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Stability of the minimal surface system and convexity of area functional


Authors: Yng-Ing Lee and Mao-Pei Tsui
Journal: Trans. Amer. Math. Soc. 366 (2014), 3357-3371
MSC (2010): Primary 53A10
DOI: https://doi.org/10.1090/S0002-9947-2014-06223-5
Published electronically: February 24, 2014
MathSciNet review: 3192598
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Abstract: We study the convexity of the area functional for the graphs of maps with respect to the singular values of their differentials. Suppose that $ f$ is a solution to the Dirichlet problem for the minimal surface system and the area functional is convex at $ f$. Then the graph of $ f$ is stable. New criteria for the stability of minimal graphs in any co-dimension are derived in the paper by this method. Our results in particular generalize the co-dimension one case, and improve the condition in the 2003 paper of the first author and M.-T. Wang from $ \vert\bigwedge ^{2} df\vert\leq \frac {1}{p-1}$ to $ \vert\bigwedge ^{2} df\vert\leq \frac {1}{\sqrt {p-1}},$ where $ p$ is an upper bound of the rank of $ df$, and the condition in the 2008 paper of the first author and M.-T. Wang from $ \sqrt {det(I+(df)^Tdf)} \leq \frac {43}{40}$ to $ \sqrt {det(I+(df)^Tdf)} \leq 2 $.


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Additional Information

Yng-Ing Lee
Affiliation: Department of Mathematics, National Taiwan University, Taipei, Taiwan — and — National Center for Theoretical Sciences, Taipei Office, Taipei, Taiwan
Email: yilee@math.ntu.edu.tw

Mao-Pei Tsui
Affiliation: Department of Mathematics and Statistics, University of Toledo, Toledo, Ohio 43606
Email: Mao-Pei.Tsui@Utoledo.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06223-5
Received by editor(s): November 23, 2011
Published electronically: February 24, 2014
Article copyright: © Copyright 2014 American Mathematical Society