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The isoperimetric inequality for a minimal surface with radially connected boundary. (English) Zbl 0745.53004

As an extension of the classical isoperimetric inequality for domains in Euclidean space, it is conjectured that for any \(k\)-dimensional compact minimal submanifold \(M\) of \(R^ n\), \(\hbox{volume}(M)^{k-1}\leq k^{- k}w_ k^{-1}\hbox{volume}(\partial M)^ k\), where \(\omega_ k\) is the volume of the \(k\)-dimensional unit ball. In this paper it is proved that the isoperimetric inequality holds for every minimal surface \(\Sigma\) in \(R^ n\) whose boundary \(\partial\Sigma\) is radially connected from a point in \(\Sigma\), i.e. if there exists a point \(p\in\Sigma\) such that \(I:=\{r\mid r=dist(p,q),\;q\in \partial \Sigma\}\) is a connected interval.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49Q05 Minimal surfaces and optimization
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References:

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