Sharp local upper bound of the first non-zero Neumann eigenvalue in Riemannian manifolds

Details Sharp local upper bound of the first non-zero Neumann eigenvalue in Riemannian manifolds Sharp local upper bound of the first non-zero Neumann eigenvalue in Riemannian manifolds

2011-10-21

arxiv.org/abs/1110.4770v1

Mathematics

Differential Geometry

15 pages

Scientific paper

We study the local Szeg\"o-Weinberger isoperimetric profile in a geodesic ball $B_g(y_0,r_0)$ centered at a point $y_0$ in a Riemannian manifold $(\M,g)$. This profile is obtained by maximizing the first nontrivial Neumann eigenvalue $\mu_2$ of the Laplace-Beltrami Operator $\Delta_g$ on $\M$ among subdomains of $B_g(y_0,r_0)$ with fixed volume. We derive a sharp asymptotic upper bound of this profile in terms of the scalar curvature of $\M$ at $y_0$. As a corollary, we deduce an isoperimetric comparison principle relative to the corresponding profile in a space form of constant sectional curvature. The results are related to similar work for the profile corresponding to the minimization of the first Dirichlet eigenvalue of $\Delta_g$, see \cite{Fall-eigen}. However, the methods used here are quite different since the variational problem consists in maximizing an eigenvalue. Moreover, in space forms of constant sectional curvature the optimal domains give rise to degeneracy of $\mu_2$.

Affiliated with

Also associated with

No associations

Rating

Sharp local upper bound of the first non-zero Neumann eigenvalue in Riemannian manifolds does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Sharp local upper bound of the first non-zero Neumann eigenvalue in Riemannian manifolds, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Sharp local upper bound of the first non-zero Neumann eigenvalue in Riemannian manifolds will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-85265