Uniform approximation of near-singular surfaces

Uniform Approximation on Riemann Surfaces

This thesis consists of three contributions to the theory of complex approximation on Riemann surfaces. It is known that if E is a closed subset of an open Riemann surface R and f is a holomorphic function on a neighbourhood of E, then it is “usually” not possible to approximate f uniformly by functions holomorphic on all of R. In Chapter 2, we show, however, that for every open Riemann surface...

Quantitative Uniform Approximation by Generalized Discrete Singular Operators

Here we study the approximation properties with rates of generalized discrete versions of Picard, Gauss-Weierstrass, and Poisson-Cauchy singular operators. We treat both the unitary and non-unitary cases of the operators above. We establish quantitatively the pointwise and uniform convergences of these operators to the unit operator by involving the uniform higher modulus of smoothness of a uni...

Uniform Approximation by Algebraic Minimal Surfaces in R

An approximation theorem for minimal surfaces by complete minimal surfaces of finite total curvature in R is obtained. This result can be extended to the family of complete minimal surfaces of weak finite total curvature, that is to say, having finite total curvature on proper regions of finite conformal type. We deal only with the orientable case.

On Singular Cubic Surfaces

We study global log canonical thresholds of cubic surfaces with canonical singularities, and we prove the existence of a Kähler–Einstein metric on two singular cubic surfaces.

On Constrained Uniform Approximation