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Complete minimal surfaces of finite total curvature
Abstract
Let $X$: $M \to \IR\sp3$ be a complete minimal immersion. Suppose that M has finite total curvature. Then a result of Osserman says that the Gauss map G of M can omit at most 3 directions. The problem is that there is no known examples whose Gauss map omits 3 directions. The Gauss map of the Catenoid omits 2 directions. Osserman proved that if G omits 3 directions, then the total curvature C(M) will be less than $-8\pi$. Weitsman and Xavier proved that C(M) $<$ -12$\pi$. In this thesis we will prove that C(M) $<$ -16$\pi$ with the possible exception of the case when it is a minimal surface of genus-2 with 3 flat ends. The existence problem of such an exceptional example is still an open problem. Other general results concerning the necessary conditions for an example which omits 3 directions are obtained in this thesis. Also, we construct a family of Enneper-type minimal surfaces and an example of singly periodic complete minimal surface whose Gauss map omits 3 directions.
Subject Area
Mathematics
Recommended Citation
Fang, Yi, "Complete minimal surfaces of finite total curvature" (1990). Doctoral Dissertations Available from Proquest. AAI9110132.
https://scholarworks.umass.edu/dissertations/AAI9110132