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Relative isoperimetric inequality for minimal submanifolds in space forms

Authors
서검교
Issue Date
Jun-2010
Publisher
강원경기수학회
Keywords
isoperimetric inequality; minimal submanifold; convex set
Citation
한국수학논문집, v.18, no.2, pp.195 - 200
Journal Title
한국수학논문집
Volume
18
Number
2
Start Page
195
End Page
200
URI
https://scholarworks.sookmyung.ac.kr/handle/2020.sw.sookmyung/13497
ISSN
1976-8605
Abstract
Let C be a closed convex set in Sm or Hm. Assume that ∑ is an n-dimensional compact minimal submanifold outside C such that ∑ is orthogonal to ∂C along ∂∑ ∩ ∂C and ∂∑ lies on a geodesic sphere centered at a fixed point p ∈ ∂∑ ∩ ∂C and that r is the distance in Sm or Hm from p. We make use of a modified volume Mp(∑) of ∑ and obtain a sharp relative isoperimetric inequality ½nnωnMp(∑)n-1 ≤ Vol(∂∑ ~ ∂C)n,where ωn is the volume of a unit ball in Rn. Equality holds if and only if ∑ is a totally geodesic half ball centered at p.
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