Simple proof of Calabi-Bernstein’s Theorem on maximal surfaces

Abstract

(Communicated by Christopher Croke) Dedicated to the memory of Professor P. Bobillo The following uniqueness result, called Calabi-Bernstein's Theorem on maximal surfaces, is well known ([Ca], [C-Y], [K], [E-R]). Theorem. The only entire solutions to the maximal surface equation div Du 1 − |Du| 2 = 0, |Du| < 1, are affine functions. In the references cited above, this result appears as either a particular case of some much more general theorems or stated in terms of local complex representation of the surface. However, a direct simple proof would be desirable to be easily understood for beginning researchers. The proof we present here uses only Liouville's Theorem on harmonic functions on R 2. Thus, it is simple and complex function theory is not needed. This proof is inspired by [Ch]. Roughly, the key steps of our proof are: (1) On any maximal surface there exists a positive harmonic function, which is constant if and only if the surface is totally geodesic. (2) The metric of any spacelike graph is globally conformally related to a metric g * , which is complete when the graph is entire. (3) On any maximal graph the metric g * is flat. 1. Preliminaries Consider the Lorentz-Minkowski space L 3 with its Lorentzian metric , = dx 2 1 + dx 2 2 − dx 2 3 , given in the usual coordinate system. Let x : M → L 3 be a spacelike immersion of a two-dimensional manifold M in L 3. Note that M must be orientable. Let N be a globally defined unit timelike normal vector field on M. Suppose that the mean curvature of x vanishes; then M is called a maximal surface in L 3. For each vector a ∈ L 3 we can consider on M the smooth function N, a. Let a T be the vector field on M induced from the tangential component of a at any