Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone

2024/08/15 by Gao, Ya, Mao, Jing
#35K10 #53E10 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2408.07949

Abstract

For a given smooth convex cone in the Euclidean (n+1)-space ℝⁿ⁺¹ which is centered at the origin, we investigate the evolution of strictly mean convex hypersurfaces, which are star-shaped with respect to the center of the cone and which meet the cone perpendicularly, along an inverse curvature flow with the speed equal to (f(r)H)^-1, where f is a positive function of the radial distance parameter r and H is the mean curvature of the evolving hypersurfaces. The evolution of those hypersurfaces inside the cone yields a fully nonlinear parabolic Neumann problem. Under suitable constraints on the first and the second derivatives of the radial function f, we can prove the long-time existence of this flow, and moreover the evolving hypersurfaces converge smoothly to a piece of the round sphere.

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