A dichotomy for minimal hypersurfaces in manifolds thick at infinity

Abstract

Let $(M^{n+1},g)$ be a complete $(n+1)$-dimensional Riemannian manifold with $2\leq n\leq 6$. Our main theorem generalizes the solution of Yau's conjecture for minimal surfaces and builds on a result of Gromov. Suppose that $(M,g)$ is thick at infinity, i.e. any connected finite volume complete minimal hypersurface is closed. Then the following dichotomy holds for the space of closed hypersurfaces in $M$: either there are infinitely many saddle points of the $n$-volume functional, or there is none. Additionally, we give a new short proof of the existence of a finite volume minimal hypersurface if $(M,g)$ has finite volume, we check Yau's conjecture for finite volume hyperbolic 3-manifolds and we extend the density result of Irie-Marques-Neves when $(M,g)$ is shrinking to zero at infinity.