A global weak solution to the Lorentzian harmonic map flow

TL;DR

This paper proves the existence of a unique global weak solution to a Lorentzian harmonic map flow system, with limited singularities, advancing understanding of harmonic maps into Lorentzian manifolds.

Contribution

It establishes the first global weak solution existence result for Lorentzian harmonic map flow with finite singularities.

Findings

Existence of a unique global weak solution

Solution is regular except for finitely many points

Advances mathematical understanding of Lorentzian harmonic maps

Abstract

We investigate a parabolic-elliptic system which is related to a harmonic map from a compact Riemann surface with a smooth boundary into a Lorentzian manifold with a warped product metric. We prove that there exists a unique global weak solution for this system which is regular except for at most finitely many singular points.