Chen-Gackstatter type surfaces in R^4_1: deformation, symmetry, and embeddedness

TL;DR

This paper explicitly constructs and analyzes deformations of Chen-Gackstatter surfaces in four-dimensional Lorentzian space, exploring their symmetry, embeddedness, and uniqueness properties.

Contribution

It introduces explicit 2- and 4-parameter deformation families of Chen-Gackstatter surfaces in R^4_1, and establishes a uniqueness theorem under symmetry assumptions.

Findings

Explicit 2-parameter deformation family constructed.

Existence of a larger 4-parameter deformation family shown.

Partial results on embeddedness discussed.

Abstract

We find a 2-parameter family of deformations in R^4_1 of the classical Chen-Gackstatter surface explicitly, and show the existence of a larger 4-parameter family of deformations. Each of them still has genus one, a unique end, with total Gaussian curvature $-\int K=8\pi$. On the other hand, a uniqueness theorem is obtained when we assume that the surface has more than 4 symmetries. The problem of embeddedness is also discussed with some partial results.