Abstract
This paper shows that a simply connected, oriented, time-like surface S in E31 that is complete as a surface in Euclidean 3-space E3 is C∞ conformally equivalent to the Minkowski plane E21 provided that the integral of the absolute value of E31 mean curvature on S with respect to the E3 area element is finite. This provides the broadest generalization to date for the conformal Bernstein's theorem, which states that any entire timelike minimal surface in E31 is C∞ conformally equivalent to E31.
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