Existence of proper minimal surfaces of arbitrary topological type

4:00 pm Thursday, April 22, 2010
Francisco Martin (Universidad de Granada)

A natural question in the global theory of minimal surfaces, first raised by Calabi and later revisited by Yau, asks whether or not there exists a complete immersed minimal surface in a bounded domain D in Euclidean space. Consider a domain D which is convex (possibly all R^3) or which is smooth and bounded. Given any open surface M, we prove that there exists a complete, proper minimal immersion f: M ---> D. Moreover, if D is smooth and bounded, then we prove that the immersion f can be chosen so that the limit sets of distinct ends of M are disjoint, connected compact sets in the boundary of D. Finally, we will prove that the results above are sharp, in the sense that they fail to be true when D is neither convex or smooth and bounded.


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