Inverse mean curvature evolution of entire graphs

TL;DR

This paper investigates the inverse mean curvature flow of entire graphs in Euclidean space, establishing global existence results for starshaped graphs and analyzing finite-time singularity formation in asymptotically conical convex graphs.

Contribution

It provides new existence and singularity results for inverse mean curvature flow of entire graphs, especially in the critical asymptotically conical case.

Findings

Global existence for starshaped entire graphs with superlinear growth

Finite-time convergence to a flat plane for asymptotically conical convex graphs

Flow's behavior influenced by asymptotic growth at infinity

Abstract

We study the evolution of strictly mean-convex entire graphs over $R^n$ by Inverse Mean Curvature flow. First we establish the global existence of starshaped entire graphs with superlinear growth at infinity. The main result in this work concerns the critical case of asymptotically conical entire convex graphs. In this case we show that there exists a time $ T < +\infty$, which depends on the growth at infinity of the initial data, such that the unique solution of the flow exists for all $t < T$. Moreover, as $t \to T$ the solution converges to a flat plane. Our techniques exploit the ultra-fast diffusion character of the fully-nonlinear flow, a property that implies that the asymptotic behavior at spatial infinity of our solution plays a crucial influence on the maximal time of existence, as such behavior propagates infinitely fast towards the interior.