Liouville Properties and Dimensionality Bounds for Harmonic and Caloric Functions

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.