Interior gradient bounds for solutions to the minimal surface system

Mu-Tao Wang Columbia University

Differential Geometry mathscidoc:1608.10068

American Journal of Mathematics , 126, (4), 921-934, 2004
this article we generalize the classical gradient estimate for the minimal surface equation to higher codimension. We consider a vector-valued function $u : \Omega \subset \math{R}^n \to \math{R}^m$ that satisfies the minimal surface system, see equation (1.1) in §1. The graph of u is then a minimal submanifold of $\math{R}^{n+m}$. We prove an a priori gradient bound under the assumption that the Jacobian of $ du: \math{R}^n \to \math{R}^m$ on any two dimensional subspace of $\mtah{R}^n$ is less than or equal to one. This assumption is automatically satisfied when $du$ is of rank one and thus the estimate covers the case when m=1, i.e., the original minimal surface equation. This is applied to Bernstein type theorems for minimal submanifolds of higher codimension.
minimal surface system
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@inproceedings{mu-tao2004interior,
  title={Interior gradient bounds for solutions to the minimal surface system},
  author={Mu-Tao Wang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160822230209910943385},
  booktitle={American Journal of Mathematics },
  volume={126},
  number={4},
  pages={921-934},
  year={2004},
}
Mu-Tao Wang. Interior gradient bounds for solutions to the minimal surface system. 2004. Vol. 126. In American Journal of Mathematics . pp.921-934. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160822230209910943385.
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