Geometry and Rigidity in the Complex Plane and in Surfaces

ongoing

The main objective of the proposal is to develop novel methods for the study of the quasiconformal geometry of metric surfaces and of subsets of the complex plane. Geometry of metric surfaces: The uniformization problem asks for conditions on a fractal metric space so that it can be transformed to a smooth space with a well-behaved transformation that preserves the geometry. In joint work with Romney the PI has resolved a major open problem and proved a general quasiconformal uniformization result for 2-dimensional spheres of finite area. The current project expects to exploit this recent breakthrough and develop an analytic theory for 2-dimensional surfaces of locally finite area under no other assumption; the classical approaches in the field require instead several geometric assumptions. In particular, the PI proposes the study of the following problems on fractal surfaces: quasiconformal classification, embedding in Euclidean space, uniformization of surfaces of infinite area, and connections between quasiconformal geometry and rectifiability. Uniformization and rigidity in the plane: A long-standing conjecture of Koebe asserts that every domain in the plane can be conformally transformed to a circle domain. This proposal introduces a wide class of domains to test the conjecture, using techniques recently developed by the PI and collaborators, in combination with the transboundary modulus of Schramm. The PI will also study the problem of uniqueness of this conformal transformation and connections to the problem of conformal removability. The latter is a rigidity problem asking whether a given set in the plane is negligible from the domain of a conformal map. The PI has recently displayed several examples of (non)removable sets and has identified a new general class of sets that are conjectured to provide a characterization of removable sets. The PI will study this conjecture and several related deep open problems.