The smooth Riemannian extension problem

Given a metrically complete Riemannian manifold \((M,g)\) with smooth nonempty boundary and assuming that one of its curvatures is subject to a certain bound, we address the problem of whether it is possibile to realize \((M,g)\) as a domain inside a geodesically complete Riemannian manifold \((M�...

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Bibliographic Details
Published inarXiv.org
Main Authors Pigola, Stefano, Giona Veronelli
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 30.06.2016
Subjects
Brownian motion
Convexity
Curvature
Existence theorems
Obstructions
Riemann manifold
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Summary:Given a metrically complete Riemannian manifold \((M,g)\) with smooth nonempty boundary and assuming that one of its curvatures is subject to a certain bound, we address the problem of whether it is possibile to realize \((M,g)\) as a domain inside a geodesically complete Riemannian manifold \((M',g')\) without boundary, by preserving the same curvature bounds. In this direction we provide three kind of results: (1) a general existence theorem showing that it is always possible to obtain a geodesically complete Riemannian extension without curvature constraints; (2) various topological obstructions to the existence of a complete Riemannian extension with prescribed sectional and Ricci curvature bounds; (3) some existence results of complete Riemannian extensions with sectional and Ricci curvature bounds, mostly in the presence of a convexity condition on the boundary.
Bibliography:content type line 50
SourceType-Working Papers-1
ObjectType-Working Paper/Pre-Print-1
ISSN:2331-8422