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The Asymptotic Geometry of Genus-g Helicoids
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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