Contracting convex hypersurfaces by functions of the mean curvature

2016/10/26 by Guo, Shunzi
#35B40 (Primary) #35K55 #53C44 #58J35 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1610.08209

Abstract

This paper concerns the evolution of a closed convex hypersurface in ℝⁿ⁺¹, in direction of its inner unit normal vector, where the speed is given by a smooth function depending only on the mean curvature, and satisfies some further restrictions, without requiring homogeneity. It is shown that the flow exists on a finite maximal interval, convexity is preserved and the hypersurfaces shrink down to a single point as the final time is approached. This result covers and generalises the corresponding result of Schulze \citeSch05 for the positive power mean curvature flow to a much larger possible class of flows by the functions depending only on the mean curvature.

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