Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A Bernstein type theorem for minimal volume preserving maps


Author: Lei Ni
Journal: Proc. Amer. Math. Soc. 130 (2002), 1207-1210
MSC (2000): Primary 58E20
DOI: https://doi.org/10.1090/S0002-9939-01-06448-6
Published electronically: November 9, 2001
MathSciNet review: 1873798
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that any minimal volume preserving map from the Euclidean plane into itself is a linear diffeomorphism. We derive this from a similar result on minimal diffeomorphisms. We also show that the classical Bernstein theorem on minimal graphs is a corollary of our result.


References [Enhancements On Off] (What's this?)

  • [1] F. Almgren, Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem, Ann. Math. 74(1966), 277-292. MR 34:702
  • [2] S. Bernstein, Sur un theoreme de geometrie et ses applications aux equations aux derivees partielle du type elliptique, Comm. de la Soc. Math de Kharkov 15(1915-17), 38-45.
  • [3] E. Bombieri, E. De Giorgi, E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7(1969), 243-268. MR 40:3445
  • [4] D. Fischer-Colbrie, R. Schoen, The structure of complete stable minimal surfaces in $3$-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33(1980), no. 2, 199-211. MR 81i:53044
  • [5] H. Fujimoto, Modified defect relations for the Gauss map of minimal surfaces, J. Differential Geom. 29(1989), no. 2, 245-262. MR 89m:53012
  • [6] J. Nitsche, Elementary proof of Bernstein's theorem on minimal surfaces, Ann. of Math. (2) 66(1957), 543-544. MR 19:878f
  • [7] R. Osserman, A survey of minimal surfaces, Van Nostrand-Reinhold, New York, 1969. MR 41:934
  • [8] R. Schoen, The role of harmonic mappings in rigidity and deformation problems, Complex geometry (Osaka, 1990), 179-200, Lecture Notes in Pure and Appl. Math., 143, Dekker, New York, 1993. MR 94g:58055
  • [9] J. Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88(1968), 62-105. MR 38:1617
  • [10] N. Trudinger, X. Wang, The Bernstein problem for affine maximal hypersurfaces Invent. Math. 140(2000), no. 2, 399-422. MR 2001h:53016
  • [11] F. Xavier, The Gauss map of a complete nonflat minimal surface cannot omit $7$ points of the sphere, Ann. of Math. (2) 113(1981), no. 1, 211-214. MR 82b:53015; Erratum MR 83h:53016
  • [12] J. Wolfson, Minimal Lagrangian diffeomorphisms and the Monge-Ampère equation, J. Differential Geom. 46 (1997), no. 2, 335-373. MR 99e:58045

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 58E20

Retrieve articles in all journals with MSC (2000): 58E20


Additional Information

Lei Ni
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
Email: lni@math.stanford.edu

DOI: https://doi.org/10.1090/S0002-9939-01-06448-6
Keywords: Minimal maps, volume preserving, lagrangian submanifolds
Received by editor(s): August 18, 2000
Published electronically: November 9, 2001
Additional Notes: This research was partially supported by an NSF grant.
Communicated by: Bennett Chow
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society