The smooth Riemannian extension problem

Stefano Pigola; Giona Veronelli

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arXiv · 1606.08320

The smooth Riemannian extension problem

Stefano Pigola·Giona Veronelli

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Abstract

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.

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BibTeX RIS

Stefano Pigola, Giona Veronelli. 2016-06-27. The smooth Riemannian extension problem. https://arxiv.org/abs/1606.08320

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