Perfect Non-Extremal Riemann Surfaces

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 Length and Removable Boundaries of Riemann Surfaces

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...

Spin Structures on Riemann Surfaces and the Perfect Numbers

The equality between the number of odd spin structures on a Riemann surface of genus g, with 2 g − 1 being a Mersenne prime, and the even perfect numbers suggests that properties of the set of spin structures might provide an indication of whether the sequence of perfect numbers continues indefinitely. A method for determining whether Mersenne numbers are primes is developed by using a geometri...

Non-abelian vortices on compact Riemann surfaces

We consider the vortex equations for a U(n) gauge field A coupled to a Higgs field φ with values on the n × n matrices. It is known that when these equations are defined on a compact Riemann surface Σ, their moduli space of solutions is closely related to a moduli space of τ -stable holomorphic n-pairs on that surface. Using this fact and a local factorization result for the matrix φ, we show t...