Lifespan theorem for simple constrained surface diffusion flows

TL;DR

This paper establishes a Lifespan Theorem for constrained surface diffusion flows of hypersurfaces in 3D and 4D, providing bounds on existence time and curvature concentration based on initial conditions.

Contribution

It introduces an improved Lifespan Theorem for constrained surface diffusion flows, removing area assumptions and linking lifespan to initial curvature concentration.

Findings

Provides a positive lower bound on solution existence time.

Establishes an upper bound on total curvature during the lifespan.

Improves previous results by eliminating area assumptions.

Abstract

We consider closed immersed hypersurfaces in $\R^3$ and $\R^4$ evolving by a special class of constrained surface diffusion flows. This class of constrained flows includes the classical surface diffusion flow. In this paper we present a Lifespan Theorem for these flows, which gives a positive lower bound on the time for which a smooth solution exists, and a small upper bound on the total curvature during this time. The hypothesis of the theorem is that the surface is not already singular in terms of concentration of curvature. This turns out to be a deep property of the initial manifold, as the lower bound on maximal time obtained depends precisely upon the concentration of curvature of the initial manifold in $L^2$ for $M^2$ immersed in $R^3$ and additionally on the concentration in $L^3$ for $M^3$ immersed in $R^4$. This is stronger than a previous result on a different class of constrained surface diffusion flows, as here we obtain an improved lower bound on maximal time, a better estimate during this period, and eliminate any assumption on the area of the evolving hypersurface.