Some partial differential equations and conformal surfaces of the 4-dimensional Minkowski space

TL;DR

This paper develops a complex representation for spacelike surfaces in 4D Minkowski space, deriving PDEs like the Riccati equation, and constructs explicit solutions to classify certain conformal immersions and their geometric properties.

Contribution

It introduces a new complex representation for spacelike surfaces in Minkowski space and links solutions of PDEs to geometric classifications of conformal immersions.

Findings

Explicit solutions to the partial Riccati equation

Characterization of conformal totally umbilical spacelike immersions

Correspondence between solutions and Bryant immersions in hyperbolic space

Abstract

This paper introduces a complex representation for spacelike surfaces in the Lorentz-Minkowski space $L^4$, based in two complex valued functions which can be assumed to be holomorphic or anti-holomorphic. When the immersion is contained in quadrics of $L^4$, the representation then allows us to obtain interesting partial differential equations with holomorphic or anti-holomorphic parameters, within which we find the partial Riccati Equation. Using then theory of holomorphic complex functions we construct explicitly new local solutions for those PDEs together with its associated geometric solutions. So, several explicit examples are given. As geometric consequence, through of our approach we characterize all conformal totally umbilical spacelike immersions into $L^4$, and in addition, we also show that for each conformal immersion in $L^4$ which satisfies the partial Riccati equation there exists a Bryant immersion in $H^3$, both immersions being congruent by a translation vector.