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Discs area-minimizing in mean convex Riemannian n-manifolds

Part of: Global differential geometry

Published online by Cambridge University Press:  06 September 2021

Ezequiel Barbosa Open the ORCID record for Ezequiel Barbosa [Opens in a new window]  and
Franciele Conrado
Ezequiel Barbosa
Affiliation:
Universidade Federal de Minas Gerais (UFMG), Caixa Postal 702, 30123-970 Belo Horizonte, MG, Brazil ([email protected])
Franciele Conrado
Affiliation:
Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, 30161-970 Belo Horizonte, MG, Brazil ([email protected])
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Abstract

In this work, we consider oriented compact manifolds which possess convex mean curvature boundary, positive scalar curvature and admit a map to $\mathbb {D}^{2}\times T^{n}$ with non-zero degree, where $\mathbb {D}^{2}$ is a disc and $T^{n}$ is an $n$-dimensional torus. We prove the validity of an inequality involving a mean of the area and the length of the boundary of immersed discs whose boundaries are homotopically non-trivial curves. We also prove a rigidity result for the equality case when the boundary is strongly totally geodesic. This can be viewed as a partial generalization of a result due to Lucas Ambrózio in (2015, J. Geom. Anal., 25, 1001–1017) to higher dimensions.

Keywords

Free-boundarymanifolds with boundaryminimal surfacesrigidity resultsuniversal covering

MSC classification

Primary: 53C24: Rigidity results
Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 6 , December 2022 , pp. 1361 - 1382
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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