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- Title
On the existence of min-max minimal surface of genus.
- Authors
Zhou, Xin
- Abstract
In this paper, we establish a min-max theory for minimal surfaces using sweepouts of surfaces of genus . We develop a direct variational method similar to the proof of the famous Plateau problem by Douglas [Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931) 263-321] and Rado [On Plateau's problem, Ann. Math. 31 (1930) 457-469]. 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- 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.
- Subjects
PLATEAU'S problem; HARMONIC analysis (Mathematics); FINITE element method; BUBBLE measurement; STOCHASTIC convergence
- Publication
Communications in Contemporary Mathematics, 2017, Vol 19, Issue 4, p-1
- ISSN
0219-1997
- Publication type
Article
- DOI
10.1142/S0219199717500419