Timelike surfaces in the de Sitter space S-1(3) (1) subset of R-1(4)

Author(s)

Dussan, M. P. ; Franco Filho, A. P. ; Magid, M.

• Document type: Journal article
• Source: ANNALS OF GLOBAL ANALYSIS AND GEOMETRY; v. 61, n. 4, p. 20-pg., 2022-03-25.

Abstract

This paper studies timelike minimal surfaces in the De Sitter space S-1(3)(1) subset of R-1(4) via a complex variable. Using complex analysis and stereographic projection of lightlike vectors in C boolean OR{infinity}, we obtain a complex representation formula, together with some results about the existence of convenient isotropic coordinates. This allows us to construct timelike minimal surfaces in S-1(3) (1) via local solutions of a certain PDE in a complex variable which arises when investigating our geometric conditions. Specifically, we find a new kind of complex functions which generalize the classes of holomorphic and anti-holomorphic functions, which we call quasi-holomorphic functions. We show that there is a correspondence between a timelike minimal surface in S-1(3) (1) and a pair of quasi-holomorphic functions. In particular, when the two functions are holomorphic, we showthat they are related by aMobius transformation and then construct many families of minimal timelike surfaces in S-1(3)(1) whose intrinsic Gauss map will also belong to the same class of surfaces. Several explicit examples are given. (AU)

• FAPESP's process: 16/23746-6 - Algebraic, topological and analytical techniques in differential geometry and geometric analysis
• Grantee: Paolo Piccione
• Support Opportunities: Research Projects - Thematic Grants