Minimal Surfaces with Finite Total Curvature

Abstract

In this chapter we will deal exclusively with complete minimal surfaces in R 3 with finite total curvature; for the sake of convenience we will call such a surface an algebraic minimal surface. In earlier papers [Y4, Y5, Y6, Y7] the author began a study of the Puncture Number Problem for algebraic minimal surfaces. Given a compact Riemann surface M g of genus g, a positive integer r is called a puncture number of M g if M g can be conformally immersed in R 3 as an algebraic minimal surface with exactly r punctures. The set of all puncture numbers for M g is denoted by P (M g ).