# Mean curvature in manifolds with Ricci curvature bounded from below

### Jaigyoung Choe

Korea Institute for Advanced Study (KIAS), Seoul, Republic of Korea### Ailana Fraser

The University of British Columbia, Vancouver, Canada

## Abstract

Let $M$ be a compact Riemannian manifold of nonnegative Ricci curvature and $Σ$ a compact embedded 2-sided minimal hypersurface in $M$. It is proved that there is a dichotomy: If $Σ$ does not separate $M$ then $Σ$ is totally geodesic and $M∖Σ$ is isometric to the Riemannian product $Σ×(a,b)$, and if $Σ$ separates $M$ then the map $i_{∗}:π_{1}(Σ)→π_{1}(M)$ induced by inclusion is surjective. This surjectivity is also proved for a compact 2-sided hypersurface with mean curvature $H≥(n−1)k $ in a manifold of Ricci curvature Ric$_{M}≥−(n−1)k$, $k>0$, and for a free boundary minimal hypersurface in an $n$-dimensional manifold of nonnegative Ricci curvature with nonempty strictly convex boundary. As an application it is shown that a compact $(n−1)$-dimensional manifold $N$ with the number of generators of $π_{1}(N)<n−1$ cannot be minimally embedded in the flat torus $T_{n}$.

## Cite this article

Jaigyoung Choe, Ailana Fraser, Mean curvature in manifolds with Ricci curvature bounded from below. Comment. Math. Helv. 93 (2018), no. 1, pp. 55–69

DOI 10.4171/CMH/429