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The asymptotic Dirichlet problems on manifolds with unbounded negative curvature

Published online by Cambridge University Press:  30 May 2018

RAN JI
RAN JI*
Affiliation:
Yau Mathematical Sciences Center, Tsinghua University Room 317, Jing Zhai Building, Tsinghua University, Beijing 100084, China e-mail: jiran.jbm@gmail.com
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Abstract

Elton P. Hsu used probabilistic method to show that the asymptotic Dirichlet problem is uniquely solvable under the curvature condition −Ce(2−η)r(x) ≤ KM(x) ≤ −1 with η > 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 −Ce(2/3−η)r(x) ≤ KM(x) ≤ −1 with η > 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 proof is a modification of an argument due to M. T. Anderson and R. Schoen.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 167 , Issue 1 , July 2019 , pp. 133 - 157
Copyright
Copyright © Cambridge Philosophical Society 2018 

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References

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