Summability matrices and mean-convex sequences

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.

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No mass drop for mean curvature flow of mean convex hypersurfaces

Duke Mathematical Journal ◽

10.1215/00127094-2008-007 ◽

2008 ◽

Vol 142 (2) ◽

pp. 283-312 ◽

Cited By ~ 6

Author(s):

Jan Metzger ◽

Felix Schulze

Keyword(s):

Mean Curvature ◽

Mean Curvature Flow ◽

Curvature Flow ◽

Mean Convex ◽

Convex Hypersurfaces

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Capacity, quasi-local mass, and singular fill-ins

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2019-0040 ◽

2020 ◽

Vol 2020 (768) ◽

pp. 55-92 ◽

Cited By ~ 1

Author(s):

Christos Mantoulidis ◽

Pengzi Miao ◽

Luen-Fai Tam

Keyword(s):

Scalar Curvature ◽

The Other ◽

Independent Interest ◽

Asymptotically Flat ◽

Local Mass ◽

Singular Metrics ◽

Comparison Results ◽

Mean Convex ◽

New Inequalities

AbstractWe derive new inequalities between the boundary capacity of an asymptotically flat 3-manifold with nonnegative scalar curvature and boundary quantities that relate to quasi-local mass; one relates to Brown–York mass and the other is new. We argue by recasting the setup to the study of mean-convex fill-ins with nonnegative scalar curvature and, in the process, we consider fill-ins with singular metrics, which may have independent interest. Among other things, our work yields new variational characterizations of Riemannian Schwarzschild manifolds and new comparison results for surfaces in them.

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