The inverse mean curvature flow in warped cylinders of non-positive radial curvature

Abstract

We consider the inverse mean curvature flow in smooth Riemannian manifolds of the form $([R_0,\infty )\times S^n,\bar{g})$ with metric $\bar{g}=d{r}^2+{\vartheta }^2(r)\sigma$ and non-positive radial sectional curvature. We prove, that for initial mean-convex graphs over $S^n$ the flow exists for all times and remains a graph over $S^n$. Under weak further assumptions on the ambient manifold, we prove optimal decay of the gradient and that the flow leaves become umbilic exponentially fast. We prove optimal $C^2$ estimates in case that the ambient pinching improves.

29 pages. Thoroughly checked revised version. Fixed some computational errors and added another small technical assumption to make the main theorem work. Comments and discussions are very welcome

References in corpus (1)

• Pinching and asymptotical roundness for inverse curvature flows in Euclidean space

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