Communications in Analysis and Geometry

Volume 29 (2021)

Number 2

Self-expanders to inverse curvature flows by homogeneous functions

Pages: 329 – 362

DOI: https://dx.doi.org/10.4310/CAG.2021.v29.n2.a3

Authors

Tsz-Kiu Aaron Chow (Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong)

Ka-Wing Chow (Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong)

Frederick Tsz-Ho Fong (Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong)

Abstract

In this paper, we study self-expanding solutions to a large class of parabolic inverse curvature flows by homogeneous symmetric functions of principal curvatures in Euclidean spaces. These flows include the inverse mean curvature flow and many nonlinear flows in the literature.

We first show that the only compact self-expanders to any of these flows are round spheres. Secondly, we show that complete non-compact self-expanders to any of these flows with asymptotically cylindrical ends must be rotationally symmetric. Thirdly, we show that when such a flow is uniformly parabolic, there exist complete rotationally symmetric self-expanders which are asymptotic to two round cylinders with different radii. These extend some earlier results in [15, 16, 29] to a wider class of curvature flows.

Received 8 February 2017

Accepted 13 June 2018