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The Asymptotic Geometry of Genus-g Helicoids

Bay Area Differential Geometry Seminar November 21, 2009 - November 21, 2009

November 21, 2009 (11:00 AM PST - 12:00 PM PST)
Speaker(s): Jacob Bernstein (Johns Hopkins University)
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Abstract We discuss the problem of classifying the asymptotic geometry of complete, properly embedded minimal surfaces in $R^3$ with finite topology (i.e. topologically a finitely punctured compact surface). The techniques needed to address this question are dramatically different depending on whether there is one puncture or more than one. The former has proven substantially more challenging, requiring very deep results of Colding and Minicozzi, and will be the main focus of the talk. Using very directly Colding and Minicozzi's work, we will sketch a proof of the result - due to Meeks and Rosenberg - that the plane and helicoid are the only such simply connected surfaces. We will then indicate how this argument can be generalized to positive genus. That is, such a surface is asymptotic to a helicoid and hence may be called a "genus-g helicoid".
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