Complete nonorientable minimal surfaces in a ball of $\mathbb {R}^3$

Abstract

The existence of complete minimal surfaces in a ball was proved by N. Nadirashvili in 1996. However, the construction of such surfaces with nontrivial topology remained open. In 2002, the authors showed examples of complete orientable minimal surfaces with arbitrary genus and one end. In this paper we construct complete bounded nonorientable minimal surfaces in R 3 with arbitrary finite topology. The method we present here can also be used to construct orientable complete minimal surfaces with arbitrary genus and number of ends.