Abstract
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. 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.
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