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Interpolation by maximal and minimal surfaces

Rukmini Dey, Rahul Kumar Singh

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.

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 in Lorentz-Minkowski space to another real analytic spacelike curve , which is ``close" enough to in a certain sense by constructing a maximal surface containing them. Next we apply the same method to interpolate two given real analytic curve in Euclidean space and a real analytic curve , 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.

Paper Structure

This paper contains 15 sections, 15 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Inverting the Björling-Schwarz Function
  3. Interpolation by maximal surface
  4. Existence of sup-norm open ball containing an appropriate $x_0$ in $V^{\Omega}$
  5. Local interpolation by maximal surface
  6. Interpolation by minimal surfaces
  7. Future Directions
  8. Appendix
  9. $\tilde{d}(u,0)$ is real
  10. Definition of $V_1$ is consistent
  11. Definition of $\widetilde{V}$ on all of $\Omega$:
  12. Solving the ODE
  13. Crossproduct
  14. Expression for $K_2$
  15. Acknowledgement

Key Result

Proposition 2.1

We have the following proposition.

Theorems & Definitions (33)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 23 more