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The existence and topology of properly embedded minimal surfaces in R('3)
Abstract
This dissertation consists of two parts. In the first part, we study the geometry and topology of properly embedded doubly-periodic minimal surfaces. Our first result is the construction of a one-parameter family of such surfaces that have--after dividing out by all translational symmetries--genus two and four ends in the quotient T $\times$ $\IR$. These new examples are the first examples of this type having genus greater than one. Previously known examples of embedded doubly-periodic minimal surfaces are Scherk's genus-zero examples and Karcher's genus-one examples. The second result concerns the classification of the genus-zero properly embedded minimal surfaces in T $\times$ $\IR$. We prove that such a surface can not have an equal number n of top ends and bottom ends, where n is greater than two and, if it has only two top or two bottom ends, then it is covered by a Scherk's surface. As by-products, we find a two-parameter family of new examples of embedded triply-periodic minimal surfaces of genus four and an interesting example of an immersed nonembedded doubly-periodic minimal surface of genus zero that is not covered by a Scherk's surface. In the second part of the dissertation, we study the asymptotic behavior of the Costa-Hoffman-Meeks three-end examples of minimal surfaces with finite total curvature. We prove that the two catenoid-type ends of these surfaces are asymptotic to the ends of two catenoid that differ by a nontrival translation.
Subject Area
Mathematics
Recommended Citation
Wei, Fusheng, "The existence and topology of properly embedded minimal surfaces in R('3)" (1991). Doctoral Dissertations Available from Proquest. AAI9207471.
https://scholarworks.umass.edu/dissertations/AAI9207471