Width, Ricci curvature, and minimal hypersurfaces. (English) Zbl 1359.53051

The main result of the paper under review is the following bound on the volume of minimal hypersurfaces for certain conformal classes of Riemannian metrics.
Theorem 1.1. Suppose \(M_0\) is a closed Riemannian manifold of dimension \(n\), for \(3\leq n\leq 7\). If \(M\) is in the conformal class of \(M_0\) then \(M\) contains a smooth closed embedded minimal hypersurface \(\Sigma \) with volume bounded above by \(C(M_0)\mathrm{Vol}(M)^{\frac{n-1}{n}}\). When \(\mathrm{Ricci}(M_0)\geq 0\), the constant \(C(M_0)\) is an absolute constant that depends only on \(n\). In general, for \(M_0\) with \(\mathrm{Ricci}(M_0)\geq -(n-1)a^2\) we can take \(C(M_0)=C(n)\max\{1, a\mathrm{Vol}(M_0)^{\frac{1}{n}}\}\). If \(n>7\) the same upper bound holds for the \((n-1)\)-volume of a closed minimal hypersurface with singularities of dimension at most \(n-8\).
This result follows from a bound on the width of \(M\).


53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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