Relative isoperimetric inequalities for minimal submanifolds outside a convex set
✍ Scribed by Keomkyo Seo
- Publisher
- John Wiley and Sons
- Year
- 2012
- Tongue
- English
- Weight
- 156 KB
- Volume
- 285
- Category
- Article
- ISSN
- 0025-584X
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✦ Synopsis
Abstract
Let C be a closed convex set in a complete simply connected Riemannian manifold M with sectional curvature bounded above by a positive constant K. Assume that Σ is a compact minimal surface outside C such that Σ is orthogonal to ∂C along ∂Σ∩∂C and ∂Σ ∼ ∂C is radially connected from a point p ∈ ∂Σ∩∂C. We introduce a modified volume M~p~(Σ) of Σ and obtain a sharp isoperimetric inequality
where equality holds if and only if Σ is a geodesic half disk with constant Gaussian curvature K. We also prove higher dimensional isoperimetric inequalities for minimal submanifolds outside a closed convex set in a Riemannian manifold using the modified volume.