Archiv der Mathematik

, Volume 110, Issue 6, pp 629–635 | Cite as

A volume decreasing theorem for \(\mathbf{V}\)-harmonic maps and applications

  • Guangwen ZhaoEmail author


We establish a volume decreasing result for V-harmonic maps between Riemannian manifolds. We apply this result to obtain corresponding results for Weyl harmonic maps from conformal Weyl manifolds to Riemannian manifolds. We also obtain corresponding results for holomorphic maps from almost Hermitian manifolds to quasi-Kähler manifolds, which generalize or improve the partial results in Goldberg and Har’El (Bull Soc Math Grèce 18(1):141–148, 1977, J Differ Geom 14(1):67–80, 1979).


Volume decreasing V-harmonic map Holomorphic map Almost Hermitian manifold 

Mathematics Subject Classification

58E20 32Q60 32H02 


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The author gratefully acknowledges the much helpful guidance of Professor Qun Chen.


  1. 1.
    Q. Chen, J. Jost, and G. Wang, A maximum principle for generalizations of harmonic maps in Hermitian, affine, Weyl, and Finsler geometry, J. Geom. Anal. 25 (2015), 2407–2426.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Q. Chen and H. Qiu, Rigidity of self-shrinkers and translating solitons of mean curvature flows, Adv. Math. 294 (2016), 517–531.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Q. Chen and G. Zhao, A Schwarz lemma for \(v\)-harmonic maps and their applications, Bull. Aust. Math. Soc. 96 (2017), 504–512.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    S.-s. Chern and S. I. Goldberg, On the volume decreasing property of a class of real harmonic mappings, Amer. J. Math. 97 (1975), 133–147.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    S. I. Goldberg, On the distance-decreasing property of a class of real harmonic mappings, Geom. Dedicata 4 (1975), 61–69.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    S. I. Goldberg and Z. Har’El, A general schwarz lemma for riemannian manifolds, Bull. Soc. Math. Grèce 18 (1977), 141–148.MathSciNetzbMATHGoogle Scholar
  7. 7.
    S. I. Goldberg and Z. Har’El, Mappings of almost Hermitian manifolds, J. Differential Geom. 14 (1979), 67–80.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Z. Har’El, Harmonic mappings of negatively curved manifolds, Canad. J. Math. 30 (1978), 631–637.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    G. Kokarev, On pseudo-harmonic maps in conformal geometry, Proc. London Math. Soc. 99 (2009), 168–194.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    S.-T. Yau, A general Schwarz lemma for Kähler manifolds, Amer. J. Math. 100 (1978), 197–203.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanPeople’s Republic of China
  2. 2.School of Mathematical SciencesFudan UniversityShanghaiPeople’s Republic of China

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