Rigidity of complete hyperbolic manifolds and Riemann surfaces

This project 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 our results depend on the existence of nice maps between the given Riemann surfaces. For example, we prove that if R and Sare non-simply connected Riemann surfaces and there is an analytic map f:R → S having an analytic homotopy inverse g, then R is conformally equivalent to S.

We will obtain this result 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.