The smooth Riemannian extension problem

TL;DR

This paper investigates the problem of extending a Riemannian manifold with boundary into a complete boundaryless manifold while preserving curvature bounds, providing existence results, obstructions, and conditions for such extensions.

Contribution

It introduces new theorems on the existence and obstructions of curvature-preserving Riemannian extensions of manifolds with boundary.

Findings

Existence of geodesically complete extensions without curvature constraints.

Topological obstructions to extensions with prescribed curvature bounds.

Conditions under which curvature-preserving extensions exist, especially with boundary convexity.

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.