In this paper we will discuss the geometry of finite topology properly embedded minimal surfaces M in R 3 . M of finite topology means M is homeomorphic to a compact surface M (of genus k and empty boundary) minus a finite number of points p 1 , ..., p j M , called the punctures. A closed neighborhood E of a puncture in M is called an end of M . We will choose the ends sufficiently small so they are topologically S 1 [0, 1) and hence, annular. We remark that M is orientable since M is properly embedded in R 3 .
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Meeks et al. (2005) studied this question.
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