Gap results for compact quasi-Einstein metrics

Bakry D, Qian Z M. Volume comparison theorems without Jacobi fields in current trends in potential theory. Theta Ser Adv Math, 2005, 4: 115–122 Besse A L. Einstein Manifolds. Volume 10 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Berlin: Springer-Verlag, 1987 Cao H D, Zhu X P. A complete proof of the Poincar´e and geometrization conjectures—application of the Hamilton-Perelman theory of the Ricci flow. Asian J Math, 2006, 10: 165–492 Case J. On the nonexistence of quasi-Einstein metrics. Pacific J Math, 2010, 248: 277–284 Case J, Shu Y J, Wei G. Rigidity of quasi-Einstein metrics. Differential Geom Appl, 2011, 29: 93–100 Chavel I. Eigenvalues in Riemannian Geometry. New York: Academic Press, 1984 Cheeger J, Yau S T. A lower bound for the heat kernel. Comm Pure Appl Math, 1981, 34: 465–480 Fernández-López M, García-Río E. Some gap theorems for gradient Ricci solitons. Internat J Math, 2012, 23: 1250072 Grigor′yan A. Heat kernels on weighted manifolds and applications. Contemp Math, 2006, 398: 93–191 Hamilton R S. The Formation of Singularities in the Ricci Flow. Surveys in Differential Geometry. Cambridge: International Press, 1995 Kim D S, Kim Y H. Compact Einstein warped product spaces with nonpositive scalar curvature. Proc Amer Math Soc, 2003, 131: 2573–2576 Li H. Gap theorems for Kähler-Ricci solitons. Arch Math (Basel), 2008, 91: 187–192 Li X D. Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J Math Pures Appl (9), 2005, 84: 1295–1361 Li X D. Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry-Émery Ricci curvature. Math Ann, 2012, 353: 403–437 Lichnerowicz A. Geometrie des groupes de transformations. Paris: Dunod, 1958 Lott J. Some geometric properties of the Bakry-Émery Ricci tensor. Comment Math Helv, 2003, 78: 865–883 Lü H, Page D N, Pope C N. New inhomogeneous Einstein metrics on sphere bundles over Einstein-Kähler manifolds. Phys Lett B, 2004, 593: 218–226 Perelman G. The entropy formula for the Ricci flow and its geometric applications. ArXiv:math/0211159v1, 2002 Schoen R, Yau S T. Lectures on Differential Geometry. Conference Proceedings and Lecture Notes in Geometry and Topology, I. Cambridge: International Press, 1994 Wang L F. The upper bound of the L 2 μ spectrum. Ann Global Anal Geom, 2010, 37: 393–402 Wang L F. Rigidity properties of quasi-Einstein metrics. Proc Amer Math Soc, 2011, 139: 3679–3689 Wang L F. On noncompact τ-quasi-Einstein metrics. Pacific J Math, 2011, 254: 449–464 Wang L F. On L p f -spectrum and τ-quasi-Einstein metric. J Math Anal Appl, 2012, 389: 195–204 Wang L F. Monotonicity formulas via the Bakry-Émery curvature. Nonlinear Anal, 2013, 89: 230–241 Wang L F. Diameter estimate for compact quasi-Einstein metrics. Math Z, 2013, 273: 801–809 Wang L F. Potential functions estimates for quasi-Einstein metrics. J Funct Anal, 2014, 267: 1986–2004 Wei G, Wylie W. Comparison geometry for the Bakry-Émery Ricci tensor. J Differential Geom, 2009, 83: 377–405