On extremal properties for annular radial and circular slit mappings of bordered Riemann surfaces

Nonlinear Riemann-hilbert Problem for Bordered Riemann Surfaces

Let Σ be a bordered Riemann surface with genus g and m boundary components. Let {γz}z∈∂Σ be a smooth family of smooth Jordan curves in C which all contain the point 0 in their interior. Then there exists a holomorphic function f(z) on Σ smooth up to the boundary with at most 2g +m− 1 zeros on Σ such that f(z) ∈ γz for every z ∈ ∂Σ.

Bounded Outdegree and Extremal Length on Discrete Riemann Surfaces

Let T be a triangulation of a Riemann surface. We show that the 1-skeleton of T may be oriented so that there is a global bound on the outdegree of the vertices. Our application is to construct extremal metrics on triangulations formed from T by attaching new edges and vertices and subdividing its faces. Such refinements provide a mechanism of convergence of the discrete triangulation to the cl...

Extremal Hermitian Metrics on Riemann Surfaces with Singularities

0. Introduction. It is a well-known consequence of the classical uniformization theorem that there is a metric with constant Gaussian curvature in each conformal class of any compact Riemann surface. It is natural to ask how to generalize this classical uniformization theory to compact surfaces with conical singularities and with nonempty boundary, or to find a “best metric” on such surfaces. H...

Extremal Length and Removable Boundaries of Riemann Surfaces

Projections in the Space H∞ and the Corona Theorem for Coverings of Bordered Riemann Surfaces

Let M be a non-compact connected Riemann surface of finite type, and R ⊂⊂ M be a relatively compact domain such that H1(M,Z) = H1(R,Z). Let R̃ −→ R be a covering. We study the algebra H∞(U) of bounded holomorphic functions defined in some domains U ⊂ R̃. Our main result is a Forelli type theorem on projections in H∞(D).