Extension of Krust theorem and deformations of minimal surfaces

TL;DR

This paper extends Krust's theorem to a new deformation family connecting minimal and maximal surfaces, and explores the implications for various deformation families and isotropic space, using harmonic mapping theory.

Contribution

It introduces a novel deformation family linking minimal and maximal surfaces and proves Krust-type theorems within this framework, expanding the theorem's applicability.

Findings

Krust-type theorem established for the new deformation family.

Graphness of minimal surfaces in isotropic space influences deformed surfaces.

Results rely on advances in planar harmonic mapping theory.

Abstract

In the minimal surface theory, the Krust theorem asserts that if a minimal surface in the Euclidean 3-space $\mathbb{E}^3$ is the graph of a function over a convex domain, then each surface of its associated family is also a graph. The same is true for maximal surfaces in the Minkowski 3-space $\mathbb{L}^3$. In this article, we introduce a new deformation family that continuously connects minimal surfaces in $\mathbb{E}^3$ and maximal surfaces in $\mathbb{L}^3$, and prove a Krust-type theorem for this deformation family. This result induces Krust-type theorems for various important deformation families containing the associated family and the L\'opez-Ros deformation. Furthermore, minimal surfaces in the isotropic 3-space $\mathbb{I}^3$ appear in the middle of the above deformation family. We also prove another type of Krust's theorem for this family, which implies that the graphness of such minimal surfaces in $\mathbb{I}^3$ strongly affects the graphness of deformed surfaces. The results are proved based on the recent progress of planar harmonic mapping theory.