Approximation and Interpolation Theorems for Minimal Surfaces

Abstract

This chapter begins with the development of some of the principal tools used in the book. It continues with the first major collection of new results on oriented minimal surfaces in Rⁿ and holomorphic null curves in Cⁿ for any n ≥ 3, obtained since 2012 by complex analytic methods. The main results of the chapter include Runge, Mergelyan, and Carleman type approximation theorems for conformal minimal surfaces, analogues of the Weierstrass and Mittag-Leffler interpolation theorems, general position theorems for minimal surfaces and null curves, the construction of proper minimal surfaces with arbitrary conformal structure in Euclidean spaces, and the homotopy theory for the space of conformal minimal immersions from a given open Riemann surface into Rⁿ.