On the Existence of Min-Max Minimal Torus |
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Authors: | Xin Zhou |
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Institution: | 1.Department of Mathematics,Stanford University,Stanford,USA |
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Abstract: | In this paper, we will study the existence problem of min-max minimal torus. We use classical conformal invariant geometric
variational methods. We prove a theorem about the existence of min-max minimal torus in Theorem 5.1. First we prove a strong
uniformization result (Proposition 3.1) using the method of Ahlfors and Bers (Ann. Math. 72(2):385–404, 1960). Then we use this proposition to choose good parameterization for our min-max sequences. We prove a compactification result
(Lemma 4.1) similar to that of Colding and Minicozzi (Width and finite extinction time of Ricci flow, math.DG], 2007), and then give bubbling convergence results similar to that of Ding et al. (Invent. math. 165:225–242, 2006). In fact, we get an approximating result similar to the classical deformation lemma (Theorem 1.1). |
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