Interpolation by maximal and minimal surfaces

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 Björling problem and Schwarz's solution to it play pivotal roles.

Comments are welcome