A global weak solution to the Lorentzian harmonic map flow

  • Xiaoli Han
  • Lei Liu
  • Liang ZhaoEmail author


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.


harmonic map heat flow Lorentzian manifold warped product blow up weak solution 


53C43 58E20 


Unable to display preview. Download preview PDF.

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.


  1. 1.
    Al'ber S I. On n-dimensional problems in the calculus of variations in the large. Sov Math Dokl, 1964, 5: 700–704zbMATHGoogle Scholar
  2. 2.
    Al'ber S I. Spaces of mappings into a manifold with negative curvature. Sov Math Dokl, 1967, 9: 6–9zbMATHGoogle Scholar
  3. 3.
    Benci V, Fortunato D, Giannoni F. On the existence of multiple geodesics in static space-times. Ann Inst H Poincaré Anal Non Linéaire, 1985, 2: 119–141CrossRefzbMATHGoogle Scholar
  4. 4.
    Chen Y, Levine S. The existence of the heat ow of H-systems. Discrete Contin Dyn Syst, 2002, 8: 219–236MathSciNetzbMATHGoogle Scholar
  5. 5.
    Eells J, Sampson J. Harmonic mappings of Riemannian manifolds. Amer J Math, 1964, 86: 109–160MathSciNetzbMATHGoogle Scholar
  6. 6.
    Greco C. The Dirichlet-problem for harmonic maps from the disk into a lorentzian warped product. Ann Inst H Poincaré Anal Non Linéaire, 1993, 10: 239–252MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Greco C. A multiplicity result for the Dirichlet problem for harmonic maps from the disk into a Lorentzian warped product. Nonlinear Anal, 1997, 28: 1661–1668MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hamilton R. Harmonic Maps of Manifolds with Boundary. Lecture Notes in Mathematics, vol. 471. Berlin-Heidelberg: Springer, 1975CrossRefzbMATHGoogle Scholar
  9. 9.
    Han X L, Jost J, Liu L, et al. Global existence of the harmonic map heat ow into Lorentzian manifolds. Https:// 70.pdf, 2016Google Scholar
  10. 10.
    Han X L, Jost J, Liu L, et al. Bubbling analysis for approximate Lorentzian harmonic maps from Riemann surfaces. Calc Var Partial Differential Equations, 2017, 56: 175MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Han X L, Zhao L, Zhu M M. Energy identity for harmonic maps into standard stationary Lorentzian manifolds. J Geom Phys, 2017, 114: 621–630MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Isobe T. Optimal regularity of harmonic maps from a Riemannian manifold into a static Lorentzian manifold. Pacific J Math, 1997, 178: 71–93MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Isobe T. Regularity of harmonic maps into a static Lorentzian manifold. J Geom Anal, 1998, 8: 447–463MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Jost J, Liu L, Zhu M M. A global weak solution of the Dirac-harmonic map ow. Ann Inst H Poincaré Anal Non Linéaire, 2017, 34: 1851–1882CrossRefzbMATHGoogle Scholar
  15. 15.
    Kramer D, Stephani H, Hertl E, et al. Exact Solutions of Einstein's Field Equations. Cambridge: Cambridge University Press, 1980Google Scholar
  16. 16.
    Ladyzenskaja O, Solonnikov V A, Ural'ceva N N. Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs, vol. 23. Providence: Amer Math Soc, 1968CrossRefGoogle Scholar
  17. 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
  18. 18.
    O'Neill B. Semi-Riemannian Geometry: With Applications to Relativity. Pure and Applied Mathematics, vol. 103. New York: Academic Press, 1983zbMATHGoogle Scholar
  19. 19.
    Struwe M. On the evolution of harmonic mappings of Riemannian surfaces. Comment Math Helv, 1985, 60: 558–581MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zhu M M. Regularity for harmonic maps into certain pseudo-Riemannian manifolds. J Math Pures Appl (9), 2013, 99: 106–123MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingChina
  2. 2.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  3. 3.School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems of MOEBeijing Normal UniversityBeijingChina

Personalised recommendations