Ph.D Thesis | |

Ph.D Student | Mangoubi Dan |
---|---|

Subject | Topics in the Geometry of the Laplace-Beltrami Operator |

Department | Department of Mathematics |

Supervisors | PROF. Michael Entov |

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 *T** ^{2}* x

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/**
l** ^{k}*. 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.