Abstract
We prove that a map f : M → N with finite p-energy, p > 2, from a complete manifold \({\left(M,\left\langle ,\right\rangle \right)}\) into a non-positively curved, compact manifold N is homotopic to a constant, provided the negative part of the Ricci curvature of the domain manifold is small in a suitable spectral sense. The result relies on a Liouville-type theorem for finite q-energy, p-harmonic maps under spectral assumptions.
Similar content being viewed by others
References
Burstall F.: Harmonic maps of finite energy from non-compact manifolds. J. Lond. Math. Soc. 30, 361–370 (1984)
Duzaar F., Fuchs M.: On removable singularities of p-harmonic maps. Ann. Inst. H. Poincaré Anal. Non Linaire 7(5), 385–405 (1990)
Nakauchi N.: A Liouville type theorem for p-harmonic maps. Osaka J. Math. 35(2), 303–312 (1998)
Pigola S., Rigoli M., Setti A.G.: Vanishing theorems on riemannian manifolds and geometric applications. J. Funct. Anal. 229, 424–461 (2005)
Pigola S., Rigoli M., Setti A.G.: Constancy of p-harmonic maps of finite q-energy into non-positively curved manifolds. Math. Z. 258(2), 347–362 (2008a)
Pigola S., Rigoli M., Setti A.G.: Vanishing and finiteness results in geometric analysis: a generalization of the Bochner technique. Progress in mathematics, vol. 266, pp. xiv+282. Birkhäuser, Basel (2008b)
Schoen R., Yau S.T.: Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative ricci curvature. Comm. Math. Helv. 51, 333–341 (1976)
Takegoshi K.: A maximum principle for p-harmonic maps with L q finite energy. Proc. Am. Math. Soc. 126(12), 3749–3753 (1998)
Wei S.W.: The minima of the p-energy functional elliptic and parabolic methods in geometry (Minneapolis 1994), pp. 171–203. AK Peters, Wellesley (1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pigola, S., Veronelli, G. On the homotopy class of maps with finite p-energy into non-positively curved manifolds. Geom Dedicata 143, 109–116 (2009). https://doi.org/10.1007/s10711-009-9376-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-009-9376-z