2026-09-21T07:09:53Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/1515052023-08-01T03:17:11Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Minicozzi II, William P.
Gui, Feng
Massachusetts Institute of Technology. Department of Mathematics
2023-07-31T19:44:54Z
2023-07-31T19:44:54Z
2023-06
2023-06-09T18:02:26.390Z
https://hdl.handle.net/1721.1/151505
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.
Ph.D.
Massachusetts Institute of Technology
In Copyright - Educational Use Permitted
Copyright retained by author(s)
https://rightsstatements.org/page/InC-EDU/1.0/
Liouville Properties and Dimensionality Bounds for Harmonic and Caloric Functions
Thesis
application/pdf
Doctoral
Doctor of Philosophy
Publication
unknown
unknown
<Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="c4a169a5-b353-4345-8b8e-a52a9d732547">
<Type xmlns="https://www.openaire.eu/cerif-profile/vocab/COAR_Publication_Types">http://purl.org/coar/resource_type/c_1843</Type>
<Title>Liouville Properties and Dimensionality Bounds for Harmonic and Caloric Functions</Title>
<PublishedIn>
<Publication>
</Publication>
</PublishedIn>
<PublicationDate>2023-06</PublicationDate>
<Authors>
<Author>
<DisplayName>Gui, Feng</DisplayName>
<Affiliation>
<OrgUnit>
</OrgUnit>
</Affiliation>
</Author>
</Authors>
<Editors>
</Editors>
<Publishers>
<Publisher>
<DisplayName>Massachusetts Institute of Technology</DisplayName>
<OrgUnit />
</Publisher>
</Publishers>
<License>https://rightsstatements.org/page/InC-EDU/1.0/</License>
<Abstract>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.</Abstract>
<Access xmlns="http://purl.org/coar/access_right"
>
</Access>
</Publication>