Journal of Geometric Analysis

, Volume 20, Issue 4, pp 1026–1055 | Cite as

On the Existence of Min-Max Minimal Torus

Article

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, 0707.0108 [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).

Keywords

Min-max minimal torus Geometric variational methods Energy identity Deformation lemma 

Mathematics Subject Classification (2000)

58E20 58E12 

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Copyright information

© Mathematica Josephina, Inc. 2010

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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