Fundamental tone estimates for elliptic operators in divergence form and geometric applications

Details Fundamental tone estimates for elliptic operators in divergence form and geometric applications Fundamental tone estimates for elliptic operators in divergence form and geometric applications

2004-03-25

arxiv.org/abs/math/0403436v2

An. Acad. Bras. Cienc., Sept 2006, vol.78, no.3, p.391-404.

Mathematics

Differential Geometry

14 paginas, latex file This is the former article titled "Estimates for the First Nonzero Eigenvalue of Elliptic Operators in

Scientific paper

We establish a method for giving lower bounds for the fundamental tone of elliptic operators in divergence form in terms of the divergence of vector fields. We then apply this method to the $L_{r}$ operator associated to immersed hypersurfaces with locally bounded $(r+1)$-th mean curvature $H_{r+1}$ of the space forms $\mathbb{N}^{n+1}(c)$ of curvature $c$. As a corollary we give lower bounds for the extrinsic radius of closed hypersurfaces of $\mathbb{N}^{n+1}(c)$ with $H_{r+1}>0$ in terms of the $r$-th and $(r+1)$-th mean curvatures. Finally we observe that bounds for the Laplace eigenvalues essentially bound the eigenvalues of a self-adjoint elliptic differential operator in divergence form. This allows us to show that Cheeger's constant gives a lower bounds for the first nonzero $L_{r}$-eigenvalue of a closed hypersurface of $\mathbb{N}^{n+1}(c)$.

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