Elliptic differential inequalities with applications to harmonic maps

Andrea RATTO; Marco RIGOLI

doi:10.2969/jmsj/04520321

Andrea RATTO, Marco RIGOLI

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JOURNAL FREE ACCESS

1993 Volume 45 Issue 2 Pages 321-337

DOI https://doi.org/10.2969/jmsj/04520321

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Published: 1993 Received: August 13, 1991 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) E. Calabi, An extension of E. Hopf's maximum principle, Duke Math. J., 25 (1937), 45-56.
2) Q. Chen and Y. L. Xin, A generalized maximum principle and its applications in geometry, Preprint MSRI (1990), to appear in Amer. J. Math..
3) S. Y. Cheng, Liouville's theorems for harmonic maps, Proc. Sympos. Pure Math., 36 (1980), 91-94.
4) H. I. Choi and A. Treibergs, Gauss maps of space-like constant mean curvature hypersurfaces of Minkowski space, J. Differential Geom., 32 (1990), 775-817.
5) J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc., 10 (1978), 1-68.
6) J. Eells and L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc., 20 (1988), 385-524.
7) J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160.
8) R. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Math., 699, Springer, 1979.
9) L. Jorge and D. Koutroufiotis, An estimate for the curvature of bounded submanifolds, Amer. J. Math., 103 (1981), 711-725.
10) L. Karp, Differential inequalities on complete Riemannian manifolds and applications, Math. Ann., 272 (1985), 449-459.
11) A. Kasue, Estimates for solutions of Poisson equations and their applications to submanifolds, (ed. K. Kenmotsu), Lecture Notes in Math., 1090, Springer, 1984, pp. 1-14.
12) R. Osserman, On the inequality Δu_??_f(u), Pacific J. Math., 7 (1957), 1641-1647.
13) A. Ratto and M. Rigoli, On the asymptotic behaviour of rotationally symmetric harmonic maps, J. Differential Equations, to appear.
14) R. Redheffer, A classification of solutions of certain nonlinear differential inequalities with applications to theorems of Liouville type, Math. Z., 192 (1986), 453-465.
15) R. Redheffer, On the inequality Δu_??_f (u, |grad u|), J. Math. Anal. Appl., 1 (1960), 277-299.
16) A. Tachikawa, A non-existence result for harmonic mappings from Rⁿ to Hⁿ, Tokyo J. Math., 11 (1988), 311-316.
17) A. Tachikawa, Harmonic mappings from Rⁿ into an Hadamard manifold, J. Math. Soc. Japan, 42 (1990), 147-153.
18) S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math., 28 (1975), 201-228.

Right : [1] E. Calabi, An extension of E. Hopf's maximum principle, Duke Math. J., 25 (1957), 45-56.
[2] Q. Chen and Y. L. Xin, A generalized maximum principle and its applications in geometry, Preprint MSRI (1990), to appear in Amer. J. Math..
[3] S. Y. Cheng, Liouville's theorems for harmonic maps, Proc. Sympos. Pure Math., 36 (1980), 91-94.
[4] H. I. Choi and A. Treibergs, Gauss maps of space-like constant mean curvature hypersurfaces of Minkowski space, J. Differential Geom., 32 (1990), 775-817.
[5] J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc., 10 (1978), 1-68.
[6] J. Eells and L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc., 20 (1988), 385-524.
[7] J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160.
[8] R. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Math., 699, Springer, 1979.
[9] L. Jorge and D. Koutroufiotis, An estimate for the curvature of bounded submanifolds, Amer. J. Math., 103 (1981), 711-725.
[10] L. Karp, Differential inequalities on complete Riemannian manifolds and applications, Math. Ann., 272 (1985), 449-459.
[11] A. Kasue, Estimates for solutions of Poisson equations and their applications to submanifolds, (ed. K. Kenmotsu), Lecture Notes in Math., 1090, Springer, 1984, pp. 1-14.
[12] R. Osserman, On the inequality Δu≥f(u), Pacific J. Math., 7 (1957), 1641-1647.
[13] A. Ratto and M. Rigoli, On the asymptotic behaviour of rotationally symmetric harmonic maps, J. Differential Equations, to appear.
[14] R. Redheffer, A classification of solutions of certain nonlinear differential inequalities with applications to theorems of Liouville type, Math. Z., 192 (1986), 453-465.
[15] R. Redheffer, On the inequality Δu≥f (u, |grad u|), J. Math. Anal. Appl., 1 (1960), 277-299.
[16] A. Tachikawa, A non-existence result for harmonic mappings from Rⁿ to Hⁿ, Tokyo J. Math., 11 (1988), 311-316.
[17] A. Tachikawa, Harmonic mappings from Rⁿ into an Hadamard manifold, J. Math. Soc. Japan, 42 (1990), 147-153.
[18] S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math., 28 (1975), 201-228.

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