Communications in Analysis and Geometry
Volume 29 (2021)
Self-expanders to inverse curvature flows by homogeneous functions
Pages: 329 – 362
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