Global existence of the harmonic map heat flow into Lorentzian manifolds
Abstract
We investigate a parabolicelliptic 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 parabolicelliptic 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2019
 DOI:
 10.48550/arXiv.1901.00901
 arXiv:
 arXiv:1901.00901
 Bibcode:
 2019arXiv190100901H
 Keywords:

 Mathematics  Differential Geometry
 EPrint:
 to appear in J. Math. Pures Appl