{i}^{h}-\Gamma^{h}_{ij}l_{h}^{k})\rho\partial_{l}\rho \\ &\quad +r_k^l(\rho)l^{k}_{i}\partial_{j}\rho\partial_{l}\rho +r_k^l(\rho)l^{h}_{j}l^{k}_{h}\rho\partial_{i}\rho\partial_{l}\rho +{l^{k}_{j}}\partial_{i}\rho\partial_{k}\rho]. \end{align*} Here we have used \eqref{derivative of WT} and the following facts: \begin{itemize} \item $(\nu_{\sf{M}}|\partial_j\nu_{\sf{M}})=0$. \item $(\tau^{\Gamma}_{i}|\nu_{\Gamma})=0$. \item $\partial_j \nu_{\sf{M}}=-l_{ij}\tau^i$. \item $(I-\rho{L_{\sf{M}}^{\mathcal{E}}})^{-1}\nu_{\sf{M}}=\nu_{\sf{M}}$. \item $\partial_j a(\rho)=(I-\rho{L_{\sf{M}}^{\mathcal{E}}})^{-1}\partial_j(\nabla_{\sf{M}}\rho)+\partial_j[(I-\rho{L_{\sf{M}}^{\mathcal{E}}})^{-1}]\nabla_{\sf{M}}\rho$. \item $\partial_j[(I-\rho{L_{\sf{M}}^{\mathcal{E}}})^{-1}]=(I-\rho{L_{\sf{M}}^{\mathcal{E}}})^{-1} \partial_j(\rho{L_{\sf{M}}^{\mathcal{E}}}) (I-\rho{L_{\sf{M}}^{\mathcal{E}}})^{-1}$. \end{itemize} Therefore, $ {\det}(L^{\Gamma})$ can be expressed in every local chart as \[ {\det}(L^{\Gamma})=\beta^{2}(\rho)\frac{P^{\Gamma}(\rho,\partial_{j}\rho, \partial_{ij}\rho)}{Q^{\Gamma}(\rho)}. \] Here $P^{\Gamma}(\rho,\partial_{j}\rho,\partial_{ij}\rho)$ is a polynomial in $\rho$ and its derivatives up to second order with real analytic coefficients. Moreover, $Q^{\Gamma}(\rho)$ is a polynomial in $\rho$ with real analytic coefficients. In particular, we have $Q^{\Gamma}\neq{0}$. In view of the above computations, within every local chart $K_{\rho}= {\det}[G^{-1}_{\Gamma}(\rho)L^{\Gamma}]$ can be expressed locally as \begin{equation} \label{Gaussian curvature} K_{\rho}=\beta^{2}(\rho)\frac{P^{\Gamma}(\rho,\partial_{j}\rho, \partial_{ij}\rho)}{ {\det}(G^{\Gamma}(\rho)) Q^{\Gamma}(\rho)}. \end{equation} As a straightforward conclusion of the above computation, we obtain an explicit expression for $H_{\rho}:=\Psi^{\ast}_{\rho}H_{\Gamma_{\rho}}$: \begin{equation} \label{mean curvature} \begin{aligned} 2 H_{\rho} &=g^{ij}_{\Gamma}l^{\Gamma}_{ij}\\ &=\beta(\rho){g^{ij}_{\Gamma}}[{l_{ij}}-{l_{ik}}{l^{k}_{j}}\rho+{\partial_{ij}\rho}-\Gamma^{k}_{ij}\partial_{k}\rho +r_k^l(\rho)l^{k}_{i}\partial_{j}\rho\partial_{l}\rho \\ &\quad +r_k^l(\rho)(\partial_{j}l_{i}^{k}+\Gamma^{k}_{jh}l_{i}^{h} -\Gamma^{h}_{ij}l_{h}^{k})\rho\partial_{l}\rho +r_k^l(\rho)l^{h}_{j}l^{k}_{h}\rho\partial_{i}\rho\partial_{l}\rho +{l^{k}_{j}}\partial_{i}\rho\partial_{k}\rho]{.} \end{aligned} \end{equation} The reader may also find a different global expression for $H_{\rho}$ in \cite[formula~(32)]{MCH}. We can decompose $H_{\rho}$ into $H_{\rho}=P_1(\rho)\rho+F_1(\rho)$: \[ F_1(\rho)=\frac{\beta(\rho)}{2} {g^{ij}_{\Gamma}}({l_{ij}} -{l_{ik}}{l^{k}_{j}}\rho)=\frac{\beta(\rho)}{2} \operatorname{Tr}[G^{-1}_{\Gamma}(\rho)(L^{\sf{M}}-\rho{L^{\sf{M}}}L_{\sf{M}})], \] where $\operatorname{Tr}(\cdot)$ denotes the trace operator, and \begin{align*} P_1(\rho)&=\frac{\beta(\rho)}{2}\Big\{{g^{ij}_{\Gamma}}\partial_{ij} +{g^{ij}_{\Gamma}}({l^{k}_{j}}\partial_{i}\rho-\Gamma^{k}_{ij})\partial_{k} \\ &\quad +{g^{ij}_{\Gamma}}[r_k^l(\rho)l^{k}_{i}\partial_{j}\rho + r_k^l(\rho)(\partial_{j}l_{i}^{k}+\Gamma^{k}_{jh}l_{i}^{h} -\Gamma^{h}_{ij}l_{h}^{k})\rho +r_k^l(\rho)l^{h}_{j}l^{k}_{h}\rho\partial_{i}\rho ]\partial_{l}\Big\} \end{align*} in every local chart. Note that $\operatorname{Tr}[G^{-1}_{\Gamma}(\rho)L^{\sf{M}}]$ changes like $H_{\sf{M}}$ under transition maps and thus is invariant. Analogously, we can check that $F_1$ is a well-defined global operator. Hence so is $P_1(\rho)$. In addition, it is a well-known fact that $\Psi_{\rho}^{\ast}\Delta_{\Gamma_{\rho}}=\Delta_{\rho}\Psi_{\rho}^{\ast}$, where $\Delta_{\Gamma_{\rho}}$ and $\Delta_{\rho}$ are the Laplace-Beltrami operators on $(\Gamma_{\rho},g_{\Gamma})$ and $({\sf{M}},\sigma(\rho))$, respectively. Here $\sigma(\rho):=\Psi_{\rho}^{\ast}g_{\Gamma}$ stands for the pull-back metric of $g_{\Gamma}$ on ${\sf{M}}$ by $\Psi_{\rho}$. Then in every local chart, the Laplace-Beltrami operator $\Delta_{\rho}$ can be expressed as \begin{equation} \label{Laplacian} \Delta_{\rho}=\sigma^{jk}(\rho)(\partial_j\partial_k -\gamma^i_{jk}(\rho)\partial_i). \end{equation} Here $\sigma^{jk}(\rho)$ are the components of the induced metric $\sigma^{\ast}(\rho)$ of $\sigma(\rho)$ on the cotangent bundle. Note that $\sigma^{jk}(\rho)$ involves the derivatives of $\rho$ merely up to order one. $\gamma^i_{jk}(\rho)$ are the corresponding Christoffel symbols of $\sigma(\rho)$, which contain the derivatives of $\rho$ up to second order. There exists a global operator $R(\rho)\in{\mathcal{L}(h^{3+\alpha}({\sf{M}}),E_0)}$ such that $R(\cdot)$ is well defined on $\mho$ and \[ R(\rho)\rho=\frac{1}{2\beta(\rho)}\Delta_{\rho} [\beta(\rho)\operatorname{Tr}(G^{-1}_{\Gamma}(\rho)L^{\sf{M}})] -\frac{\rho}{2\beta(\rho)}\Delta_{\rho}[\beta(\rho) \operatorname{Tr}(G^{-1}_{\Gamma}(\rho){L^{\sf{M}}}L_{\sf{M}})]. \] We set \begin{gather*} P(\rho):=\frac{1}{\beta(\rho)}\Delta_{\rho}P_1(\rho)+R(\rho),\quad \rho\in\mho, \\ F(\rho):=-\frac{1}{\beta(\rho)}\Delta_{\rho}F_1(\rho) +R(\rho)\rho-\frac{2}{\beta(\rho)}H_{\rho}(H_{\rho}^{2}-K_{\rho}),\quad \rho\in\mho\cap{h^{3+\alpha}({\sf{M}})}. \end{gather*} Note that third order derivatives of $\rho$ do not appear in $F(\rho)$. Hence it is actually well-defined on $\mho$. Based on the above discussion, these two maps enjoy the following smoothness properties: \[ P\in{C}^{\omega}(\mho,\mathcal{L}(E_1,E_0)),\quad F\in{C}^{\omega}(\mho,E_0). \] Here $\omega$ is the symbol for real analyticity. Please refer to \cite[Appendix]{URMGSYS} for a proof. \begin{definition} \label{differential operators} \rm Let $l\in{\mathbb{N}}_0$. A linear operator $\mathcal{A}:\mathcal{D}({\sf{M}})\to \mathbb{R}({\sf{M}})$ is called a linear differential operator of order $l$ on ${\sf{M}}$ if in every local chart $({\sf{O}_{\kappa}},\varphi_{\kappa})$, there exists some linear differential operator \[ \mathcal{A}_{\kappa}=\sum_{|\alpha|\leq{l}}a^{\kappa}_{\alpha}\partial^{\alpha} \] with $a^{\kappa}_{\alpha}\in \mathbb{R}^{\mathbb{B}^2}$ defined on ${\mathbb{B}^2}$ such that for any $u\in\mathcal{D}({\sf{M}})$ it holds \[ \psi_{\kappa}^{\ast}(\mathcal{A}u)=\mathcal{A}_{\kappa}(\psi_{\kappa}^{\ast}u) \] Moreover, at least one of the $\mathcal{A}_{\kappa}$'s is of order $l$. In particular, when $l=0$, $\mathcal{A}u=au$ for some $a\in \mathbb{R}^{\sf{M}}$. \end{definition} By the above definition, $P(\rho)$ is a fourth order linear differential operator on ${\sf{M}}$ for each $\rho\in\mho$. In every local chart $({\sf{O}_{\kappa}},\varphi_{\kappa})$, the principal part of the local expression of $P(\rho)$ can be written as \[ P^{\pi}_{\kappa}(\rho):=\frac{1}{2} \sigma^{kl}(\rho) g^{ij}_{\Gamma}\partial_{ijkl}. \] Given $\xi\in{T^{\ast}{\sf{M}}}$, we estimate the symbol of $P^{\pi}_{\kappa}(\rho)$ as follows. \[ P^{\pi}_{\kappa}(\rho)(\xi)=\frac{1}{2}\sigma^{\ast}(\rho)(\xi,\xi) g_{\Gamma}^{\ast}(\xi,\xi)\geq{c|\xi|^4} \] for some $c>0$, and $g_{\Gamma}^{\ast}$ denotes the induced metric of $g_{\Gamma}$ on the cotangent bundle of ${\sf{M}}$. Hence, $P(\rho)$ is a normally elliptic fourth order operator acting on functions over ${\sf{M}}$ for each $\rho\in\mho$. By \cite[Theorem 3.4]{URMGSYS}, $P(\rho)\in\mathcal{H}(E_1,E_0)$, namely, $-P(\rho)$ generates an analytic semigroup on $E_0$ with $D(-P(\rho))=E_1$, $\rho\in\mho$. Now the Willmore flow \eqref{original eq 1.1} can be rewritten as \begin{equation} \label{transformed eq 1.1} \begin{gathered} \rho_t+P(\rho)\rho=F(\rho),\\ \rho(0)=\rho_0, \end{gathered} \end{equation} where $\rho_0\in\mho$. See \cite{CMA,SDF,GSWF} for related work. Applying \cite[Theorem 4.1]{MCS}, the existence and regularity result in \cite{GSWF} can be restated as follows. \begin{theorem}[{\cite[Theorem 1.1]{GSWF}}] \label{well-posedness} Suppose that $\rho_{0}\in\mho$. Then equation \eqref{transformed eq 1.1} has a unique solution $\rho$ in the interval of maximal existence $J(\rho_0):=[0,T(\rho_0))$ for some $T(\rho_0)>0$ such that \[ \rho\in{C^1_{\frac{1}{2}}(J(\rho_0),E_0)\cap{C_{\frac{1}{2}}(J(\rho_0),E_1)} \cap{C(J(\rho_0),\mho)}\cap{C^{\frac{1}{2}-\beta_0}(J(\rho_0),E_{\beta_0})}} \] for any $\beta_0\in[0,\frac{1}{2}]$. Moreover, each hypersurface $\Gamma(t)$ is of class $C^{\infty}$ for $t\in\dot{J}(\rho_0)$. \end{theorem} \section{Parameter-dependent diffeomorphisms} The main purpose of the last two sections is to show that the classical solution obtained in Theorem \ref{well-posedness} is in fact real analytic jointly in time and space. To this end, I will construct a family of parameter-dependent diffeomorphisms acting on functions over ${\sf{M}}$. Because the construction applies to manifolds of arbitrary dimensions, in this section we assume that ${\sf{M}}$ is a m-dimensional manifold with the properties imposed in Section 1. For a given point $p\in{\sf{M}}$, we choose a normalized atlas $\mathfrak{K}$ for ${\sf{M}}$ such that $\varphi_{1}(p)=0\in\mathbb{R}^m$. Choose several open subsets $B_{i}$ in ${\mathbb{B}^m}$, the open unit ball centered at the origin in $\mathbb{R}^m$, in such a manner that \begin{itemize} \item $B_{i}:=\mathbb{B}^{m}(0,i\varepsilon_{0})$, for $i=1,2,3$ and some $\varepsilon_{0}>0$. \item ${B_{3}}\subset\subset{B_{4}}\subset\subset{\mathbb{B}^m}$. \end{itemize} Next, We further select two cut-off functions on ${\mathbb{B}^m}$: \begin{itemize} \item $\chi\in{\mathcal{D}(B_{2},[0,1])}$ such that $\chi|_{\overline{B}_{1}}\equiv{1}$. We write $\chi_{\kappa}=\varphi^{\ast}_{\kappa}\chi$. \item $\zeta\in\mathcal{D}(B_{4},[0,1])$ such that $\zeta|_{\overline{B}_{3}}\equiv{1}$. We write $\zeta_{\kappa}=\varphi^{\ast}_{\kappa}\zeta$. \end{itemize} We define a re-scaled translation on ${\mathbb{B}^m}$ for any ${\mu}\in{\mathbb{B}(0,r)}\subset\mathbb{R}^m$ with $r$ sufficiently small: \[ \theta_{\mu}(x):=x+\chi{(x)}\mu,\quad x\in{\mathbb{B}^m}. \] This localization technique in Euclidean spaces is first introduced in \cite{ARP} by Escher, Pr\"uss and Simonett to establish regularity for solutions to parabolic and elliptic equations. Given a function $v\in{L_{1,loc}({\mathbb{B}^m})}$, its pull-back and push-forward induced by $\theta_{\mu}$ are defined as \[ {\theta^{\ast}_{\mu}}v:=v\circ{\theta_{\mu}}, \quad \theta^{\mu}_{\ast}v:=v\circ{\theta_{\mu}^{-1}}. \] The diffeomorphism $\theta_{\mu}$ induces a transformation $\Theta_{\mu}$ on ${\sf{M}}$ by \[ \Theta_{\mu}(q)= \begin{cases} \psi_1(\theta_{\mu}(\varphi_1(q))) & q\in{\sf{O}}_1,\\ q & q\notin{\sf{O}}_1. \end{cases} \] It can be shown that $\Theta_{\mu} \in \operatorname{Diff}^{\infty}({\sf{M}})$ for $\mu\in{\mathbb{B}(0,r)}$ with sufficiently small $r>0$. See \cite{YS1P} for details. For any $u\in{L_{1,\rm loc}({\sf{M}})}$, we can define its pull-back and push-forward induced by $\Theta_{\mu}$ analogously as \[ {{\Theta}^{\ast}_{\mu}}u:=u\circ{\Theta_{\mu}}, \quad \Theta^{\mu}_{\ast}u:=u\circ{\Theta_{\mu}^{-1}}. \] We may find an explicit global expression for the transformation ${{\Theta}^{\ast}_{\mu}}$ on ${\sf{M}}$, \[ {{\Theta}^{\ast}_{\mu}}u={\varphi^{\ast}_{1}}{\theta^{\ast}_{\mu}}{\psi_{1}^{\ast}}({\zeta_{1}}u) +(1-{\zeta_{1}})u. \] Here and in the following it is understood that a partially defined and compactly supported function is automatically extended over the whole base manifold by identifying it to be zero outside its original domain. Likewise, we can express ${\Theta}^{\mu}_{\ast}$ as \[ {\Theta}^{\mu}_{\ast}={\varphi^{\ast}_{1}}{\theta^{\mu}_{\ast}} {\psi_{1}^{\ast}}({\zeta_{1}}u)+(1-{\zeta_{1}})u. \] Let $I=[0,T]$, $T>0$. Assuming that $J\subset \mathring{I}$ is an open interval and $t_{0}\in{J}$ is a fixed point, we choose $\varepsilon_{0}$ so small that $\mathbb{B}(t_{0},3\varepsilon_{0})\subset{J}$. Next we pick another auxiliary function \[ \xi\in\mathcal{D}(\mathbb{B}(t_{0},2\varepsilon_{0}),[0,1])\quad \text{with } \xi|_{\mathbb{B}(t_{0},\varepsilon_{0})}\equiv{1}. \] The above construction now engenders a parameter-dependent transformation in terms of the time variable: \[ \varrho_{\lambda}(t):=t+\xi(t)\lambda,\quad\text{for any $t\in{I}$ and $\lambda\in{\mathbb{R}}$}. \] Now we define a family of parameter-dependent transformations on $I\times{\sf{M}}$. Given a function $u:I\times{\sf{M}}\to {\mathbb{R}}$, we set \[ {{u}_{\lambda,\mu}}(t,\cdot):={{\Theta}^{\ast}_{\lambda,\mu}}u(t,\cdot):={T_\mu }(t){{\varrho}^{\ast}_{\lambda}}u(t,\cdot), \] where ${T_\mu }(t)={\Theta}^{\ast}_{\xi(t)\mu}$ and $(\lambda,\mu)\in{\mathbb{B}(0,r)}$. It is important to note that ${{u}_{\lambda,\mu}}(0,\cdot)=u(0,\cdot)$ for any $(\lambda,\mu)\in{\mathbb{B}(0,r)}$ and any function $u$. The importance of this family of parameter-dependent diffeomorphisms lies in the following theorems. Their proofs as well as additional properties of this technique can be found in \cite{YS1P}. \begin{theorem} \label{inverse} Let $k\in{\mathbb{N}}_{0}\cup\{\infty,\omega\}$. Suppose that $u\in{C(I\times{\sf{M}})}$. Then we have that $u\in{C^{k}(\mathring{I}\times{\sf{M}})}$ if and only if for any $(t_0,p)\in\mathring{I}\times{\sf{M}}$, there exists $r=r(t_0,p)>0$ and a corresponding family of parameter-dependent diffeomorphisms $\{{{\Theta}^{\ast}_{\lambda,\mu}}:(\lambda,\mu)\in\mathbb{B}(0,r)\}$ such that \[ [(\lambda,\mu)\mapsto{{\Theta}^{\ast}_{\lambda,\mu}}u]\in{C^{k}({\mathbb{B}(0,r)},C(I\times{\sf{M}}))}. \] \end{theorem} \begin{proposition} \label{time derivatives} Suppose that $u\in{\mathbb{E}_1}(I)$. Then $u_{\lambda,\mu} \in{\mathbb{E}_1}(I)$, and \[ \partial_t[u_{\lambda,\mu}]=(1+\xi'\lambda){{\Theta}^{\ast}_{\lambda,\mu}}u_t +B_{\lambda,\mu}(u_{\lambda,\mu}), \] where \[ [(\lambda,\mu)\mapsto{B}_{\lambda,\mu}]\in{C}^{\omega} ({\mathbb{B}(0,r)},C(I,\mathcal{L}(E_1,E_0))). \] Furthermore, $B_{\lambda,0}=0$. \end{proposition} \begin{proposition} \label{regularity of differential operators} Let $s\in (0,t)$ and $l\in\mathbb{N}_0$. Suppose that $\mathcal{A}$ is a linear differential operator of order $l$ on ${\sf{M}}$ satisfying $a^{\kappa}_{\alpha}\in{BC^t({\mathbb{B}^m})}$ and $a^{1}_{\alpha}\in{BC^t({\mathbb{B}^m})}\cap{C}^{\omega}(\sf{O})$ for some open subset $\sf{O}$ such that $B_3\subset\subset{\sf{O}}\subset\subset{\mathbb{B}^m}$. Then \[ [\mu\mapsto{T}_{\mu}\mathcal{A}T_{\mu}^{-1}] \in{C}^{\omega}({\mathbb{B}(0,r)},C(I,\mathcal{L}(h^{s+l}({\sf{M}}),h^{s}({\sf{M}})))). \] \end{proposition} \begin{proposition} \label{regularity of transformed functions} Let $s> 0$. Suppose that $u\in {C^{\omega}(\psi_1(\sf{O}))\cap \mathit{h^{s}}({\sf{M}})}$, where $\sf{O}$ is defined in Proposition \ref{regularity of differential operators}. Then \[ [\mu\mapsto{T_\mu }u]\in{C^{\omega}({\mathbb{B}(0,r)},C(I,{h}^{s}({\sf{M}})))}. \] \end{proposition} \section{Real analyticity} By setting $G(\rho):=P(\rho)\rho-F(\rho)$, we may rewrite \eqref{transformed eq 1.1} as \begin{equation} \label{final eq 1.1} \begin{gathered} \rho_t+G(\rho)=0,\\ \rho(0)=\rho_0. \end{gathered} \end{equation} \begin{theorem} \label{main theorem 2} Let $0<\alpha<1$. Suppose that $\rho_{0}\in \mho$. Then \eqref{final eq 1.1} has a unique local solution $\rho$ in the interval of maximal existence $J(\rho_{0})$ such that \[ \rho\in{C^{\omega}(\dot{J}(\rho_{0})\times{\sf{M}})}. \] \end{theorem} \begin{proof} The key steps of the proof are indicated here, while the details can be found in \cite{YS1P}. For any $(t_0,p)\in\dot{J}(\rho_0)\times{\sf{M}}$ and sufficiently small $r>0$, a family of parameter-dependent diffeomorphisms ${{\Theta}^{\ast}_{\lambda,\mu}}$ can be defined for $(\lambda,\mu)\in{\mathbb{B}(0,r)}$. Henceforth, we always use the notation $\rho$ exclusively for the solution to \eqref{transformed eq 1.1} and hence to \eqref{final eq 1.1}. Set $u:={\rho}_{\lambda,\mu}$. Then as a consequence of Proposition \ref{time derivatives}, $u$ satisfies the equation \begin{align*} u_t&=\partial_t[{\rho}_{\lambda,\mu}]=(1+\xi'\lambda){{\Theta}^{\ast}_{\lambda,\mu}}\rho_t+{B}_{\lambda,\mu}(u)\\ &=-(1+\xi'\lambda){{\Theta}^{\ast}_{\lambda,\mu}}G(\rho)+{B}_{\lambda,\mu}(u)\\ &=-(1+\xi'\lambda){T_\mu }G({{\varrho}^{\ast}_{\lambda}}\rho)+{B}_{\lambda,\mu}(u)\\ &=-(1+\xi'\lambda){T_\mu }G({T}^{-1}_{\mu}u)+{B}_{\lambda,\mu}(u):=-H_{\lambda,\mu}(u). \end{align*} Select $I:[\varepsilon,T]\subset\subset{\dot{J}(\rho_{0})}$ such that $t_{0}\in\mathring{I}$ and $\mathbb{B}(t_0,3\varepsilon_0) \subset\subset\mathring{I}$. Then we define ${\mathbb{E}_0}(I)$ and ${\mathbb{E}_1}(I)$ as in Section~1 by moving the initial point from $0$ to $\varepsilon$. Set \[ \mathbb{E}_1^a (I):=\{v\in{\mathbb{E}_1}(I):\|v\|_{\infty}