Liouville Properties and Dimensionality Bounds for Harmonic and Caloric Functions

Classical Liouville type theorems claim that solutions to certain elliptic or parabolic PDE are trivial provided some generic constraints about the function and the underlying space. When the solution space is not trivial, one can ask whether it is a linear space with finite dimension. In this thesis, we study several Liouville properties in geometric analysis. First, we prove a Hamilton type and a Souplet-Zhang type gradient estimates which imply a strong Liouville theorem for ancient f-caloric functions with certain growth assumption on smooth metric measure spaces. Second, we generalize Colding-Minicozzi’s result to estimate the dimension of polynomial growth f-caloric functions. We apply some of these results to gradient shrinking Ricci solitons. Lastly, we prove a dimensionality bound for exponential growth solutions to a parabolic type equation on an infinite strip.

Name thesis:degree_name

Doctoral

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Mathematics

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2023

Author dc:creator

• Gui, Feng

Advisor dc:contributor.advisor

• Minicozzi II, William P.

Statement dc:rights

• In Copyright - Educational Use Permitted
• Copyright retained by author(s)

Licence dc:rights.uri

• https://rightsstatements.org/page/InC-EDU/1.0/

Handle dc:identifier.uri

https://hdl.handle.net/1721.1/151505

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/151505