Spacelike Surfaces in L-4 with null mean curvature vector and the nonlinear Riccati partial differential equation

Author(s)

Dussan, M. P. ^[1] ; Franco Filho, A. P. ^[1] ; Simoes, P. ^[1]

Affiliation

^[1] Univ Sao Paulo, Dept Matemat IME, BR-05508090 Sao Paulo - Brazil

• Document type: Journal article
• Source: NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS; v. 207, JUN 2021.
• Web of Science Citations: 0

Abstract

This paper introduces a complex Weierstrass representation for weakly conformal spacelike immersions in the Lorentz-Minkowski space L-4 involving holomorphic or anti-holomorphic functions. Factoring through the light cone, we parametrize these immersions by three complex functions ( mu, a, b). We prove the existence of a correspondence between those immersions for which a is holomorphic and b satisfies a Riccati partial differential equation, with conformal immersions in the hyperbolic space H-3 with lightlike mean curvature vector. As a principal application of this correspondence, we obtain a powerful tool to construct new, explicit solutions of nonlinear Riccati partial differential equation and their explicit, associated surfaces. We also use our complex representation to classify the totally umbilic, weakly conformal immersions in L-4, when a and b are both anti-holomorphic functions. (C) 2021 Elsevier Ltd. All rights reserved. (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