Sharp Area Bounds for Free Boundary Minimal Surfaces in Conformally Euclidean Balls

We prove that the area of a free boundary minimal surface \(\Sigma^2 \subset B^n\), where \(B^n\) is a geodesic ball contained in a round hemisphere \(\mathbb{S}^n_+\), is at least as big as that of a geodesic disk with the same radius as \(B^n\); equality is attained only if \(\Sigma\) coincides wi...

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Bibliographic Details
Published in:arXiv.org 2018-07
Main Authors: Freidin, Brian, McGrath, Peter
Format: Article
Language:English
Subjects:
Free boundaries
Minimal surfaces
Online Access:Get full text
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Summary:We prove that the area of a free boundary minimal surface \(\Sigma^2 \subset B^n\), where \(B^n\) is a geodesic ball contained in a round hemisphere \(\mathbb{S}^n_+\), is at least as big as that of a geodesic disk with the same radius as \(B^n\); equality is attained only if \(\Sigma\) coincides with such a disk. More generally, we prove analogous results for a class of conformally euclidean ambient spaces. This follows work of Brendle and Fraser-Schoen in the euclidean setting.
ISSN:2331-8422