# Global existence of the harmonic map heat flow into Lorentzian manifolds

## TL;DR

This paper proves the global existence of solutions for a harmonic map heat flow into Lorentzian manifolds, under certain geometric or energy conditions, leading to the existence of Lorentzian harmonic maps.

## Contribution

It introduces a novel parabolic-elliptic system for maps into Lorentzian manifolds and establishes global existence results under new geometric or energy assumptions.

## Key findings

- Global existence of solutions under geometric conditions
- Global existence with small initial energy
- Existence of Lorentzian harmonic maps in given homotopy classes

## Abstract

We investigate a parabolic-elliptic system for maps $(u,v)$ from a compact Riemann surface $M$ into a Lorentzian manifold $N\times{\mathbb{R}}$ with a warped product metric. That system turns the harmonic map type equations into a parabolic system, but keeps the $v$-equation as a nonlinear second order constraint along the flow. We prove a global existence result of the parabolic-elliptic system by assuming either some geometric conditions on the target Lorentzian manifold or small energy of the initial maps. The result implies the existence of a Lorentzian harmonic map in a given homotopy class with fixed boundary data.