On the Bj ̈orling problem for Born-Infeld solitons and the interpolation problem for timelike minimal surfaces

Abstract

Minimal surfaces are zero mean curvature surfaces that appear in nature as idealized soap films. The minimal surface theory is filled with lots of beautiful geometric results, bridging various mathematical branches such as complex analysis, functional analysis, PDE theory, and so on. This thesis is a combination of mainly two completed research works and one ongoing research work about zero mean curvature surfaces. The Bj ̈orling problem and its solution is a well-known result for minimal surfaces in Euclidean three-space. The minimal surface equation is similar to the Born-Infeld equation, which is naturally studied in physics. For the first research work, we ask the question of the Bj ̈orling problem for Born-Infeld solitons. This begins with the case of locally Born-Infeld soliton surfaces and later moves on to graph-like surfaces. We also present some results about their representation formulae. The singular Bj ̈orling problem and its solution for timelike minimal surfaces is another famous result in minimal surface theory. In the second research work, we give different proofs 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 timeline 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. The third and ongoing research work is about understanding the geometry behind the interpolation problems of minimal surfaces. Jesse Douglas earlier gave some existence results for interpolation problems of minimal surfaces, based on area. We try to make these results more concrete by studying the relationship between the existence of minimal surfaces inter-polating two curves with the distance between them and giving the explicit parametrization of such minimal surfaces.