{"paper":{"title":"The Asymptotic Dirichlet Problems on manifolds with unbounded negative curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Ran Ji","submitted_at":"2014-01-10T22:14:32Z","abstract_excerpt":"Elton P. Hsu used probabilistic method to show that the asymptotic Dirichlet problem is uniquely solvable under the curvature conditions $-C e^{2-\\eta}r(x) \\leq K_M(x)\\leq -1$ 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 und"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.2476","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}