Manifolds with a weighted Poincaré inequality
HTML articles powered by AMS MathViewer
- by Nguyen Thac Dung and Chiung-Jue Anna Sung PDF
- Proc. Amer. Math. Soc. 142 (2014), 1783-1794 Request permission
Abstract:
We study complete manifolds satisfying a weighted Poincaré type property. We establish a splitting and vanishing theorem for $L^2$ harmonic forms provided that the weight function $\rho$ is of exponential growth of the distance function. Our theory generalizes the results of Li-Wang, Lam and Chen-Sung.References
- Mingliang Cai and Gregory J. Galloway, Boundaries of zero scalar curvature in the AdS/CFT correspondence, Adv. Theor. Math. Phys. 3 (1999), no. 6, 1769–1783 (2000). MR 1812136, DOI 10.4310/ATMP.1999.v3.n6.a4
- Jui-Tang Ray Chen and Chiung-Jue Anna Sung, Harmonic forms on manifolds with weighted Poincaré inequality, Pacific J. Math. 242 (2009), no. 2, 201–214. MR 2546710, DOI 10.2140/pjm.2009.242.201
- Harold Donnelly, Eigenforms of the Laplacian on complete Riemannian manifolds, Comm. Partial Differential Equations 9 (1984), no. 13, 1299–1321. MR 764665, DOI 10.1080/03605308408820364
- Nguyen Thac Dung and Chiung Jue Anna Sung, Smooth metric measure spaces with weighted Poincaré inequality, Math. Z. 273 (2013), no. 3-4, 613–632. MR 3030670, DOI 10.1007/s00209-012-1023-y
- Kwan-Hang Lam, Results on a weighted Poincaré inequality of complete manifolds, Trans. Amer. Math. Soc. 362 (2010), no. 10, 5043–5062. MR 2657671, DOI 10.1090/S0002-9947-10-04894-4
- P. Li, Harmonic functions and applications to complete manifolds, “Lecture Notes” given at the University of California, Irvine, 2004. Available on the author’s webpage.
- Peter Li and Jiaping Wang, Complete manifolds with positive spectrum, J. Differential Geom. 58 (2001), no. 3, 501–534. MR 1906784
- Peter Li and Jiaping Wang, Weighted Poincaré inequality and rigidity of complete manifolds, Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 6, 921–982 (English, with English and French summaries). MR 2316978, DOI 10.1016/j.ansens.2006.11.001
- Xiaodong Wang, On conformally compact Einstein manifolds, Math. Res. Lett. 8 (2001), no. 5-6, 671–688. MR 1879811, DOI 10.4310/MRL.2001.v8.n5.a9
- Edward Witten and S.-T. Yau, Connectedness of the boundary in the AdS/CFT correspondence, Adv. Theor. Math. Phys. 3 (1999), no. 6, 1635–1655 (2000). MR 1812133, DOI 10.4310/ATMP.1999.v3.n6.a1
Additional Information
- Nguyen Thac Dung
- Affiliation: Department of Mathematics, National Tsing Hua University, Kuang-Fu Road, Hsinchu, Taiwan 30013
- MR Author ID: 772632
- Email: dungmath@yahoo.co.uk
- Chiung-Jue Anna Sung
- Affiliation: Department of Mathematics, National Tsing Hua University, Kuang-Fu Road, Hsinchu, Taiwan 30013
- MR Author ID: 357591
- Email: cjsung@math.nthu.edu.tw
- Received by editor(s): June 8, 2012
- Published electronically: February 14, 2014
- Additional Notes: The first author was partially supported by the grant NAFOSTED 101.01-2011.13
The second author was partially supported by the NSC - Communicated by: Lei Ni
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 142 (2014), 1783-1794
- MSC (2010): Primary 53C40
- DOI: https://doi.org/10.1090/S0002-9939-2014-11971-X
- MathSciNet review: 3168484