Even though the classification of genus zero, embedded minimal surfaces is not complete, W. H. Meeks J. Perez and A. Ros [14], [15], [16] have made progress concerning the question of the uniqueness of the Riemann examples in the class of genus zero embedded minimal surfaces which have an infinite number of ends. They conjecture in [15] that every embedded minimal surface of finite genus and wi...

We establish a 1–to–1 relation between metrics on compact Riemann surfaces without boundary, and mechanical systems having those surfaces as configuration spaces.

We continue our study of quantum Riemann surfaces initiated in 1, 2, 3]. We construct a one parameter family of deformations of compact Riemann surfaces of genus g 2. Our construction does not require any discretness condition on the value of Planck's constant. It coincides with the construction of 2] in the case when Planck's constant assumes the discrete set of values dictated by geometric qu...

“Analytic invariants” of complex manifolds that the generalizations of the genus of curves, and their birationally invariant nature. Blow-up of a surface at a point. Birational classification of complex surfaces via minimal models. Enriques-Kodaira classification. Canonical models. Calabi-Yau manifolds and K3 surfaces. Fano manifolds and del Pezzo surfaces. A Crash Course onCompact Complex Surf...

We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincaré metrics (i.e., complete metrics of constant negative curvature) by solving the equation ∆u−e = K0(z) on general open surfaces. A few other topics are discussed, including boundary behavior of the conformal factor e giving t...

Laplace operators perturbed by meromorphic potential on the Riemann and separated type Klein surfaces are constructed and their indices are calculated by two different ways. The topological expressions for the indices are obtained from the study of spectral properties of the operators. Analytical expressions are provided by the Heat Kernel approach in terms of the functional integrals. As a res...

The moduli spaceMg of compact Riemann surfaces of genus g has orbifold structure and the set of singular points of the orbifold is the branch locus Bg. In this article we show that Bg is connected for genera three, four, thirteen, seventeen, nineteen and fiftynine, and disconnected for any other genus. In order to prove this we use Fuchsian groups, automorphisms of order 5 and 7 of Riemann surf...