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

TL;DR

This paper constructs deformations of hemispheres with positive Ricci curvature that preserve minimal boundary but lower the first boundary eigenvalue, indicating that Ricci bounds alone may not control eigenvalues of minimal hypersurfaces.

Contribution

It demonstrates that minimal hypersurfaces with small first eigenvalue can exist in manifolds with positive Ricci curvature, challenging previous assumptions.

Findings

Deformations of hemispheres with preserved Ricci curvature bounds

Existence of minimal hypersurfaces with small first eigenvalue

Implication that Ricci bounds alone are insufficient for eigenvalue control

Abstract

In this paper we exhibit deformations of the hemisphere $S^{n+1}_+$, $n\geq 2$, for which the ambient Ricci curvature lower bound $\text{Ric}\geq n $ 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.