A stochastic approach to a Liouville property for plurisubharmonic functions

Hiroshi KANEKO

doi:10.2969/jmsj/04120291

Hiroshi KANEKO

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JOURNAL FREE ACCESS

1989 Volume 41 Issue 2 Pages 291-299

DOI https://doi.org/10.2969/jmsj/04120291

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Published: 1989 Received: December 07, 1987 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. Bedford and B. A. Taylor, Variational properties of the complex Monge-Ampère equation I, Dirichlet principle, Duke Math. J., 45 (1978), 375-403.
2) E. Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math., 149 (1982), 1-44.
3) D. Burns, Curvature of Monge-Ampère foliation and parabolic manifolds, Ann. Math., 115 (1982), 349-373.
4) E. D. Dynkin, Markov Processes, Springer, 1965.
5) M. Fukushima, Dirichlet forms and Markov processes, North-Holland and Kodansha, 1980.
6) M. Fukushima, On holomorphic diffusions and plurisubharmonic functions, in “Geometry of Random Motion” Contemporary Mathematics, to appear.
7) M. Fukushima, On the continuity of plurisubharmonic functions along conformal diffusions, Osaka J. Math., 23 (1986), 69-75.
8) M. Fukushima and M. Okada, On conformal martingale diffusions and pluripolar set, J. Funct. Anal., 55 (1984), 377-388.
9) M. Fukushima and M. Okada, On Dirichlet forms for plurisubharmonic functions, Acta Math., 159 (1988), 171-214.
10) R. H. Greene and H. Wu, Function theory on manifolds which possesses a pole, Lecture Notes in Math., 699, Springer, 1979.
11) R. H. Greene and H. Wu, Gap theorems for non-compact Riemannian manifolds, Duke Math. J., 49 (1982), 731-756.
12) N. Mok, Y. T. Siu and S. T. Yau, The Poincaré-Lelong equation on complete Kähler manifolds, Comp. Math., 44 (1981), 183-218.
13) Y. T. Siu and S. T. Yau, Complete Kähler manifolds with non-positive curvature of faster than quadratic decay, Ann. Math., 105 (1977), 235-264.
14) W. Stoll, The Ahlfors-Weyl theorem of meromorphic maps on parabolic manifolds, Lecture Notes in Math., 981, Springer, 1983.
15) K. Takegoshi, A non-existence theorem for plurisubharmonic maps of finite energy, Math. Z., 192 (1986), 21-27.
16) K. Takegoshi, Energy estimates and Liouville theorems for harmonic maps, Max-Planck-Institut für Mathematik, preprint.
17) H. Wu, On a problem concerning the intrinsic characterization of Cⁿ, Math. Ann., 246 (1979), 15-22.

Right : [1] E. Bedford and B. A. Taylor, Variational properties of the complex Monge-Ampère equation I, Dirichlet principle, Duke Math. J., 45 (1978), 375-403.
[2] E. Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math., 149 (1982), 1-44.
[3] D. Burns, Curvature of Monge-Ampère foliation and parabolic manifolds, Ann. Math., 115 (1982), 349-373.
[4] E. D. Dynkin, Markov Processes, Springer, 1965.
[5] M. Fukushima, Dirichlet forms and Markov processes, North-Holland and Kodansha, 1980.
[6] M. Fukushima, On holomorphic diffusions and plurisubharmonic functions, in “Geometry of Random Motion” Contemporary Mathematics, to appear.
[7] M. Fukushima, On the continuity of plurisubharmonic functions along conformal diffusions, Osaka J. Math., 23 (1986), 69-75.
[8] M. Fukushima and M. Okada, On conformal martingale diffusions and pluripolar set, J. Funct. Anal., 55 (1984), 377-388.
[9] M. Fukushima and M. Okada, On Dirichlet forms for plurisubharmonic functions, Acta Math., 159 (1988), 171-214.
[10] R. H. Greene and H. Wu, Function theory on manifolds which possesses a pole, Lecture Notes in Math., 699, Springer, 1979.
[11] R. H. Greene and H. Wu, Gap theorems for non-compact Riemannian manifolds, Duke Math. J., 49 (1982), 731-756.
[12] N. Mok, Y. T. Siu and S. T. Yau, The Poincaré-Lelong equation on complete Kähler manifolds, Comp. Math., 44 (1981), 183-218.
[13] Y. T. Siu and S. T. Yau, Complete Kähler manifolds with non-positive curvature of faster than quadratic decay, Ann. Math., 105 (1977), 235-264.
[14] W. Stoll, The Ahlfors-Weyl theorem of meromorphic maps on parabolic manifolds, Lecture Notes in Math., 981, Springer, 1983.
[15] K. Takegoshi, A non-existence theorem for plurisubharmonic maps of finite energy, Math. Z., 192 (1986), 21-27.
[16] K. Takegoshi, Energy estimates and Liouville theorems for harmonic maps, Max-Planck-Institut für Mathematik, preprint.
[17] H. Wu, On a problem concerning the intrinsic characterization of Cⁿ, Math. Ann., 246 (1979), 15-22.

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