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arXiv · 1111.6206

On The Existence of Min-Max Minimal Surface of Genus $g\geq 2$

Xin Zhou
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Abstract

In this paper, we build up a min-max theory for minimal surfaces using sweepouts of surfaces of genus $g\geq 2$. We develop a direct variational methods similar to the proof of the famous Plateau problem by J. Douglas and T. Rado. As a result, we show that the min-max value for the area functional can be achieved by a bubble tree limit consisting of branched genus-$g$ minimal surfaces with nodes, and possibly finitely many branched minimal spheres. We also prove a Colding-Minicozzi type strong convergence theorem similar to the classical mountain pass lemma. Our results extend the min-max theory developed by Colding-Minicozzi and the author to all genera.

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Xin Zhou. 2011-11-27. On The Existence of Min-Max Minimal Surface of Genus $g\geq 2$. https://arxiv.org/abs/1111.6206

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