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Which ambient spaces admit isoperimetric inequalities for submanifolds? (English) Zbl 1179.53061

Let \(N\) be a compact \((n+1)\)-dimensional Riemannian manifold with mean convex boundary. Can one bound the \(n\)-dimensional area of a minimal hypersurface in \(N\) in terms of the \((n-1)\)-dimensional area of its boundary? The absence of any closed minimal hypersurface in \(N\) is certainly a necessary condition, since such a hypersurface would contradict any such bound. In this paper, the author shows that this necessary condition is also sufficient. In fact, he proves that the absence of such a hypersurface implies the existence of a \(c=c_N<\infty\) such that \(|M|\leq c(|\partial M|+\int_M|H|)\), for every \(n\)-dimensional variety \(M\) in \(N\), where \(H(x)\) is the mean curvature of \(M\) at \(x\). He also proves an analogous but weaker result about isoperimetric inequalities for surfaces of any codimension.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C20 Global Riemannian geometry, including pinching
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