Harmonic functions of polynomial growth on complete manifolds II
ジャーナル
フリー
1995 年 47 巻 1 号 p. 37-65
詳細
- Published: 1995 Received: 1993/05/19 Available on J-STAGE: 2006/10/20 Accepted: - Advance online publication: - Revised: -
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訂正情報
訂正日: 2006/10/20 訂正理由: - 訂正箇所: 論文サブタイトル 訂正内容: Wrong : Dedicated to Professor S. Kobayashi for his 60th birthday
訂正日: 2006/10/20 訂正理由: - 訂正箇所: 引用文献情報 訂正内容: Wrong : 1) W. Ballmann, M. Gromov and V. Schreoder, Manifolds of Nonpositive Curvature, Birkhäuser, 1985.
2) S. Bando, A. Kasue and H. Nakajima, On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth, Invent. Math., 97 (1989), 313-349.
3) R. Bartnik, The mass of an asymptotically flat manifold, Comm. Pure Appl. Math., 39 (1986), 661-693.
4) S.-Y. Cheng, Liouville theorems for harmonic maps, Proc. Sympos. Pure Math., 36 (1980), 147-151.
5) J. Cheeger, K. Fukaya and M. Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. of Amer. Math. Soc., 5 (1992), 327-372.
6) J. Cheeger, M. Gromov and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom., 17 (1982), 15-53.
7) K. Fukaya, A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters, J. Differential Geom., 28 (1988), 1-21.
8) D. Gilberg and N. S. Trudinger, Elliptic Differential Equations of Second Order, (second edition), Springer-Verlag, 1983.
9) R. E. Greene and H. Wu, Lipschitz convergence of Riemannian manifolds, Pacific J. Math., 131 (1988), 119-141.
10) M. Gromov, Structure métrique pour les variétés riemannienes, (rédige par J. Lafontaine et P. Pansu), Cedic/Fernand Nathan, Paris, 1981.
11) J. Jost, Harmonic Mappings between Riemannian Manifolds, Proc. Centre Math. Anal. Austral. Nat. Univ., 4, 1983.
12) A. Kasue, A Laplacian comparison theorem and function theoretic properties of a complete Riemannian manifold, Japan. J. Math., 8 (1982), 309-341.
13) A. Kasue, A convergence theorem for Riemannian manifolds and some applications, Nagoya Math. J., 114 (1989), 21-51.
14) A. Kasue, Harmonic functions with growth conditions on a manifold of asymptotically nonnegative curvature II, Recent Topics in Differential and Analytic Geometry, (ed. T. Ochiai), Adv. Stud. Pure Math., 18-I (1990), 283-301.
15) A. Kasue, Harmonic functions of polynomial growth on complete manifolds I, Differential Geometry, (eds. R. E. Greene and S.-T. Yau), Proc. Sympos. Pure Math., 54 (1993), Part 1, 281-290.
16) A. Kasue, Measured Hausdorff convergence of Riemannian manifolds and Laplace operators, Osaka J. Math., 30 (1993), 613-651.
17) A. Kasue and H. Kumura, Spectral convergence of Riemannian manifolds, Tôhoku Math. J., 46 (1994), 147-179.
18) A. Kasue and T. Washio, Growth of equivariant harmonic maps and harmonic morphisms, Osaka J. Math., 27 (1990), 899-928.
19) J. L. Kazdan, Parabolicity and the Liouville property on complete Riemannian manifolds, Seminar on New Results in Nonlinear Partial Differential Equations, (ed. A. Tromba), A Publication of the Max-Planck-lnst, für Math. Bonn, 1987, pp. 153-166,
20) P. Li and L. F. Tam, Linear growth harmonic functions on a complete manifold, J. Differential Geom., 29 (1989), 421-425.
21) P. Li and L. F. Tam, Complete surfaces with finite total curvature, J. Differential Geom., 33 (1991), 139-168.
22) R. Lockhart and R. C. McOwen, Elliptic differential operators on noncompact manifolds, Ann. Scuola Norm. Sup. Pisa CI. Sci. (4), 12 (1985), 409-447.
23) Y. Ogura, Convergence of Riemannian metrics and diffusion processes, preprint.
24) B. O'Neill, The fundamental equations of a submersion, Michigan Math. J., 13 (1966), 459-469.
25) S. Peters, Convergence of Riemannian manifolds, Compositio Math., 62 (1987), 3-16.
26) H. Wu, The Bochner Technique in Differential Geometry, Mathematical Reports, 3, Harwood Academic Publishers, 1988, pp. 298-542.
27) H. Wu, Subharmonic functions and the volume of a noncompact manifold, Differential Geometry, Pitman Monographs Surveys Pure Appl. Math., 52, Longman Sci. Tech. Harlow, 1991, pp. 351-368.
28) S.-T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math., 28 (1975), 201-228.
Right : [1] W. Ballmann, M. Gromov and V. Schreoder, Manifolds of Nonpositive Curvature, Birkhäuser, 1985.
[2] S. Bando, A. Kasue and H. Nakajima, On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth, Invent. Math., 97 (1989), 313-349.
[3] R. Bartnik, The mass of an asymptotically flat manifold, Comm. Pure Appl. Math., 39 (1986), 661-693.
[4] S. -Y. Cheng, Liouville theorems for harmonic maps, Proc. Sympos. Pure Math., 36 (1980), 147-151.
[5] J. Cheeger, K. Fukaya and M. Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. of Amer. Math. Soc., 5 (1992), 327-372.
[6] J. Cheeger, M. Gromov and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom., 17 (1982), 15-53.
[7] K. Fukaya, A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters, J. Differential Geom., 28 (1988), 1-21.
[8] D. Gilberg and N. S. Trudinger, Elliptic Differential Equations of Second Order, (second edition), Springer-Verlag, 1983.
[9] R. E. Greene and H. Wu, Lipschitz convergence of Riemannian manifolds, Pacific J. Math., 131 (1988), 119-141.
[10] M. Gromov, Structure métrique pour les variétés riemannienes, (rédige par J. Lafontaine et P. Pansu), Cedic/Fernand Nathan, Paris, 1981.
[11] J. Jost, Harmonic Mappings between Riemannian Manifolds, Proc. Centre Math. Anal. Austral. Nat. Univ., 4, 1983.
[12] A. Kasue, A Laplacian comparison theorem and function theoretic properties of a complete Riemannian manifold, Japan. J. Math., 8 (1982), 309-341.
[13] A. Kasue, A convergence theorem for Riemannian manifolds and some applications, Nagoya Math. J., 114 (1989), 21-51.
[14] A. Kasue, Harmonic functions with growth conditions on a manifold of asymptotically nonnegative curvature II, Recent Topics in Differential and Analytic Geometry, (ed. T. Ochiai), Adv. Stud. Pure Math., 18-I (1990), 283-301.
[15] A. Kasue, Harmonic functions of polynomial growth on complete manifolds I, Differential Geometry, (eds. R. E. Greene and S. -T. Yau), Proc. Sympos. Pure Math., 54 (1993), Part 1, 281-290.
[16] A. Kasue, Measured Hausdorff convergence of Riemannian manifolds and Laplace operators, Osaka J. Math., 30 (1993), 613-651.
[17] A. Kasue and H. Kumura, Spectral convergence of Riemannian manifolds, Tôhoku Math. J., 46 (1994), 147-179.
[18] A. Kasue and T. Washio, Growth of equivariant harmonic maps and harmonic morphisms, Osaka J. Math., 27 (1990), 899-928.
[19] J. L. Kazdan, Parabolicity and the Liouville property on complete Riemannian manifolds, Seminar on New Results in Nonlinear Partial Differential Equations, (ed. A. Tromba), A Publication of the Max-Planck-lnst, für Math. Bonn, 1987, pp. 153-166,
[20] P. Li and L. F. Tam, Linear growth harmonic functions on a complete manifold, J. Differential Geom., 29 (1989), 421-425.
[21] P. Li and L. F. Tam, Complete surfaces with finite total curvature, J. Differential Geom., 33 (1991), 139-168.
[22] R. Lockhart and R. C. McOwen, Elliptic differential operators on noncompact manifolds, Ann. Scuola Norm. Sup. Pisa CI. Sci. (4), 12 (1985), 409-447.
[23] Y. Ogura, Convergence of Riemannian metrics and diffusion processes, preprint.
[24] B. O'Neill, The fundamental equations of a submersion, Michigan Math. J., 13 (1966), 459-469.
[25] S. Peters, Convergence of Riemannian manifolds, Compositio Math., 62 (1987), 3-16.
[26] H. Wu, The Bochner Technique in Differential Geometry, Mathematical Reports, 3, Harwood Academic Publishers, 1988, pp. 298-542.
[27] H. Wu, Subharmonic functions and the volume of a noncompact manifold, Differential Geometry, Pitman Monographs Surveys Pure Appl. Math., 52, Longman Sci. Tech. Harlow, 1991, pp. 351-368.
28) S.-T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math., 28 (1975), 201-228.
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