|Ph.D Thesis||Department of Mathematics|
|Supervisors:||Assoc. Prof. Entov Michael|
|Prof. Leonid Polterovich|
In this thesis we are interested in properties of the spectrum and of the eigenfunctions of the Laplace-Beltrami operator on a closed Riemannian manifold. We discuss three main problems.
In the first chapter we consider Riemannian metrics compatible with the symplectic structure on T2 x M, where T2 is a symplectic 2-torus and M is a closed symplectic manifold. To each such metric we attach the corresponding Laplacian and consider its first positive eigenvalue l1. We show that l1 can be made arbitrarily large by deforming the metric structure, keeping the symplectic structure fixed. This extends a theorem of L. Polterovich on T4 x M. The conjecture is that the same is true for any symplectic manifold of dimension r 4.
In the second chapter we let M be a closed Riemannian manifold. We consider the inner radius of a nodal domain for a large eigenvalue l. We give upper and lower bounds on the inner radius of the type C/ lk. Our proof is based on a local behavior of eigenfunctions discovered by H. Donnelly and C. Fefferman and a Poincaré type inequality proved by V. Maz'ya. Sharp lower bounds are known only in dimension two. We give an account of this case too.
In the last chapter we consider the problem of extending a conformal metric of negative curvature, given outside of a neighborhood of 0 in the unit disk D, to a conformal metric of negative curvature in D. We give conditions under which such an extension is possible, and also give obstructions to such an extension. The methods we use are based on a maximum principle and the Ahlfors-Schwarz Lemma. We also give an example in which no extension is possible, even when the conformality condition is dropped. We apply these considerations to compactification of Riemann surfaces.