Jing Mao, Qiang Tu
Published 2021-04-18Version 1
Given a smooth convex cone in the Euclidean $(n+1)$-space ($n\geq2$), we consider strictly mean convex hypersurfaces with boundary which are star-shaped with respect to the center of the cone and which meet the cone perpendicularly. If those hypersurfaces inside the cone evolve by a class of inverse curvature flows, then, by using the convexity of the cone in the derivation of the gradient and H\"{o}lder estimates, we can prove that this evolution exists for all the time and the evolving hypersurfaces converge smoothly to a piece of a round sphere as time tends to infinity.
Comments: 17 pages. This paper was finished in January 2019 when the first author visited IST, University of Lisbon. We just put it on arXiv very recently (the status of two references has been updated). Several related works will also be put on arXiv soon. Comments are welcome
Round spheres are Hausdorff stable under small perturbation of entropy
Monotone quantities involving a weighted $σ_k$ integral along inverse curvature flows
Uniqueness Theorems of Self-Conformal Solutions to Inverse Curvature Flows
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