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On the Existence of Min-Max Minimal Torus

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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, 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).

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Authors and Affiliations

  1. Department of Mathematics, Stanford University, building 380, Stanford, CA, 94305, USA

    Xin Zhou

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  1. Xin Zhou
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Correspondence to Xin Zhou.

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Communicated by Jiaping Wang.

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Zhou, X. On the Existence of Min-Max Minimal Torus. J Geom Anal 20, 1026–1055 (2010). https://doi.org/10.1007/s12220-010-9137-0

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  • DOI: https://doi.org/10.1007/s12220-010-9137-0

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