A global weak solution to the Lorentzian harmonic map flow
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.
Keywordsharmonic map heat flow Lorentzian manifold warped product blow up weak solution
Unable to display preview. Download preview PDF.
This work was supported by National Natural Science Foundation of China (Grant Nos. 11471014 and 11471299) and the Fundamental Research Funds for the Central Universities. The authors are grateful to the anonymous referees for their careful reading of the manuscript and many helpful suggestions.
- 9.Han X L, Jost J, Liu L, et al. Global existence of the harmonic map heat ow into Lorentzian manifolds. Https://www.mis.mpg.de/preprints/2016/preprint2016 70.pdf, 2016Google Scholar
- 15.Kramer D, Stephani H, Hertl E, et al. Exact Solutions of Einstein's Field Equations. Cambridge: Cambridge University Press, 1980Google Scholar
- 17.Li J Y, Liu L. Partial regularity of harmonic maps from a Riemannian manifold into a Lorentzian manifold. ArX-iv:1704.08673, 2017Google Scholar