A stability criterion for nonparametric minimal submanifolds

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.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 det(I + (df ) T df ) ≤ 43 40 to det(I + (df ) T df ) ≤ 2.

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“…In the following, we prove another stability result in terms of the area functional which improves a similar result in [LW1]. …”

Section: Stability Of Minimal Graphssupporting

confidence: 66%

“…This result follows easily from the convexity of the area functional. In higher co-dimension, the uniqueness and stability of the minimal surface systems were studied in [LW1] and [LW2]. A counterexample was constructed by Lawson and Osserman in [LO] when n = m = 2.…”

Section: Introductionmentioning

confidence: 99%

Stability of the minimal surface system and convexity of area functional

Lee

Tsui

2014

Trans. Amer. Math. Soc.

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“…Lawson and Osserman [22] studied non-existence, non-uniqueness and irregularity of solutions of the minimal surface system. There have been extensive work in extending Bernstein's Theorem to higher codimension [11,14,18,19,25,28,34,35,36,40] and studying the minimal surface system [23,24,37,38].…”

Section: Motivation and Main Resultsmentioning

confidence: 99%

Minimal surface systems, maximal surface systems and special Lagrangian equations

Lee

2012

Trans. Amer. Math. Soc.

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Abstract. We extend Calabi's correspondence between minimal graphs in Euclidean space R 3 and maximal graphs in Lorentz-Minkowski space L 3 . We establish the twin correspondence between 2-dimensional minimal graphs in Euclidean space R n+2 carrying a positive area-angle function and 2-dimensional maximal graphs in pseudo-Euclidean space R n+2 n carrying the same positive area-angle function.We generalize Osserman's Lemma on degenerate Gauss maps of entire 2-dimensional minimal graphs in R n+2 and offer several Bernstein-Calabi type theorems. A simultaneous application of Harvey-Lawson Theorem on special Lagrangian equation and our extended Osserman's Lemma yields a geometric proof of Jörgens' Theorem on 2-variables unimodular Hessian equation.We introduce the correspondence from 2-dimensional minimal graphs in R n+2 to special Lagrangian graphs in C 2 , which induces an explicit correspondence from 2-variables symplectic Monge-Ampére equations to 2-variables unimodular Hessian equation.

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“…andṼ is an arbitrary vector field of Γ((T (L)) ⊥ ) having compact support on L (see, for example, [3]). Theorem 1.4 If F is a minimal foliation of a manifold M without boundary and the orthogonal distribution D ⊥ is integrable, then any leaf L of F is stable.…”

Section: Stability Resultsmentioning

confidence: 99%