Global existence of the harmonic map heat flow into Lorentzian manifolds

Abstract

We investigate a parabolic–elliptic system for maps (u,v) from a compact Riemann surface M into a Lorentzian manifold N×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.