New Examples and Monotonicity Formula for Mean Curvature Flow

Citation

Zhang, Yongzhe (2022) New Examples and Monotonicity Formula for Mean Curvature Flow. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/1y4h-a625. https://resolver.caltech.edu/CaltechTHESIS:03232022-214542574

Abstract

The first main result of this thesis is the proof of the superconvexity of the heat kernel on hyperbolic space. We prove a conjecture of Bernstein that the heat kernel on hyperbolic space of any dimension is supercovex in a suitable coordinate and, hence, there is an analog of Huisken’s monotonicity formula for mean curvature flow in hyperbolic space of all dimensions.

In the second part of the thesis, we construct an ancient solution to planar curve shortening. The solution is at all times compact and embedded. For t ≪ 0 it is approximated by the rotating Yin-Yang soliton, truncated at a finite angle α(t) = -t, and closed off by a small copy of the Grim Reaper translating soliton.

Thesis Files