Dissertations, Theses, and Capstone Projects
Date of Degree
6-2014
Document Type
Dissertation
Degree Name
Ph.D.
Program
Mathematics
Advisor
Jozef Dodziuk
Subject Categories
Mathematics
Keywords
Dirichlet Problem, Martin Boundary, Negative curvature
Abstract
Elton P. Hsu used probabilistic method to show that the asymptotic Dirichlet problem is uniquely solvable if the curvature satisfies the condition $-C e^{(2-\eta)r(x)} \leq K_M(x)\leq -1$ with $\eta>0$. We give an analytical proof of the same statement. In addition, using this new approach we are able to establish two boundary Harnack inequalities under the curvature condition $-C e^{(2/3-\eta)r(x)} \leq K_M(x)\leq -1$ with $\eta>0$. This implies that there is a natural homeomorphism between the Martin boundary and the geometric boundary of $M$. As far as we know, this is the first result of this kind under unbounded curvature conditions. Our proofs are modifications of arguments due to M. T. Anderson and R. Schoen.
Recommended Citation
Ji, Ran, "The Asymptotic Dirichlet Problems on manifolds with unbounded negative curvature" (2014). CUNY Academic Works.
https://academicworks.cuny.edu/gc_etds/232