A class of inverse curvature flows for star-shaped hypersurfaces evolving in a cone
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 Hölder 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.
- Pub Date:
- April 2021
- Mathematics - Differential Geometry;
- 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. Two typos have been corrected in Theorem 1.1