arXiv subjects
Publications and source records attributed to Dan Mangoubi.
Let M be a closed Riemannian manifold. We consider the inner radius of a nodal domain for a large eigenvalue λ. We give upper and lower bounds on the inner radius of the type C/λ^k. Our proof is based on a local behavior of eigenfunctions discovered by Donnelly and Fefferman, and a Poincaré type inequality proved by Maz'ya. Sharp lower bounds are known only in dimension two. We give an account of this case too.
We consider the problem of extending a conformal metric of negative curvature, given outside a neighbourhood of 0 in the unit disk $\DD$, to a conformal metric of negative curvature in $\DD$. 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 apply these considerations to compactification of Riemann surfaces.
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 > 2π, 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 2π, which cannot be so filled in. Such curves can even be taken to be convex.