Skip to main content
SpringerLink
Log in

Quantitative gradient estimates for harmonic maps into singular spaces

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we show the Yau’s gradient estimate for harmonic maps into a metric space (X, dX) with curvature bounded above by a constant κ (κ ⩾ 0) in the sense of Alexandrov. As a direct application, it gives some Liouville theorems for such harmonic maps. This extends the works of Cheng (1980) and Choi (1982) to harmonic maps into singular spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Breiner C, Fraser A, Huang L-H, et al. Regularity of harmonic maps from polyhedra to CAT(1) spaces. ArXiv: 1610.07829, 2017

    MATH  Google Scholar 

  2. Burago D, Burago Y, Ivanov S. A Course in Metric Geometry. Graduate Studies in Mathematics, vol. 33. Providence: Amer Math Soc, 2001

    MATH  Google Scholar 

  3. Chen J. On energy minimizing mappings between and into singular spaces. Duke Math J, 1995, 79: 77–99

    MathSciNet  MATH  Google Scholar 

  4. Cheng S Y. Liouville theorem for harmonic maps. Proc Sympos Pure Math, 1980, 36: 147–151

    MathSciNet  Google Scholar 

  5. Choi H I. On the Liouville theorem for harmonic maps. Proc Amer Math Soc, 1982, 85: 91–94

    MathSciNet  MATH  Google Scholar 

  6. Crandall M, Ishii H, Lions P-L. User’s guide to viscosity sulutions of second order partial differential equations. Bull Amer Math Soc (NS), 1992, 27: 1–67

    MATH  Google Scholar 

  7. Daskalopoulos G, Mese C. Harmonic maps from a simplicial complex and geometric rigidity. J Differential Geom, 2008, 78: 269–293

    MathSciNet  MATH  Google Scholar 

  8. Daskalopoulos G, Mese C. Harmonic maps between singular spaces I. Comm Anal Geom, 2010, 18: 257–337

    MathSciNet  MATH  Google Scholar 

  9. Eells J, Fuglede B. Harmonic Maps between Riemannian Polyhedra. Cambridge Tracts in Mathematics, vol. 142. Cambridge: Cambridge University Press, 2001

    MATH  Google Scholar 

  10. Freidin B. A Bochner formula for harmonic maps into non-positively curved metric spaces. ArXiv:1605.08461v2, 2016

    Google Scholar 

  11. Freidin B, Zhang Y. A Liouville-type theorem and Bochner formula for harmonic maps into metric spaces. ArXiv: 1805.04192v1, 2018

    Google Scholar 

  12. Fuglede B. Hölder continuity of harmonic maps from Riemannian polyhedra to spaces of upper bounded curvature. Calc Var Partial Differential Equations, 2003, 16: 375–403

    MathSciNet  MATH  Google Scholar 

  13. Fuglede B. Harmonic maps from Riemannian polyhedra to geodesic spaces with curvature bounded from above. Calc Var Partial Differential Equations, 2008, 31: 99–136

    MathSciNet  MATH  Google Scholar 

  14. Giaquinta M, Giusti E. On the regularity of the minima of variational integrals. Acta Math, 1982, 148: 31–46

    MathSciNet  MATH  Google Scholar 

  15. Giaquinta M, Giusti E. The singular set of the minima of certain quadratic functionals. Ann Sc Norm Super Pisa Cl Sci (5), 1984, 11: 45–55

    MathSciNet  MATH  Google Scholar 

  16. Gromov M, Schoen R. Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one. Publ Math Inst Hautes Études Sci, 1992, 76: 165–246

    MathSciNet  MATH  Google Scholar 

  17. Guo C-Y. Harmonic mappings between singular metric spaces. ArXiv:1702.05086, 2017

    Google Scholar 

  18. Heinonen J, Koskela P, Shanmugalingam N, et al. Sobolev classes of Banach space-valued functions and quasiconformal mappings. J Anal Math, 2001, 85: 87–139

    MathSciNet  MATH  Google Scholar 

  19. Hildebrandt S, Kaul H, Widman K-O. An existence theorem for harmonic mappings of Riemannian manifolds. Acta Math, 1977, 138: 1–16

    MathSciNet  MATH  Google Scholar 

  20. Hua B, Liu S, Xia C. Liouville theorems for f-harmonic maps into Hadamard spaces. Pacific J Math, 2017, 290: 381–402

    MathSciNet  MATH  Google Scholar 

  21. Ishizuka W, Wang C Y. Harmonic maps from manifolds of L1-Riemannian metrics. Calc Var Partial Differential Equations, 2008, 32: 287–405

    Google Scholar 

  22. Jäger W, Kaul H. Rotationally symmetric harmonic maps from a ball into a sphere and the regularity problem for weak solutions of elliptic systems. J Reine Angew Math, 1983, 343: 146–161

    MathSciNet  MATH  Google Scholar 

  23. Jost J. Equilibrium maps between metric spaces. Calc Var Partial Differential Equations, 1994, 2: 173–204

    MathSciNet  MATH  Google Scholar 

  24. Jost J. Generalized harmonic maps between metric spaces. In: Geometric Analysis and the Calculus of Variations for Stefan Hildebrandt. Somerville: Int Press, 1996, 143–174

    MATH  Google Scholar 

  25. Jost J. Generalized Dirichlet forms and harmonic maps. Calc Var Partial Differential Equations, 1997, 5: 1–19

    MathSciNet  MATH  Google Scholar 

  26. Jost J. Nonlinear Dirichlet forms. In: New Directions in Dirichlet Forms. Somerville: Int Press, 1998, 1–47

    MATH  Google Scholar 

  27. Kirchheim B. Rectifiable metric spaces: local structure and regularity of the Hausdorff measure. Proc Amer Math Soc, 1994, 121: 113–123

    MathSciNet  MATH  Google Scholar 

  28. Korevaar N, Schoen R. Sobolev spaces and harmonic maps for metric space targets. Comm Anal Geom, 1993, 1: 561–659

    MathSciNet  MATH  Google Scholar 

  29. Kuwae K, Shioya T. Sobolev and Dirichlet spaces over maps between metric spaces. J Reine Angew Math, 2003, 555: 39–75

    Google Scholar 

  30. Lin F H. Analysis on singular spaces. In: Collection of Papers on Geometry, Analysis and Mathematical Physics. River Edge: World Scientific, 1997, 114–126

    MATH  Google Scholar 

  31. Lin F H, Wang C Y. The Analysis of Harmonic Maps and Their Heat Flows. River Edge: World Scientific, 2008

    Google Scholar 

  32. Mese C. The curvature of minimal surfaces in singular spaces. Comm Anal Geom, 2001, 9: 3–34

    MathSciNet  Google Scholar 

  33. Mese C. Harmonic maps into spaces with an upper curvature bound in the sense of Alexandrov. Math Z, 2002, 242: 633–661

    MathSciNet  MATH  Google Scholar 

  34. Morrey C B. The problem of Plateau on an Riemannian manifold. Ann of Math (2), 1948, 49: 807–851

    MathSciNet  MATH  Google Scholar 

  35. Ohta S-I. Cheeger type Sobolev spaces for metric space targets. Potential Anal, 2004, 20: 149–175

    MathSciNet  MATH  Google Scholar 

  36. Schoen R, Uhlenbeck K. A regularity theory for harmonic maps. J Differential Geom, 1982, 17: 307–335

    MathSciNet  MATH  Google Scholar 

  37. Schoen R, Uhlenbeck K. Regularity of minimizing harmonic maps into the sphere. Invent Math, 1984, 78: 89–100

    MathSciNet  MATH  Google Scholar 

  38. Schoen R, Yau S T. Lectures on Differential Geometry. Boston: Int Press, 1994

    MATH  Google Scholar 

  39. Serbinowski T. Harmonic Maps into Metric Spaces with Curvature Bounded from above. PhD Thesis. Salt Lake City: The University of Utah, 1995

    Google Scholar 

  40. Sidler H, Wneger S. Harmonic quasi-isometric maps into Gromov hyperbolic CAT(0)-spaces. ArXiv:1804.06286v1, 2018

    Google Scholar 

  41. Sinaei Z. Riemannian polyhedra and Liouville-type theorems for harmonic maps. Anal Geom Metric Spaces, 2014, 2: 294–318

    MathSciNet  MATH  Google Scholar 

  42. Sturm K-T. A semigroup approach to harmonic maps. Potential Anal, 2005, 23: 225–277

    MathSciNet  MATH  Google Scholar 

  43. Yau S-T. Harmonic functions on complete Riemannian manifolds. Comm Pure Appl Math, 1975, 28: 201–228

    MathSciNet  MATH  Google Scholar 

  44. Zhang H C, Zhu X P. Local Li-Yau’s estimates on RCD(K;N) metric measure spaces. Calc Var Partial Differential Equations, 2016, 55: 93

    MathSciNet  MATH  Google Scholar 

  45. Zhang H C, Zhu X P. Lipschitz contunuity of harmonic maps between Alexandrov spaces. Invent Math, 2018, 211: 863–934

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The first and third authors were supported by National Natural Science Foundation of China (Grant No. 11521101). The first author was also supported by National Natural Science Foundation of China (Grant No. 11571374) and National Program for Support of Top-Notch Young Professionals. The second author was supported by the Academy of Finland. Part of the work was done when the first author visited the Department of Mathematics and Statistics, University of Jyväskylä for one month in 2016. The first author thanks the department for the hospitality.

Author information

Authors and Affiliations

  1. Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

    Hui-Chun Zhang & Xi-Ping Zhu

  2. Department of Mathematics and Statistics, University of Helsinki, Helsinki, FI-00014, Finland

    Xiao Zhong

Authors
  1. Hui-Chun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiao Zhong
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xi-Ping Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xi-Ping Zhu.

Additional information

Dedicated to Professor Lo Yang on the Occasion of His 80th Birthday

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, HC., Zhong, X. & Zhu, XP. Quantitative gradient estimates for harmonic maps into singular spaces. Sci. China Math. 62, 2371–2400 (2019). https://doi.org/10.1007/s11425-018-9493-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-018-9493-1

Keywords

MSC(2010)

Search

Navigation