Abstract
Using the bubbling argument of Sacks and Uhlenbeck, we prove the existence of n-harmonic maps from the n-sphere to Riemannian manifolds. An application is made to a problem concerning manifolds with strongly pth moment stable stochastic dynamical systems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Duzaar, F. and Fuchs, M.: On removable singularities of pharmonic maps, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), 385–405.
Duzaar, F and Fuchs, M.: Existence and regularity of functions which minimize certain energies in homotopy classes of mappings, Asymptot. Anal. 5 (1991), 129–144.
Eells, J. and Lemaire, L.: A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1–68.
Eells, J. and Lemaire, L.: Another report on harmonic maps, Bull. London Math Soc. 20 (1988), 385–524.
Elworthy, K.D. and Rosenberg, S.: Homotopy and homology vanishing theorems and the stability of stochastic flows, Geom. Funct. Anal. 6(1) (1996), 51–78.
Eells, J. and Sampson, J.H.: Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160.
Hardt, R. and Lin, F.H.: Mappings minimizing the L p norm of the gradient, Comm. Pure Appl. Math. 40 (1987), 555–588.
Jost, J.: A conformally invariant variational problem for mappings between Riemannian manifolds, Preprint of Centre for Math. Analysis, Australian National Univ., 1984.
Mou, L. and Yang, P.: Regularity for n-narmonic maps, J. Geom. Anal. 6(1) (1996), 91–112.
Meeks, W. H. III and Yau, S.T.: Topology of three dimensional manifolds and the embedding problems in minimal surface theory, Ann. of Math. 112 (1980), 441–484.
Nakauchi, N. and Takakuwa, S.: A remark on p-harmonic maps, Nonlinear Anal. 25(2) (1995), 169–185.
Ochiai, T.: On properties of Douglas—Rad´o—Morrey solutions with some topics on existence of minimal surfaces (in Japanese). In: Minimal Surfaces, Reports on Global Analysis 4(3) (1982), 1–311.
Sacks, J. and Uhlenbeck, K.: The existence of minimal immersions of 2spheres, Ann. of Math. 113 (1981), 1–24.
Takeuchi, H.: Stability and Liouville theorems of P-harmonic maps, Japan. J. Math. 17 (1991), 317–332.
Takeuchi, H.: Some conformal properties of pharmonic maps and a regularity for spherevalued p-harmonic maps, J. Math. Soc. Japan 46 (1994), 217–234.
Tolksdorf, P.: Everywhere-regularity for some quasilinear systems with a lack of ellipticity, Ann. Mat. Pure Appl. 134 (1983), 241–266.
Wei, S. W.: The minima of the p-energy functional, Elliptic and Parabolic Method in Geometry (Minneapolis, MN, 1994) pp. 171–203.
Wei, S. W.: On pharmonic maps and their applications to geometry, topology, and analysis (preprint).
White, B.: Homotopy classes in Sobolev spaces and the existence of energy minimizing maps, Acta Math. 160 (1988), 1–17
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kawai, S., Nakauchi, N. & Takeuchi, H. On the Existence of n-Harmonic Spheres. Compositio Mathematica 117, 35–46 (1999). https://doi.org/10.1023/A:1000644910148
Issue Date:
DOI: https://doi.org/10.1023/A:1000644910148