Geometric flows with rough initial data

TL;DR

This paper proves the existence of global unique analytic solutions for several geometric flows starting from rough initial data, including Lipschitz graphs and bounded metrics close to Euclidean space.

Contribution

It establishes new existence and uniqueness results for geometric flows with less regular initial conditions than previously considered.

Findings

Global solutions for mean curvature, surface diffusion, and Willmore flows from Lipschitz initial data.

Existence of solutions to Ricci-DeTurck flow with initial metrics close to Euclidean.

Solutions to harmonic map flow with initial maps in small geodesic balls.

Abstract

We show the existence of a global unique and analytic solution for the mean curvature flow, the surface diffusion flow and the Willmore flow of entire graphs for Lipschitz initial data with small Lipschitz norm. We also show the existence of a global unique and analytic solution to the Ricci-DeTurck flow on euclidean space for bounded initial metrics which are close to the euclidean metric in $L^\infty$ and to the harmonic map flow for initial maps whose image is contained in a small geodesic ball.