Timelike surfaces in the de Sitter space $\mathbb S^3_1(1)\subset \mathbb R^4_1$

TL;DR

This paper develops a complex analysis framework to represent and construct timelike minimal surfaces in de Sitter space, introducing quasi-holomorphic functions and explicit solutions to PDEs for surface classification.

Contribution

It introduces a novel complex variable approach and a new class of quasi-holomorphic functions for representing timelike minimal surfaces in de Sitter space.

Findings

Derived a representation formula using complex analysis and stereographic projection.

Identified special quadrics with grassmannians of 2-planes in Minkowski space.

Constructed explicit families of minimal timelike surfaces with intrinsic Gauss maps.

Abstract

This paper studies timelike minimal surfaces in the De Sitter space $\mathbb S^3_1(1) \subset \mathbb R^4_1$ via a complex variable. Using complex analysis and stereographic projection of lightlike vectors we obtain a representation formula. Real and complex special quadrics in $\mathbb CP^3$ are identified with the grassmannians of spacelike and timelike oriented 2-planes of $\mathbb R^4_1$, and the normal frame is written in terms of certain complex valued functions $x$ and $y$, which may be considered holomorphic functions as a special case. Then several results describing the analytic restrictions via solutions of certain PDE in complex variable, are shown. Finding solutions allows us to identify explicitly the representation of the associated surfaces. Moreover, using our technique we find a new kind of complex function which we call quasi-holomorphic and which satisfy a generalized version of the Cauchy-Riemann equations. Our technique allows the explicit construction of many families of minimal timelike surfaces in $\mathbb S^3_1(1)$ whose intrinsic Gauss map will also belong to the same class of surfaces.