scholarly journals $K$-mean convex and $K$-outward minimizing sets

2021 ◽  
Author(s):  
Annalisa Cesaroni ◽  
Matteo Novaga
Keyword(s):  
Mean Convex

Discs area-minimizing in mean convex Riemannian n-manifolds

2021 ◽  
pp. 1-22
Author(s):  
Ezequiel Barbosa ◽  
Franciele Conrado
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Higher Dimensions ◽  
Dimensional Torus ◽  
Rigidity Result ◽  
Equality Case ◽  
Totally Geodesic ◽  
Zero Degree ◽  
Area Minimizing ◽  
Mean Convex

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.

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

2020 ◽  
Vol 2020 (768) ◽  
pp. 55-92 ◽  
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.

Mean-convex Alexandrov embedded constant mean curvature tori in the 3-sphere

2016 ◽  
Vol 112 (3) ◽  
pp. 588-622 ◽  
Author(s):  
L. Hauswirth ◽  
M. Kilian ◽  
M. U. Schmidt
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Mean Convex

scholarly journals On integral inequalities of Hermite-Hadamard type for coordinated r-mean convex functions

2019 ◽  
Vol 20 (2) ◽  
pp. 873
Author(s):  
Dan-Dan Gao ◽  
Bo-Yan Xi ◽  
Ying Wu ◽  
Bai-Ni Guo
Keyword(s):  
Convex Functions ◽  
Integral Inequalities ◽  
Mean Convex

scholarly journals On the Construction of Closed Nonconvex Nonsoliton Ancient Mean Curvature Flows

2020 ◽  
Author(s):  
Theodora Bourni ◽  
Mathew Langford ◽  
Alexander Mramor
Keyword(s):  
Mean Curvature ◽  
The Self ◽  
Mean Convex

Abstract We construct closed, embedded, ancient mean curvature flows in each dimension $n\ge 2$ with the topology of $S^1\times S^{n-1}$. These examples are not mean convex and not solitons. They are constructed by analyzing perturbations of the self-shrinking doughnuts constructed by Drugan and Nguyen (or, alternatively, Angenent’s self-shrinking torus when $n =2$).

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