Calabi-Bernstein's Theorem on Maximal Surfaces
by
Luis J. Alias
Universidad de Murcia, Spain
Coauthors: Bennett Palmer (University of Durham, UK)

Oral Communication

A maximal surface in the Lorentz-Minkowski space L³ is a spacelike surface with zero mean curvature. One of the most important global results about maximal surfaces in L³ is Calabi-Bernstein's theorem, which states that the only entire maximal graphs in the Lorentz-Minkowski are the spacelike planes. This theorem was first proved by Calabi [1] in 1970. Later on, Cheng and Yau [2] extended it to the general n-dimensional case and gave its parametric version, proving that the only complete maximal hypersurfaces in the (n+1)-dimensional Lorentz-Minkowski space are the spacelike hyperplanes. More recently, several authors have approached Calabi-Bernstein's theorem from different viewpoints, providing diverse extensions and new proofs of the theorem. For instance, Kobayashi [4] derives Calabi-Bernstein's theorem as a simple consequence of the corresponding Weierstrass-Enneper formula for maximal surfaces in L³. On the other hand, in Estudillo and Romero [3] obtain Calabi-Bernstein's theorem as a consequence of a universal inequality for the Gaussian curvature of a maximal surface in L³. Finally, Romero [5] has recently found a simple and nice proof of Calabi-Bernstein's theorem based on Liouville's theorem for harmonic functions on R².

In this talk we will introduce a new approach to Calabi-Bernstein's theorem which is based on a local integral inequality for the Gaussian curvature of a maximal surface in L³, involving the local geometry of the surface and its hyperbolic image. As an application of this, we will provide a new proof of Calabi-Bernstein's theorem. We will also derive other consequences and applications of this inequality. In particular, at points of a maximal surface where the Gaussian curvature is non-zero we are able to estimate the maximum possible geodesic radius in terms of a local positive constant.

[1] E. Calabi, Examples of Bernstein problems for some nonlinear equations, Proc. Symp. Pure Math. 15 (1970), 223-230.

[2] S.Y. Cheng and S.T. Yau, Maximal spacelike hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math. 104 (1976), 407-419.

[3] F.J.M. Estudillo and A. Romero, On the Gauss curvature of maximal surfaces in the 3-dimensional Lorentz-Minkowski space, Comment. Math. Helv. 69 (1994), 1-4.

[4] O. Kobayashi, Maximal surfaces in the 3-dimensional Lorentz-Minkowski space L³, Tokyo J. Math. 6 (1983), 297-309.

[5] A. Romero, Simple proof of Calabi-Bernstein's theorem on maximal surfaces, Proc. Amer. Math. Soc. 124 (1996), 1315-1317.

Date received: March 22, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-20.