On vanishing-curvature extensions of Lorentzian metrics

We study the problem of extending Lorentzian metrics, defined on the boundary of a smooth domain in C, into the interior of the domain in such a way that the curvature vanishes. This amounts to solving a nonlinear partial differential equation for matrices with given boundary values. Generalizing and strengthening a result of Berndtsson, we prove that solutions do exist for certain boundary metrics, assuming the existence of a “subsolution.” The proof is based on the continuity method with a result of Coifman and Semmes concerning positive-definite metrics as the starting point.