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Rigidity of Complete Hyperbolic Manifolds and Riemann Surfaces

Masters Thesis
written under the direction of John B. Conway

University of Tennessee, 1993

This work is to understand the rigidity of Riemann surfaces. That is, what (formally) weak conditions on two Riemann surfaces imply that they are conformally equivalent? It is well known that diffeomorphism is not enough. Thus one is led to consider additional geometrical or analytical conditions.

Here the results depend on the existence of nice maps between the given Riemann surfaces. For example, we prove that if R and S are non-simply connected Riemann surfaces and there is an analytic map f from R to S and an analytic map g from S to R such that f and g are homotopy inverses, then R is conformally equivalent to S.

This result is obtained as a corollary to a more general result, where one replaces the condition that f and g be analytic with the condition that f and g decrease hyperbolic distances.

We also consider natural generalizations to complete hyperbolic manifolds of any dimension. These results give a Mostow type rigidity theorem for non-compact hyperbolic manifolds