Asymptotic convergence of evolving hypersurfaces

We consider a closed connected differentiable manifold $M^{n}$ of dimension $n\geq 1$ and $\psi:M^{n}\to\mathbb{R}^{n+1}$ a smooth immersion of $M^{n}$ in the Euclidean space $\mathbb{R}^{n+1}$. We shall usually omit the superscript $n$ denoting the dimension of $M$. For such an immersion, we always assume that $M$ is endowed with the metric tensor $g=\psi^{*}\langle\cdot,\cdot\rangle_{\mathbb{R}^{n+1}}$, induced by the immersion $\psi$ that we also sometimes simply denote as $\langle\cdot,\cdot\rangle$. The Levi–Civita connection of the Riemannian manifold $(M,g)$ is denoted by $\nabla$ and the associated volume measure by $\mu$. Then, for $m\in\mathbb{N}$ with $m\geq 1$, we consider the functional

on the smooth immersions $\psi:M^{n}\to\mathbb{R}^{n+1}$, where $\nu$ is a (locally defined) unit normal vector field along $\psi$. Let us specify that in the above definition, if $\nu=\nu^{\alpha}e_{\alpha}$ and $e_{\alpha}$ is the standard basis in $\mathbb{R}^{n+1}$, we mean $|\nabla^{m}\nu|^{2}\coloneqq\sum_{\alpha=1}^{n+1}|\nabla^{m}\nu^{\alpha}|^{2}$. Notice that $\mathcal{F}_{m}$ is independent of the local choice of the unit normal $\nu$ and it is well-defined without further hypotheses on $M$. However, by the discussion below in Remark 1.2, we shall always assume without loss of generality that $M$ is orientable and that a global choice of unit normal field $\nu$ is understood.

We observe that in case $n=m=1$, we recognize the well-known elastic energy of closed curves $\psi:\mathbb{S}^{1}\to\mathbb{R}^{2}$, i.e., $\mathcal{F}_{1}(\psi)=\int_{\mathbb{S}^{1}}1+|k|^{2}ds$, where $k$ is the curvature of $\psi$. If instead $n=2$, $m=1$, then one has $|\nabla\nu|^{2}=|{\mathrm{A}}|^{2}$, where ${\mathrm{A}}$ is the second fundamental form (see (1.1)), and then $\mathcal{F}_{1}$ yields the sum of the area and of (an equivalent form of) the Willmore energy of an immersed surface $\psi:M^{2}\to\mathbb{R}^{3}$. Let us also notice that if instead $n=m=2$, then $\mathcal{F}_{2}(\psi)=\int_{M}1+|\nabla{\mathrm{A}}|^{2}+\mathfrak{p}_{0}({\mathrm{A}},{\mathrm{A}},{\mathrm{A}},{\mathrm{A}})d\mu$ for any given immersion $\psi:M^{2}\to\mathbb{R}^{3}$, where $\mathfrak{p}_{0}$ is some polynomial as in (1.3) below.

By the formula for the first variation of $\mathcal{F}_{m}$ (see Theorem 2.1), one can prove that the associated $L^{2}$–gradient flow is defined by an evolution equation

for a smooth map $\varphi:M\times[0,T)\to\mathbb{R}^{n+1}$ (where $\varphi_{t}=\varphi(\cdot,t):M\to\mathbb{R}^{n+1}$ describes the moving hypersurface and $\nu_{t}$ is its unit normal vector field), which turns out to be a (quasilinear and degenerate) parabolic system of PDEs.
If $m>\lfloor n/2\rfloor$, the study carried out in [17] shows that for every initial smooth immersed hypersurface $\varphi_{0}:M\to\mathbb{R}^{n+1}$, there exists a unique smooth solution $\varphi_{t}$ with initial datum $\varphi_{0}$, defined for all positive times; moreover, $\varphi_{t}$ sub–converges to a critical point $\varphi_{\infty}:M\to\mathbb{R}^{n+1}$ of the functional $\mathcal{F}_{m}$, that is, such that ${\mathrm{E}}_{m}(\varphi_{\infty})=0$ (see Theorem 4.3). By sub–convergence we mean that for some sequence of times $t_{j}\to+\infty$, the sequence $\varphi_{t_{j}}$ smoothly converges to $\varphi_{\infty}$, up to diffeomorphisms and translations in $\mathbb{R}^{n+1}$. More precisely, there exist a sequence of smooth diffeomorphisms $\sigma_{j}:M\to M$ and a sequence of points $p_{j}\in\mathbb{R}^{n+1}$ such that the sequence of immersions $\varphi_{t_{j}}\circ\sigma_{j}-p_{j}$ converge to $\varphi_{\infty}$ in $C^{k}(M)$, for any $k\in\mathbb{N}$. From such a sub–convergence result it is anyway not possible to immediately deduce that the flow fully converges, i.e., that there exists the full limit of $\varphi_{t}$ as $t\to+\infty$ in $C^{k}(M)$ for any $k$ (up to diffeomorphisms). Actually, the sub–convergence of the flow does not even guarantee that the limits of the flow along different diverging sequences of times coincide. Moreover, as the evolution equations involved here are of order greater or equal than four with respect to the parametrization, it is not even possible to conclude that the flow stays in a compact set of $\mathbb{R}^{n+1}$ for all times by means of comparison arguments, as maximum principles are not applicable.

In this work we address this issue, that is, we prove that the gradient flow of $\mathcal{F}_{m}$ does actually converge, for any initial hypersurface. Our main result is the following theorem.

We remark that the assumption $m>\lfloor n/2\rfloor$ is sharp, in fact if $m\leq\lfloor n/2\rfloor$ then one gets flows that may develop singularities in finite time.

A relevant motivation for the study of the gradient flow of the functionals $\mathcal{F}_{m}$ goes back to Ennio De Giorgi. In one of his last papers, he conjectured that any compact $n$–dimensional hypersurface in $\mathbb{R}^{n+1}$, evolving by the gradient flow of certain functionals depending on sufficiently high derivatives of the curvature does not develop singularities during the flow (see [10] and [11, Section 5, Conjecture 2] for an English translation, see also [17, Section 9]). This result was central in his program to approximate singular geometric flows, as the mean curvature flow, with sequences of smooth ones (see [17, Sec. 9] and [2] for a result in this direction). The functionals $\mathcal{F}_{m}$ are strictly related to the ones proposed by De Giorgi since, roughly speaking, the derivative of the normal field yields the curvature of $M$ (see (1.1)). Though not exactly the same, the energies $\mathcal{F}_{m}$ can then play the same role in the approximation process he suggested and the analysis of the asymptotic behavior of their gradient flow is another step in understanding such process.

The main tool in the proof of Theorem 1.1 is a Łojasiewicz–Simon gradient inequality for the functional $\mathcal{F}_{m}$ (see Corollary 4.2). Such an estimate bounds a less–than–$1/2$ power of the difference in “energy” (the value of the functional) between a critical point and a point sufficiently close to it in terms of a suitable norm of the first variation of the functional. The use of this kind of inequalities in the study of the convergence of parabolic equations of gradient–type goes back to Łojasiewicz [15, 16] and to the seminal paper of Simon [23]. More recently, useful sufficient hypotheses implying a Łojasiewicz–Simon gradient inequality have been derived in [4] (see also [13] for several recent generalizations). Building on the abstract tools developed in [4], a first recent application of the inequality to get convergence of an extrinsic geometric flow is contained in [5], where the authors investigate the Willmore flow of surfaces in neighborhoods of critical points. In the last years and in the context of higher order geometric gradient flows, the Łojasiewicz–Simon inequality appeared as a tool for “promoting” the sub–convergence of a flow to its full convergence. As applications of this method we mention [9], in which it is proved the full convergence of the elastic flow of open clamped curves, and [20], in which the sub–convergence of the $p$–elastic flow of closed curves on Riemannian manifolds is shown to imply the full convergence. The analysis in [20] led to a further simplification and deeper understanding of the method, which is exposed in [19]. In this work we essentially generalize the strategy employed in [19] to the gradient flows of the functionals $\mathcal{F}_{m}$. Moreover we tried to keep most of the arguments as general as possible, in order that the method could be possibly applied also to other geometric gradient flows, also in the context of extrinsic geometric flows in higher codimension possibly in Riemannian manifolds.

In the broad framework of geometric flows, Łojasiewicz–Simon gradient inequalities found many other notable applications. For example, the study of singularities of mean curvature flow can be reconducted to the study of the smooth convergence of a suitable extrinsic geometric flow. Smooth convergence of such flow has been proved exploiting Łojasiewicz–Simon inequalities in some relevant particular cases in [22, 8, 6]. Let us also mention [7], where a classification of ancient solutions to a family of geometric flows in Riemannian manifolds is derived. Łojasiewicz–Simon inequalities have been employed also in the context of intrinsic geometric flows. We refer, for instance, to the study of the rate of convergence of Yamabe flows in [3], or to the deep investigation on the Yang–Mills flow contained in [12] (see also references therein).

Let us recall the first variation formula for the functional $\mathcal{F}_{m}$.

The next lemma states the evolution formulae for the geometric quantities that we need in the computation of the second variation of the functional $\mathcal{F}_{m}$.

We can now compute the second variation of $\mathcal{F}_{m}$.

It follows that, by polarization, we can define the bilinear form

for any pair of smooth functions $f_{1},f_{2}:M\to\mathbb{R}$ and $\Omega$ is as in Theorem 2.3.

Suppose that $\varphi:M\to\mathbb{R}^{n+1}$ is a smooth critical point of $\mathcal{F}_{m}$, i.e., ${\mathrm{E}}_{m}(\varphi)=0$. The formula for the second variation given above shows that $(\delta^{2}\mathcal{F}_{m})_{\varphi}(f_{1},f_{2})$ is well-defined for $f_{1}\in W^{2m+2,2}(M,g)$ and $f_{2}\in L^{2}(\mu)$. This means that

for any $f\in W^{2m+2,2}(M,g)$ and it is well-defined the map

We are going to exploit the theory of Fredholm operators between Banach spaces. For definitions and results on the subject we refer the reader to [14, Section 19.1]. We recall that if $T:V_{1}\to V_{2}$ is a Fredholm operator between Banach spaces, its index is defined to be the integer number

where $\dim$ denotes the dimension of a finite dimensional vector space.

In order to prove Proposition 3.1 we need the following commutation rule.

We are now ready to prove Proposition 3.1. A relevant property about Fredholm operators that we are going to use is the following. If $T:V_{1}\to V_{2}$ is a Fredholm operator between Banach spaces and $K:V_{1}\to V_{2}$ is a compact operator, then $T+K$ is Fredholm and ${\rm index}(T+K)={\rm index}\,T$ (see [14, Corollary 19.1.8]).