Horgan Minimal Non-Surface
Horgan Minimal Non-Surface
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.
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