Relative isoperimetric inequalities for minimal submanifolds outside a convex set

Keomkyo Seo

Mathematische Nachrichten, 2012, vol. 285, issue 10, 1264-1273

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 Mp(Σ) of Σ and obtain a sharp isoperimetric inequality \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} $$ 2\pi M_p (\Sigma ) \le {\rm Length}(\partial \Sigma \sim \partial C)^2, $$ \end{document} 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.

Date: 2012
References: Add references at CitEc
Citations:

Downloads: (external link)
https://doi.org/10.1002/mana.201100078

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:bla:mathna:v:285:y:2012:i:10:p:1264-1273

Ordering information: This journal article can be ordered from
http://www.blackwell ... bs.asp?ref=0025-584X

Access Statistics for this article

Mathematische Nachrichten is currently edited by Robert Denk

More articles in Mathematische Nachrichten from Wiley Blackwell
Bibliographic data for series maintained by Wiley Content Delivery ().