Methods and Applications of Analysis

Volume 21 (2014)

Number 3

Special issue dedicated to the 60th birthday of Stephen S.-T. Yau: Part I

Guest editors: John Erik Fornæss, Norwegian University of Science and Technology; Xiaojun Huang, Rutgers University; Song-Ying Li, University of California, Irvine; Yat Sun Poon, University of California, Riverside; Wing Shing Wong, The Chinese University of Hong Kong; and Zhouping Xin, The Institute of Mathematical Sciences, CUHK.

Some remarks on Yau’s conjecture and complex plateau problem

Pages: 357 – 364

DOI: http://dx.doi.org/10.4310/MAA.2014.v21.n3.a5

Authors

Rong Du (Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, China)

Yun Gao (Department of Mathematics, Shanghai Jiao Tong University, Shanghai, China)

Abstract

The invariant $g^{(1,1)}$ was introduced by Du and Yau for solving the complex Plateau problem. In this paper, we prove that this invariant never vanishes for any minimal elliptic surface singularity, which confirms Yau’s conjecture for the strictly positivity of $g^{(1,1)}$. We also show that this invariant can be arbitrarily large. As an application of this invariant, we give a new criterion for the regularity problem of the Harvey-Lawson solution to the complex Plateau problem for a strongly pseudoconvex compact $CR$ manifold of dimension 3.

Keywords

complex plateau problem, $CR$ manifold, Harvey-Lawson solution

2010 Mathematics Subject Classification

32S05, 32S20, 32V15

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