Given a metrically complete Riemannian manifold (M, g) with smooth non-empty 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.

Pigola, S., Veronelli, G. (2020). The smooth Riemannian extension problem. ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE, 20(4), 1507-1551 [10.2422/2036-2145.201802_013].

The smooth Riemannian extension problem

Pigola, S
;
Veronelli, G
2020

Abstract

Given a metrically complete Riemannian manifold (M, g) with smooth non-empty 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.
Articolo in rivista - Articolo scientifico
Manifolds with boundary; Extension problem; Curvature bounds
English
2020
20
4
1507
1551
partially_open
Pigola, S., Veronelli, G. (2020). The smooth Riemannian extension problem. ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE, 20(4), 1507-1551 [10.2422/2036-2145.201802_013].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/230262
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