scholarly journals Generic regularity of homologically area minimizing hypersurfaces in eight dimensional manifolds

1993 ◽  
Vol 1 (2) ◽  
pp. 217-228 ◽  
Author(s):  
Nathan Smale
Keyword(s):  
Generic Regularity ◽  
Area Minimizing

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 Area-minimizing surfaces in cones

2002 ◽  
Vol 10 (5) ◽  
pp. 971-983 ◽  
Author(s):  
Frank Morgan
Keyword(s):  
Area Minimizing
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