Inverse mean curvature evolution of entire graphs

Abstract

We study the evolution of strictly mean-convex entire graphs over \({{\mathbb {R}}}^n\) 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 \(t \rightarrow T\) 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.