Optimal isoperimetric inequalities for complete proper minimal submanifolds in hyperbolic space

Sung-Hong Min; Keomkyo Seo

doi:10.1515/crelle-2012-0119

Published by De Gruyter March 7, 2013

Optimal isoperimetric inequalities for complete proper minimal submanifolds in hyperbolic space

Sung-Hong Min and Keomkyo Seo

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2012-0119

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Abstract

Let Σ be a k-dimensional complete proper minimal submanifold in the Poincaré ball model Bⁿ of hyperbolic geometry. If we consider Σ as a subset of the unit ball Bⁿ in Euclidean space, we can measure the Euclidean volumes of the given minimal submanifold Σ and the ideal boundary ∂∞Σ, say Volℝ(Σ) and Volℝ(∂∞Σ), respectively. Using this concept, we prove an optimal linear isoperimetric inequality. We also prove that if Volℝ(∂∞Σ)≥Volℝ(𝕊k-1), then Σ satisfies the classical isoperimetric inequality. By proving the monotonicity theorem for such Σ, we further obtain a sharp lower bound for the Euclidean volume Volℝ(Σ), which is an extension of Fraser–Schoen and Brendle's recent results to hyperbolic space. Moreover we introduce the Möbius volume of Σ in Bⁿ to prove an isoperimetric inequality via the Möbius volume for Σ.

Funding source: National Research Foundation of Korea (NRF); Ministry of Education, Science and Technology; Basic Science Research Program

Award Identifier / Grant number: 20110005520

The authors would like to thank the referee for helpful comments and suggestions, Professor Jaigyoung Choe and Professor Xiang Ma for their valuable comments on this work, and Professor Frank Morgan for informing us of the related paper [Geom. Funct. Anal. 22 (2012), no. 3, 621–626]. The second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (20110005520).

Received: 2012-2-27

Revised: 2012-10-22

Published Online: 2013-3-7

Published in Print: 2014-9-1

Optimal isoperimetric inequalities for complete proper minimal submanifolds in hyperbolic space

Abstract

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