On The Existence of Min-Max Minimal Surface of Genus $ggeq 2$
On The Existence of Min-Max Minimal Surface of Genus $ggeq 2$
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关于亏格$ggeq 2$最小-最大极小曲面的存在性
DOI：
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发表时间：
2011
期刊：
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通讯作者：
Xin Zhou
中科院分区：
文献类型：
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作者：
Xin Zhou
本文利用g≥2的曲面的清扫，建立了最小曲面的最小-最大理论。我们发展了一种直接变分方法，类似于道格拉斯证明著名的高原问题[高原问题的解，译]。阿米尔。数学。《社会科学》，33(1931)263-321]和Rado[论高原问题，安。]数学。31(1930)457-469]。结果表明，面积泛函的最小最大值可以通过气泡树极限来实现，气泡树极限由带有节点的分支的属-g极小曲面和可能的有限个分支极小球体组成。我们还证明了一个类似于经典山口引理的Colding-Minicozzi型强收敛定理。我们的结果将Colding-Minicozzi和作者的最小最大理论推广到所有的属。
In this paper, we establish a min-max theory for minimal surfaces using sweepouts of surfaces of genus g ≥ 2. 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-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.