{"paper":{"title":"Escape Metrics and its Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fengyan Yang, Xiaopeng Zhao, Zhen-Hu Ning","submitted_at":"2018-11-30T08:25:17Z","abstract_excerpt":"Geodesics escape is widely used to study the scattering of hyperbolic equations. However, there are few progresses except in a simply connected complete Riemannian manifold with nonpositive curvature. We propose a kind of complete Riemannian metrics in $\\mathbb{R}^n$, which is called as escape metrics. We expose the relationship between escape metrics and geodesics escape in $\\mathbb{R}^n$. Under the escape metric $g$, we prove that each geodesic of $(\\mathbb{R}^n,g)$ escapes, that is, $\\lim_{t\\rightarrow +\\infty} |\\gamma (t)|=+\\infty$ for any $x\\in \\mathbb{R}^n$ and any unit-speed geodesic $\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.12668","kind":"arxiv","version":1},"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"}