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

Abstract

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

30 pages; changes to sections 3.2, 3.3, typos fixed; to appear in J. Topol. Anal

References in corpus (1)

• Laplacian eigenvalues functionals and metric deformations on compact manifolds

Cited by in corpus (1)

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