Ricerche di Matematica

, Volume 60, Issue 2, pp 219–235 | Cite as

A variational problem for pullback metrics



Let (Mg) and (Nh) be Riemannian manifolds without boundary and let f : MN be a smooth map. Let \({\|f^*h\|}\) denote the norm of the pullback metric of h by f. In this paper, we consider the functional \({{\Phi (f) = \int_M \|f^*h\|^2 dv_g}}\). We prove the existence of minimizers of the functional Φ in each 3-homotopy class of maps, where maps f 1 and f 2 are 3-homotopic if they are homotopic on the three dimensional skeltons of a triangulation of M. Furthermore, we give a monotonicity formula and a Bochner type formula.


Variational problem Pullback metric Monotonicity formula Bochner type formula 

Mathematics Subject Classification (2000)

53C43 58E20 


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  1. 1.
    Bethuel F.: The approximation problem for Sobolev maps between two manifolds. Acta Math. 167, 153–206 (1991)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Eells J., Lemaire L.: A report on harmonic maps. Bull. Lond. Math. Soc. 10, 1–68 (1978)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Eells J., Lemaire L.: Another report on harmonic maps. Bull. Lond. Math. Soc. 20, 385–524 (1988)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Kawai, S., Nakauchi, N.: Some results for stationary maps of a functional related to pullback metrics. Nonlinear Anal. (to appear)Google Scholar
  5. 5.
    Kawai, S., Nakauchi, N.: Weak conformality of stable stationary maps for a functional related to conformality (preprint)Google Scholar
  6. 6.
    Nakauchi, N.: A variational problem related to conformal maps. Osaka J. Math. (to appear)Google Scholar
  7. 7.
    Price P.: A monotonicity formula for Yang-Mills fields. Manuscripta Math. 43, 131–166 (1983)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Urakawa, H.: Calculus of Variations and Harmonic Maps, Translations of Monographs, vol. 132. American Mathematical Society (1993)Google Scholar
  9. 9.
    White B.: Infima of energy functionals in homotopy classes of mappings. J. Diff. Geom. 23, 127–142 (1986)MATHGoogle Scholar
  10. 10.
    White B.: Homotopy classes in Sobolev spaces and the existence of energy minimizing maps. Acta Math. 160, 1–17 (1988)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Xin Y.: Geometry of Harmonic Maps. Birkhäuser, Boston (1996)MATHCrossRefGoogle Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2011

Authors and Affiliations

  1. 1.Graduate School of Science and EngineeringYamaguchi UniversityYamaguchiJapan

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