Properly embedded minimal surfaces with finite total curvature

Abstract

Among properly embedded minimal surfaces in Euclidean three space, those that have finite total curvature from a natural and important subclass. The first nontrivial examples, other than the plane and the Catenoid, were constructed only some years ago by Costa [9], and Hoffman and Meeks [16], [17]. These examples began the study of existence, uniqueness and structure theorems for minimal surfaces of finite total curvature, usually attending to their topology. Several methods compete to solve the main problems in this theory, although up to now, the structure of the space of such kind of surfaces with a fixed topology is not well understood. However, we dispose today of a certain number of partial results, and some of them will be explained in these notes.

On the other hand, there are other aspects of the theory which are not covered by these notes. We refer the interested reader to the following literature:

The classic book of Osserman [39] is considered nowadays as one of the obliged sources for the beginner. The texts by Meeks [30,32], Hoffman and Karcher [14], López and Martín [27], and Colding and Minicozzi [7] review a large number of global results on minimal surfaces.

For the last progresses in constructions techniques of properly embedded minimal examples with finite total curvature, see Kapouleas [22], Pitts and Rubinstein [43] and Weber and Wolf [57], or Traizet [55] in the periodic case. Recent embedded examples with infinite total curvature can be found in Hoffman, Karcher and Wei [15] and Weber [56].

The analytical structure of the spaces of properly embedded minimal surfaces with (fixed) finite topology is studied in Pérez and Ros [42]. The paper by Mazzeo and Pollack [29] contains a comparative study between the theory of minimal surfaces and other noncompact geometric problems, like constant (nonzero) mean curvature surfaces.