Minimal hypersurfaces with small first eigenvalue in manifolds of positive Ricci curvature

Abstract

In this paper we exhibit deformations of the hemisphere [Formula: see text], [Formula: see text], for which the ambient Ricci curvature lower bound [Formula: see text] and the minimality of the boundary are preserved, but the first Laplace eigenvalue of the boundary decreases. The existence of these metrics suggests that any resolution of Yau’s conjecture on the first eigenvalue of minimal hypersurfaces in spheres would likely need to consider more geometric data than a Ricci curvature lower bound.