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New Examples and Monotonicity Formula for Mean Curvature Flow


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


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.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:mean curvature flow, curve shortening flow, heat kernels
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Ni, Yi
Thesis Committee:
  • Yu, Tony Yue (chair)
  • Ni, Yi
  • Hovik, Melissa
  • Smillie, Peter
  • Wang, Lu
Defense Date:23 March 2022
Non-Caltech Author Email:yongzhezhang.zyz (AT)
Record Number:CaltechTHESIS:03232022-214542574
Persistent URL:
Related URLs:
URLURL TypeDescription adapted for Chapter 2.
Zhang, Yongzhe0000-0001-6725-3086
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:14526
Deposited By: Yongzhe Zhang
Deposited On:20 Apr 2022 19:43
Last Modified:05 Jul 2022 19:01

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