Horgan Minimal Non-Surface

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The Horgan surface is similar to the Costa minimal surface, which is an embedded surface of genus 2 with finite total curvature. (Roughly, the genus is the number of holes.) The Horgan surface can be described as the gluing of a plane with two handles at the top and the bottom, connecting to catenoid ends; but actually, it does not exist. Its existence as a surface would contradict the Hoffman–Meeks conjecture: for a complete embedded minimal surface of genus , the number of ends must be . When the Weierstrass representation is used to solve the period problem, the periods appear in a one-parameter family and seem to be zero, but one of them becomes vanishingly small; in the limit, the resulting surface degenerates, so that the equations cannot be completely solved. For small values of the parameter , a gap can be seen, reflecting this situation.

Contributed by: Enrique Zeleny (January 2015)
Open content licensed under CC BY-NC-SA


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References

[1] C. J. Costa, "Example of a Complete Minimal Immersion in of Genus One and Three-Embedded Ends," Bulletin/Brazilian Mathematical Society, 15(1–2), 1984, pp. 47–54. doi:10.1007/BF02584707.

[2] M. Weber, "On the Horgan Minimal Non-Surface," Calculus of Variations and Partial Differential Equations, 7(4), 1998, pp. 373–379. doi:10.1007/s005260050112.

[3] M. Weber. "The Horgan Surface." Bloomington's Virtual Minimal Surface Museum. (Dec 31, 2014) www.indiana.edu/~minimal/essays/horgan/index.html.



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