A class of curvature flows expanded by support function and curvature function

Ding, Shanwei; Li, Guanghan

doi:10.1090/proc/15189

A class of curvature flows expanded by support function and curvature function
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by Shanwei Ding and Guanghan Li

Proc. Amer. Math. Soc. 148 (2020), 5331-5341

DOI: https://doi.org/10.1090/proc/15189

Published electronically: September 18, 2020

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Abstract:

In this paper, we consider a class of expanding flows of closed, smooth, uniformly convex hypersurfaces in Euclidean $\mathbb {R}^{n+1}$ with speed $u^\alpha f^\beta$ ($\alpha , \beta \in \mathbb {R}^1$), where $u$ is the support function of the hypersurface, $f$ is a smooth, symmetric, homogenous of degree one, positive function of the principal curvature radii of the hypersurface. If $\alpha \leqslant 0<\beta \leqslant 1-\alpha$, we prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalization, to a round sphere centered at the origin.

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