Abstract
Title: Deformation of zero mean curvature surfaces and its application
Abstract: Minimal surfaces in the Euclidean 3-space and maximal surfaces in the Minkowski 3-space share many interesting properties. In this talk, we discuss a deformation family connecting these surfaces with changing ambient spaces and some topics related to such a deformation.
As one of such topics, we mainly focus on the graphness of zero mean curvature surfaces. In the minimal surface theory, the Krust theorem asserts that if a minimal surface in the Euclidean 3-space 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. In this talk, we prove a Krust-type theorem for a deformation family including various important deformation of minimal and maximal surfaces such as the associated family and the Lopez-Ros deformation and the Calabi-type duality correspondence. We also prove another Krust-type theorem which does not assume the convexity assumption.
The results are proved based on the recent progress of planar harmonic mapping theory.
This talk is mainly based on the paper below, which is the joint work with Hiroki Fujino (Nagoya University, Japan).
Reference: S. Akamine and H. Fujino, Extension of Krust theorem and deformations of minimal surfaces, Ann. Mat. Pura Appl. (4).