Ricerche di Matematica

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

A variational problem for pullback metrics

Article

Abstract

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 f1 and f2 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.

Keywords

Variational problem Pullback metric Monotonicity formula Bochner type formula 

Mathematics Subject Classification (2000)

53C43 58E20 

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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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