The Geometric Cauchy Problem for Surfaces With Lorentzian Harmonic Gauss maps

TL;DR

This paper addresses the geometric Cauchy problem for specific surfaces in pseudo-Riemannian manifolds, establishing existence, uniqueness, and construction methods for solutions using loop group techniques and harmonic map representations.

Contribution

It provides a comprehensive analysis of the Cauchy problem for pseudospherical and timelike CMC surfaces, including explicit solution formulas and conditions for characteristic initial data.

Findings

Unique solutions for non-characteristic initial curves

Necessary and sufficient conditions for characteristic initial curves

Explicit construction formulas using loop group techniques

Abstract

The geometric Cauchy problem for a class of surfaces in a pseudo-Riemannian manifold of dimension 3 is to find the surface which contains a given curve with a prescribed tangent bundle along the curve. We consider this problem for constant negative Gauss curvature surfaces (pseudospherical surfaces) in Euclidean 3-space, and for timelike constant non-zero mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space. We prove that there is a unique solution if the prescribed curve is non-characteristic, and for characteristic initial curves (asymptotic curves for pseudospherical surfaces and null curves for timelike CMC) it is necessary and sufficient for similar data to be prescribed along an additional characteristic curve that intersects the first. The proofs also give a means of constructing all solutions using loop group techniques. The method used is the infinite dimensional d'Alembert type representation for surfaces associated with Lorentzian harmonic maps (1-1 wave maps) into symmetric spaces, developed since the 1990's. Explicit formulae for the potentials in terms of the prescribed data are given, and some applications are considered.