Split-harmonic maps and the interpolation problem for timelike minimal surfaces

The singular Bj\" orling problem and its solution for timelike minimal surfaces is a well-known result in minimal surface theory. In this article, we give a different proof of this theorem using split-harmonic maps. This is motivated by a similar solution of the singular Bj\"orling problem for maximal surfaces using harmonic maps. As an application, we study the problem of interpolating a given split-Fourier curve to a point by a timelike minimal surface. This is inspired by an analogous result for maximal surfaces. We also solve the problem of interpolating a given split-Fourier curve to another specified split-Fourier curve by a timelike minimal surface.