Estimates of the first eigenvalue of minimal hypersurfaces of ${\mathbb{S}}^{n+1}$

Abstract

We consider a solution f of a certain Dirichlet Problem on a domain in $S^{(n+1)}$ whose boundary is a minimal hypersurface and we prove a Poincare type inequality for f. One have equality iff Yau's conjecture about the first non-zero eigenvalue of closed minimal hypersurfaces of $S^{(n+1)}$ is true.

arXiv commentA note of 5 pages