On the existence of min-max minimal surface of genus \(g\geq2\). (English) Zbl 1369.49059
Summary: In this paper, we establish a min-max theory for minimal surfaces using sweepouts of surfaces of genus \(g\geq2\). We develop a direct variational method similar to the proof of the famous Plateau problem by J. Douglas [Trans. Am. Math. Soc. 33, 263–321 (1931; Zbl 0001.14102)] and T. Radó [Ann. Math. (2) 31, 457–469 (1930; JFM 56.0437.02)]. 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 by Colding-Minicozzi and the author to all genera.
MSC:
| 49Q05 | Minimal surfaces and optimization |
| 49J35 | Existence of solutions for minimax problems |
| 58E20 | Harmonic maps, etc. |
| 58E12 | Variational problems concerning minimal surfaces (problems in two independent variables) |
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
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