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Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus

Daniele De Gennaro, Antonia Diana, Andrea Kubin, Anna Kubin

TL;DR

This work analyzes two volume-preserving geometric flows on the flat torus $\mathbb{T}^N$: the volume-preserving mean curvature flow with velocity $V_t=-\mathrm{H}_{E_t}+\bar{\mathrm{H}}_{E_t}$ and the surface diffusion flow with $V_t=\Delta_{E_t}\mathrm{H}_{E_t}$. For initial sets $E_0$ that are $C^{1,1}$-close to a strictly stable critical set $E$ of the perimeter, the authors prove global-in-time existence and exponential convergence to a translate $E+\tau$ of $E$ in $C^k$ for all $k$, leveraging a unified framework built on a quantitative Alexandrov-type inequality (DeKu) and a quantitative isoperimetric inequality (AFM). A key aspect is treating both flows as gradient flows of the perimeter functional, enabling decay estimates without high-derivative curvature energy control, and obtaining explicit exponential rates through a sharp Lojasiewicz-Simon-type inequality. Short-time existence is established via Schauder estimates for the corresponding quasilinear parabolic problems, with higher-order regularity controlled by time-weighted norms. The techniques generalize to other gradient-perimeter flows and provide a robust approach to stability analysis in geometric evolution problems on the flat torus.

Abstract

We prove that, in the flat torus and in any dimension, the volume-preserving mean curvature flow and the surface diffusion flow, starting $C^{1,1}-$close to a strictly stable critical set of the perimeter $E$, exist for all times and converge to a translate of $E$ exponentially fast as time goes to infinity.

Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus

TL;DR

This work analyzes two volume-preserving geometric flows on the flat torus : the volume-preserving mean curvature flow with velocity and the surface diffusion flow with . For initial sets that are -close to a strictly stable critical set of the perimeter, the authors prove global-in-time existence and exponential convergence to a translate of in for all , leveraging a unified framework built on a quantitative Alexandrov-type inequality (DeKu) and a quantitative isoperimetric inequality (AFM). A key aspect is treating both flows as gradient flows of the perimeter functional, enabling decay estimates without high-derivative curvature energy control, and obtaining explicit exponential rates through a sharp Lojasiewicz-Simon-type inequality. Short-time existence is established via Schauder estimates for the corresponding quasilinear parabolic problems, with higher-order regularity controlled by time-weighted norms. The techniques generalize to other gradient-perimeter flows and provide a robust approach to stability analysis in geometric evolution problems on the flat torus.

Abstract

We prove that, in the flat torus and in any dimension, the volume-preserving mean curvature flow and the surface diffusion flow, starting close to a strictly stable critical set of the perimeter , exist for all times and converge to a translate of exponentially fast as time goes to infinity.
Paper Structure (7 sections, 16 theorems, 147 equations)

This paper contains 7 sections, 16 theorems, 147 equations.

Table of Contents

  1. Preliminary results
  2. Short-time existence for the mean curvature flow
  3. Short-time existence for the surface diffusion flow
  4. Stability
  5. Stability of the volume preserving mean curvature flow
  6. Stability of the surface diffusion flow
  7. Sketch of a general proof of the Lemma \ref{['lemma est Jh']}

Key Result

Theorem 1

Let $E\subset \mathbb{T}^N$ be a strictly stable set and let $E_0 =E_{u_0} \subset \mathbb{T}^N$ be the normal deformation of $E$ induced by $u_0 \in C^{1,1}(\partial E)$ (see Definition normaldeformation) with $|E_0|=|E|$. There exists $\delta=\delta(E)>0$ such that if $\|u_0\|_{C^{1,1}(\partial E) Where with exponentially fast we mean that the sets $E_t$ can be written as normal deformations of

Theorems & Definitions (29)

  • Theorem 1
  • Definition 1.1
  • Definition 1.2
  • Lemma 1.3: AFM
  • Definition 1.4
  • Lemma 1.5
  • proof
  • Theorem 1.6: AFM
  • Theorem 1.7: DeKu
  • Remark 1.8
  • ...and 19 more