On graphic Bernstein type results in higher codimension

Wang, Mu-Tao

doi:10.1090/S0002-9947-02-03108-2

On graphic Bernstein type results in higher codimension
HTML articles powered by AMS MathViewer

by Mu-Tao Wang

Trans. Amer. Math. Soc. 355 (2003), 265-271

DOI: https://doi.org/10.1090/S0002-9947-02-03108-2

Published electronically: September 5, 2002

PDF | Request permission

Abstract:

Let $\Sigma$ be a minimal submanifold of $\mathbb {R}^{n+m}$ that can be represented as the graph of a smooth map $f:\mathbb {R}^n\mapsto \mathbb {R}^m$. We apply a formula that we derived in the study of mean curvature flow to obtain conditions under which $\Sigma$ must be an affine subspace. Our result covers all known ones in the general case. The conditions are stated in terms of the singular values of $df$.

References

João Lucas Marquês Barbosa, An extrinsic rigidity theorem for minimal immersions from $S^{2}$ into $S^{n}$, J. Differential Geometry 14 (1979), no. 3, 355–368 (1980). MR 594706
Klaus Ecker and Gerhard Huisken, A Bernstein result for minimal graphs of controlled growth, J. Differential Geom. 31 (1990), no. 2, 397–400. MR 1037408
D. Fischer-Colbrie, Some rigidity theorems for minimal submanifolds of the sphere, Acta Math. 145 (1980), no. 1-2, 29–46. MR 558091, DOI 10.1007/BF02414184
Lei Fu, An analogue of Bernstein’s theorem, Houston J. Math. 24 (1998), no. 3, 415–419. MR 1686614
S. Hildebrandt, J. Jost, and K.-O. Widman, Harmonic mappings and minimal submanifolds, Invent. Math. 62 (1980/81), no. 2, 269–298. MR 595589, DOI 10.1007/BF01389161
Reese Harvey and H. Blaine Lawson Jr., Calibrated geometries, Acta Math. 148 (1982), 47–157. MR 666108, DOI 10.1007/BF02392726
J. Jost and Y. L. Xin, Bernstein type theorems for higher codimension, Calc. Var. Partial Differential Equations 9 (1999), no. 4, 277–296. MR 1731468, DOI 10.1007/s005260050142
J. Jost and Y. L. Xin, A Bernstein theorem for special Lagrangian graphs, preprint, 2001.
H. B. Lawson Jr. and R. Osserman, Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system, Acta Math. 139 (1977), no. 1-2, 1–17. MR 452745, DOI 10.1007/BF02392232
Ni, Lei. A Bernstein type theorem for minimal volume preserving maps, Proc. Amer. Math. Soc. 130 (2002), 1207-1210.
M.-P. Tsui and M.-T. Wang, A Bernstein type result for special Lagrangian submanifolds, preprint, 2001.
M.-T. Wang, Mean curvature flow of surfaces in Einstein four-manifolds, J. Differential Geom. 57 (2001), 301-338.
M.-T. Wang, Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension, Invent. Math. 148 (2002) 3, 525-543.
Y. Yuan, A Bernstein problem for special Lagrangian equation, preprint, 2001.