Approximate mean curvature flows of a general varifold, and their limit spacetime Brakke flow
Blanche Buet, Gian Paolo Leonardi, Simon Masnou, Abdelmouksit Sagueni
TL;DR
The paper develops a general method to approximate mean curvature flow for very general initial data encoded as varifolds, including point clouds, by an explicit time-discrete scheme using push-forwards with velocity given by an approximate mean curvature $h_{\varepsilon}$. It proves existence and uniqueness of a limit flow $V_{\varepsilon}(t)$ as the time-step vanishes, along with stability, a Brakke-type inequality, and mass decay. By coupling with a spacetime measure $dt\otimes V_{\varepsilon}(t)$ and passing to the limit $\varepsilon\to0$, the authors obtain a spacetime measure $\lambda$ that satisfies a spacetime Brakke flow inequality under a rectifiability assumption, yielding $L^{2}$ bounds on the spacetime mean curvature and mass decay. The framework applies to arbitrary dimensions and codimensions, including discrete data like point clouds, and provides a robust bridge between Brakke-type weak flows and spacetime formulations. This has potential implications for numerical analysis of geometric flows on unstructured data and for theoretical understanding of weak mean curvature flow in broad settings.
Abstract
We propose a construction of mean curvature flows by approximation for very general initial data, in the spirit of the works of Brakke and of Kim & Tonegawa based on the theory of varifolds. Given a general varifold, we construct by iterated push-forwards an approximate time-discrete mean curvature flow depending on both a given time step and an approximation parameter. We show that, as the time step tends to $0$, this time-discrete flow converges to a unique limit flow, which we call the approximate mean curvature flow. An interesting feature of our approach is its generality, as it provides an approximate notion of mean curvature flow for very general structures of any dimension and codimension, ranging from continuous surfaces to discrete point clouds. We prove that our approximate mean curvature flow satisfies several properties: stability, uniqueness, Brakke-type equality, mass decay. By coupling this approximate flow with the canonical time measure, we prove convergence, as the approximation parameter tends to $0$, to a spacetime limit measure whose generalized mean curvature is bounded. Under an additional rectifiability assumption, we further prove that this limit measure is a spacetime Brakke flow.