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Abstract: These notes outline recent developments in classical minimal surface theorythat are essential in classifying the properly embedded minimal planar domainsM in R^3 with infinite topology equivalently, with an infinite number ofends. This final classification result by Meeks, Perez, and Ros states thatsuch an M must be congruent to a homothetic scaling of one of the classicalexamples found by Riemann in 1860. These examples {\cal R} s, 0}, are singly-periodic andintersect each horizontal plane in R^3 in a circle or a line parallel to thex-axis. Earlier work by Collin, Lopez and Ros and Meeks and Rosenbergdemonstrate that the plane, the catenoid and the helicoid are the only properlyembedded minimal surfaces of genus zero with finite topology equivalently,with a finite number of ends. Since the surfaces {\cal R} s converge to acatenoid as s tends to 0 and to a helicoid as s tends to infinity, then themoduli space {\cal M} of all properly embedded, non-planar, minimal planardomains in R^3 is homeomorphic to the closed unit interval 0,1.



Author: William H. Meeks III, Joaquin Perez

Source: https://arxiv.org/







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