An embedded genus-one helicoid
- *Department of Mathematics, Indiana University, Bloomington, IN 47405; †Department of Mathematics, Stanford University, Stanford, CA 94708; and ‡Department of Mathematics, Rice University, Houston, TX 77005
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Communicated by Richard M. Schoen, Stanford University, Stanford, CA, September 18, 2005 (received for review August 9, 2004)
Abstract
There exists a properly embedded minimal surface of genus one with a single end asymptotic to the end of the helicoid. This genus-one helicoid is constructed as the limit of a continuous one-parameter family of screw-motion invariant minimal surfaces, also asymptotic to the helicoid, that have genus equal to one in the quotient.
Footnotes
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↵ § To whom correspondence should be addressed. E-mail: mwolf{at}math.rice.edu.
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Author contributions: M. Weber, D.A.H., and M. Wolf performed research and wrote the paper.
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Conflict of interest statement: No conflicts declared.
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↵ ¶ A surface is said to have “finite topology” if it is homeomorphic to a compact surface with a finite number of points removed.
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↵ ∥ There is an additional cone point (with cone angle 6π) in these structures at a vertical point.
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↵ ** The helicoid, a surface swept out by a horizontal line rotating at a constant rate as it moves up a vertical axis at a constant rate, is clearly properly embedded and has finite topology (in fact it is simply connected). Because it is singly periodic and evidently not flat, it has infinite total curvature. Any periodic surface asymptotic to the helicoid must also have infinite total curvature.
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↵ †† Uniqueness follows from an observation of Karcher, using the proof of ref. 14 and a uniqueness result in ref. 15.
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↵ ‡‡ An animation of the genus-one helicoid is available from the authors upon request.
- Copyright © 2005, The National Academy of Sciences
