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

Gregório P. Bessa,Luquésio P. Jorge,José F. Montenegro,Barnabé P. Lima

doi:10.1590/s0001-37652006000300001

Abstract

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 Lr operator associated to immersed hypersurfaces with locally bounded (r + 1)-th mean curvature Hr + 1 of the space forms Nn+ 1(c) of constant sectional curvature c. As a corollary we give lower bounds for the extrinsic radius of closed hypersurfaces of Nn+ 1(c) with Hr + 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 Lr-eigenvalue of a closed hypersurface of Nn+ 1(c).

Highlights

Let be a domain in a smooth Riemannian manifold M and let : → End (T ) be a smooth symmetric and positive definite section of the bundle of all endomorphisms of T
Each section is associated to a second order self-adjoint elliptic operator L ( f ) = div, f ∈ C2( )
Sr +1 n r +1 arising from normal variations of a hypersurface M immersed into the (n + 1)-dimensional connected space form Nn+1(c) of constant sectional curvature c ∈ {1, 0, −1}, where Sr+1 is the (r + 1)-th elementary symmetric function of the principal curvatures k1, k2, . . . , kn, see (Reilly 1973) and (Rosenberg 1993) for details

Summary

INTRODUCTION

We give lower bounds for the Lr -fundamental tone of domains ⊂ φ−1(BNn+1(c)( p, R)), in terms of the r -th and (r + 1)-th mean curvatures Hr , Hr+1, (Theorem 3.2), where BNn+1(c)( p, R) is the geodesic ball of Nn+1(c) centered at p with radius R From these estimates we derive three geometric corollaries 3.4, 3.5 and 3.8 that should be viewed as an extension of a result of Jorge and Xavier (Jorge and Xavier 1981). Our main estimate is the following method for giving lower bounds for L -fundamental tone of arbitrary domains of Riemannian manifolds It extends the version of Barta’s theorem (Barta 1937) proved by Cheng-Yau in (Cheng and Yau 1977). Where X ( ) is the set of all smooth vector fields on

GEOMETRIC APPLICATIONS

CLOSED EIGENVALUE PROBLEM

PROOF OF THE RESULTS

Lr f