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

Glynn-Adey, Parker; Liokumovich, Yevgeny

J. Differ. Geom. 105, No. 1, 33-54 (2017).

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 \(M\).

Reviewer: Constantin Călin (Iaşi)

MSC:

53C42

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

Keywords:

width; Ricci curvature; minimal hypersurface