A dichotomy for minimal hypersurfaces in manifolds thick at infinity

Abstract

Let (M,g) be a complete (n + 1)-dimensional Riemannian manifold with 2 ≤ n ≤ 6. Our main theorem generalizes the solution of S.-T. Yau's conjecture on the abundance of minimal surfaces and builds on a result of M. Gromov. Suppose that (M,g) has bounded geometry, or more generally is thick at infinity. 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 in finite volume manifolds, we check Yau's conjecture for finite volume hyperbolic 3-manifolds and we extend the density result due to Irie-Marques-Neves when (M,g) is shrinking to zero at infinity.