|
[1] D. Bakry and M. Emery, Diffusions hypercontractives, Seminaire de probabilites, XIX, 1983/84, volume 1123 of Lecture Notes in Math., 177-206. Springer, Berlin, 1985. [2] R.L Bishop and R.J. Crittenden, Geometry of Manifolds, Academic Press, 1964. [3] R. Brooks, A relation between growth and the spectrum of the Laplacian, Math. Z. 178(1981), 501-508. [4] H.D. Cao, Recent progress on Ricci solitons, Adv. Lect. Math. 11 (2)(2010), 1-38. [5] H.D. Cao and D. Zhou, On the complete gradient shrinking Ricci solitons, J. Differential Geom. 85 (2010), no 2, 175-186. [6] J. Carillo and L. Ni, Sharp logarithmic sobolev inequalities on gradient solitons and applications. Comm. Anal. Geom. 17, 721-753(2009). [7] J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry 6 (1971), 119-128. [8] B.L. Chen, Strong uniqueness of the Ricci flow, J. Differentail Geom. 82(2009), no. 2, 362-382. [9] S.Y. Cheng, Eigenvalue comparison theorems and its geometric applications, Math. Z. 143(1975) 289-297. [10] B. Chow, P. Lu and L. Ni, Hamilton’s Ricci flow, Graduate studies in mathematics, 2006. [11] Manfredo perdig˜ao do Carmo. Riemannian Geometry, Birkauser, Boston, 1992. [12] F. Fang, X.-D. Li and Z. Zhang, Two generalizations of Cheeger-Gromoll splitting theorem via Bakry-Emery Ricci curvature, Annales de l’insitut Fourier, 59 no. 2 (2009), p.563-573. [13] L. Gross, Logarithmic Sobolev inequalities in the Ricci flow, American J. Math. 97 (1975), 1061 1083. 38 A NOTE ON VOLUME COMPARISON THEOREM ON SMOOTH METRIC MEASURE SPACE [14] R. Hamilton, The formation of singularities in the Ricci flow, Surveys in Differentail Geom. Surveys in Differential Geom.2 (1995), 7-136, [15] R. Haslhofer and R. Muller, A compactness theorem for complete Ricci shrinkers, Geom. Funct. Anal. 21 (2011), 1091-1116. [16] P. Li, Lecture Notes on Geometric Analysis, Lecture Notes Series No. 6, Research Institute of mathematics, Global Analysis Research Center, Seoul National University, Korea (1993). [17] P. Li and J. Wang, Complete manifolds with positive spectrum, J. Differentail Geo. 58 (2001), 501 534. [18] P. Li and J. Wang, Complete manifolds with positive spectrum, II, J. Differentail Geo. 62 (2002), 143-162. [19] A. Lichnerowicz, Varietes Riemanniennes a tensor C non negatif. C. R. Acad. Sci. Paris Sr. A, 271 (1970) 650-653. [20] J. Lott, Some geometric properties of the Bakry-Emery-Ricci tensor, Comm. Math. Helv. 78 (2003), 865-883. [21] F. Morgan, Manifolds with Density. Notices of the Amer. Math. Soc, 52 (005), no. 8, 853-858. [22] O. Munteanu and N. Sesum, On gradient Ricci solitons, to appear in J. Geom. Anal. [23] O. Munteanu and J. Wang, Geometry of manifolds with densties, arXiv:1211.3996v1 math. DG 16 Nov 2012 [24] O. Munteanu and J. Wang, Smooth metric measure spaces with nonnegative curvature, Comm, Anal. Geom 19 (2011), no. 3, 451-486. [25] O. Munteanu and J. Wang, Analysis of the weighted Laplacian and applications to Ricci solitons, Comm, Anal. Geom 20 (2012), no. 1, 55-94. [26] G. Perelman, The entropy formula for the Ricci flow and its geometric applocations, arXiv:math. DG/0211159. [27] R. Schoen and S. T. Yau, Lectures on Differential Geometry, International Press, 1986. [28] K. Sturm, On the grometry of metric measure spaces. I. Acta Math. 196 (2006), 65-131. [29] K. Sturm, On the grometry of metric measure spaces. II. Acta Math. 196 (2006), 133-177. A NOTE ON VOLUME COMPARISON THEOREM ON SMOOTH METRIC MEASURE SPACE 39 [30] G. Wei and W, Wylie, Comparison geometry for the Bakry-Emery Ricci tensor, J. Differential Geom (2009), 377-405. [31] Yang, A Note on nonnegative Bakry-Emery Ricci curvature, Arch. 93 (2009) , no. 5, 491- 496. [32] S. T. Yau, Some function-theoretic properties of complete Riemannian manifold and their applica tions to geometry, Indiana University Mathematics Journal, Vol. 25, no. 7, 1976 |