Harmonic functions of polynomial growth on singular spaces with nonnegative Ricci curvature
HTML articles powered by AMS MathViewer
- by Bobo Hua
- Proc. Amer. Math. Soc. 139 (2011), 2191-2205
- DOI: https://doi.org/10.1090/S0002-9939-2010-10635-4
- Published electronically: November 17, 2010
- PDF | Request permission
Abstract:
In the present paper, we will derive the Liouville theorem and the finite dimension theorem for polynomial growth harmonic functions defined on Alexandrov spaces with nonnegative Ricci curvature in the sense of Kuwae-Shioya and Sturm-Lott-Villani.References
- Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR 1835418, DOI 10.1090/gsm/033
- Yu. Burago, M. Gromov, and G. Perel′man, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992), no. 2(284), 3–51, 222 (Russian, with Russian summary); English transl., Russian Math. Surveys 47 (1992), no. 2, 1–58. MR 1185284, DOI 10.1070/RM1992v047n02ABEH000877
- Tobias H. Colding and William P. Minicozzi II, Harmonic functions on manifolds, Ann. of Math. (2) 146 (1997), no. 3, 725–747. MR 1491451, DOI 10.2307/2952459
- Tobias H. Colding and William P. Minicozzi II, Weyl type bounds for harmonic functions, Invent. Math. 131 (1998), no. 2, 257–298. MR 1608571, DOI 10.1007/s002220050204
- A. A. Grigor′yan, The heat equation on noncompact Riemannian manifolds, Mat. Sb. 182 (1991), no. 1, 55–87 (Russian); English transl., Math. USSR-Sb. 72 (1992), no. 1, 47–77. MR 1098839
- Bobo Hua, Generalized Liouville theorem in nonnegatively curved Alexandrov spaces, Chinese Ann. Math. Ser. B 30 (2009), no. 2, 111–128. MR 2487589, DOI 10.1007/s11401-008-0376-3
- Qing Han and Fanghua Lin, Elliptic partial differential equations, Courant Lecture Notes in Mathematics, vol. 1, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1997. MR 1669352
- Kazuhiro Kuwae, Yoshiroh Machigashira, and Takashi Shioya, Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces, Math. Z. 238 (2001), no. 2, 269–316. MR 1865418, DOI 10.1007/s002090100252
- K. Kuwae and T. Shioya, Laplacian comparison for Alexandrov spaces, Comm. Anal. Geom. 5 (1997), no. 2, 333–387.
- Kazuhiro Kuwae and Takashi Shioya, On generalized measure contraction property and energy functionals over Lipschitz maps, Potential Anal. 15 (2001), no. 1-2, 105–121. ICPA98 (Hammamet). MR 1838897, DOI 10.1023/A:1011218425271
- Nicholas J. Korevaar and Richard M. Schoen, Global existence theorems for harmonic maps to non-locally compact spaces, Comm. Anal. Geom. 5 (1997), no. 2, 333–387. MR 1483983, DOI 10.4310/CAG.1997.v5.n2.a4
- Peter Li, Harmonic sections of polynomial growth, Math. Res. Lett. 4 (1997), no. 1, 35–44. MR 1432808, DOI 10.4310/MRL.1997.v4.n1.a4
- Peter Wai-Kwong Li, Harmonic functions and applications to complete manifolds, XIV Escola de Geometria Diferencial. [XIV School of Differential Geometry], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2006. MR 2369440
- John Lott and Cédric Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2) 169 (2009), no. 3, 903–991. MR 2480619, DOI 10.4007/annals.2009.169.903
- Jürgen Moser, A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math. 13 (1960), 457–468. MR 170091, DOI 10.1002/cpa.3160130308
- Yukio Otsu and Takashi Shioya, The Riemannian structure of Alexandrov spaces, J. Differential Geom. 39 (1994), no. 3, 629–658. MR 1274133
- G. Perelman, DC-structure on Alexandrov space, preprint.
- A. Petrunin, Alexandrov meets Lott-Villani-Sturm, preprint.
- A. Petrunin, Subharmonic functions on Alexandrov space, preprint.
- Laurent Saloff-Coste, Aspects of Sobolev-type inequalities, London Mathematical Society Lecture Note Series, vol. 289, Cambridge University Press, Cambridge, 2002. MR 1872526
- L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities, Internat. Math. Res. Notices 2 (1992), 27–38. MR 1150597, DOI 10.1155/S1073792892000047
- Karl-Theodor Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006), no. 1, 65–131. MR 2237206, DOI 10.1007/s11511-006-0002-8
- Karl-Theodor Sturm, On the geometry of metric measure spaces. II, Acta Math. 196 (2006), no. 1, 133–177. MR 2237207, DOI 10.1007/s11511-006-0003-7
- Luen-Fai Tam, A note on harmonic forms on complete manifolds, Proc. Amer. Math. Soc. 126 (1998), no. 10, 3097–3108. MR 1459152, DOI 10.1090/S0002-9939-98-04474-8
- Cédric Villani, Optimal transport, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009. Old and new. MR 2459454, DOI 10.1007/978-3-540-71050-9
- Shing Tung Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201–228. MR 431040, DOI 10.1002/cpa.3160280203
- Shing-Tung Yau, Nonlinear analysis in geometry, Enseign. Math. (2) 33 (1987), no. 1-2, 109–158. MR 896385
- Shing-Tung Yau, Open problems in geometry, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 1–28. MR 1216573, DOI 10.1090/pspum/054.1/1216573
Bibliographic Information
- Bobo Hua
- Affiliation: School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China
- MR Author ID: 865783
- Email: 071018011@fudan.edu.cn
- Received by editor(s): April 12, 2010
- Received by editor(s) in revised form: April 19, 2010, and June 6, 2010
- Published electronically: November 17, 2010
- Communicated by: Jianguo Cao
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 2191-2205
- MSC (2010): Primary 51F99, 31C05
- DOI: https://doi.org/10.1090/S0002-9939-2010-10635-4
- MathSciNet review: 2775397