The effect of curvature on convexity properties of harmonic functions and eigenfunctions

We give a proof of the Donnelly-Fefferman growth bound of Laplace-Beltrami eigenfunctions which is probably the easiest and the most elementary one. Our proof also gives new quantitative geometric estimates in terms of curvature bounds which improve and simplify previous work by Garofalo and Lin. The proof is based on a convexity property of harmonic functions on curved manifolds, generalizing Agmon's Theorem on a convexity property of harmonic functions in R^n.Comment: 24 pages. This is a major revision. The main theorem now treats the case of pinched curvature between any two real values. The proof is simplified in several points. Ricci curvature is replaced by sectional curvature. Referee's remarks incorporate

Tubular Neighborhoods of Nodal Sets and Diophantine Approximation

We give upper and lower bounds on the volume of a tubular neighborhood of the nodal set of an eigenfunction of the Laplacian on a real analytic closed Riemannian manifold M. As an application we consider the question of approximating points on M by nodal sets, and explore analogy with approximation by rational numbers.Comment: 22 pages; revised version containing full proof of lower bound; reference added; to appear in Amer. J. Math

Riemann Surfaces and 3-Regular Graphs

In this thesis we consider a way to construct a rich family of compact Riemann Surfaces in a combinatorial way. Given a 3-regualr graph with orientation, we construct a finite-area hyperbolic Riemann surface by gluing triangles according to the combinatorics of the graph. We then compactify this surface by adding finitely many points. We discuss this construction by considering a number of examples. In particular, we see that the surface depends in a strong way on the orientation. We then consider the effect the process of compactification has on the hyperbolic metric of the surface. To that end, we ask when we can change the metric in the horocycle neighbourhoods of the cusps to get a hyperbolic metric on the compactification. In general, the process of compactification can have drastic effects on the hyperbolic structure. For instance, if we compactify the 3-punctured sphere we lose its hyperbolic structure. We show that when the cusps have lengths > 2\pi, we can fill in the horocycle neighbourhoods and retain negative curvature. Furthermore, the last condition is sharp. We show by examples that there exist curves arbitrarily close to horocycles of length 2\pi, which cannot be so filled in. Such curves can even be taken to be convex.Comment: M.Sc. Thesis (Technion, Israel Institute of Technology), 55 Pages, Under the Direction of Robert Brook