The Riemann Boundary Problem on Closed Riemann Surfaces

Abstract

A Riemann surface is a two-dimensional manifold having a complex structure. We now define these notions. A two-dimensional manifold M is a Haussdorf topological space on which every point p ∈ M has a neighbourhoodU _p homeomorphic to the unit disk |z| < 1 of the complex z-plane. The function z(q) that homeomorphically maps U _p on |z| < 1 is called a local coordinate (local parameter), and U _p is a coordinate neighbourhood. Choose a covering of the surface M by coordinate neighbourhoods possessing the following property. Let the intersection of two coordinate neighbourhoods U and U′ be nonempty. Then for every two local coordinates z and z′ in U ∩ U′, the correspondences z = z(z′) and z′ = z′ (z) are defined. We demand that these functions (called relationships of neighbourhoods) should be holomorphic. A chosen class of local coordinates {U, z(p)}, called the atlas, determines the complex structure of the manifold M. We can choose other systems of local coordinates {V, w(p)} such that all functions z = z(w) are holomorphic. Such an atlas is equivalent to the first one and determines the same complex structure.