Mathematics Colloquia and Seminars

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I will discuss the construction of a properly embedded genus-one minimal surface in ℝ³ that is asymptotic to the helicoid, and the attempt to make higher-genus examples. Brian White and I constructed such higher-genus surfaces in S² × ℝ. The compactness of the first factor helped us to get the process (induction on the genus) started and to keep the handles from drifting away. These examples are of independent interest, but our ultimate goal is to prove existence in ℝ³ by taking limits of the examples in S² × ℝ. We are working now with Martin Traizet (Tours) to control handle drift as the radius r of S² becomes infinite.

I hope to avoid technicalities as much as possible and to indicate why the helicoid and its relatives are so important to minimal surface theory.