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

TL;DR

This paper introduces split-harmonic maps to provide a new proof of the singular Björling problem for timelike minimal surfaces and explores interpolation problems for split-Fourier curves within this context.

Contribution

It offers a novel proof of the singular Björling problem using split-harmonic maps and applies this to interpolation problems for split-Fourier curves in timelike minimal surfaces.

Findings

New proof of the singular Björling problem using split-harmonic maps

Solutions for interpolating split-Fourier curves by timelike minimal surfaces

Extension of harmonic map techniques to split-harmonic maps in minimal surface theory

Abstract

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.