Interpolation by maximal and minimal surfaces

TL;DR

This paper develops a method using the inverse function theorem to interpolate between real analytic curves in Lorentz-Minkowski and Euclidean spaces via maximal and minimal surfaces, leveraging classical solutions to the Björling problem.

Contribution

It introduces a novel approach to interpolate real analytic curves with maximal and minimal surfaces using the inverse function theorem, extending classical geometric methods.

Findings

Successful interpolation of curves using maximal and minimal surfaces.

Application of the inverse function theorem in Lorentz-Minkowski and Euclidean spaces.

Utilization of the Björling problem in the interpolation process.

Abstract

In this article, we use the inverse function theorem for Banach spaces to interpolate a given real analytic spacelike curve $a$ in Lorentz-Minkowski space $\mathbb{L}^3$ to another real analytic spacelike curve $c$, which is ``close" enough to $a$ in a certain sense by constructing a maximal surface containing them. Next we apply the same method to interpolate two given real analytic curve $a$ in Euclidean space $\mathbb{E}^3$ and a real analytic curve $c$, which is also ``close" enough to ``a" in a certain sense with a minimal surface. Throughout this study, the Bj\"orling problem and Schwarz's solution to it play pivotal roles.