Some curvature pinching results for Riemannian manifolds with density. (English) Zbl 1334.53036
Summary: In this note we consider versions of both Ricci and sectional curvature pinching for Riemannian manifolds with density. In the Ricci curvature case the main result implies a diameter estimate that is new even for compact shrinking Ricci solitons. In the case of sectional curvature we prove a new sphere theorem.
MSC:
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
| 53C20 | Global Riemannian geometry, including pinching |
References:
| [1] | Bakry, D.; \'Emery, Michel, Diffusions hypercontractives. S\'eminaire de probabilit\'es, XIX, 1983/84, Lecture Notes in Math. 1123, 177-206 (1985), Springer, Berlin · Zbl 0561.60080 · doi:10.1007/BFb0075847 |
| [2] | Berger, Marcel, Sur certaines vari\'et\'es riemanniennes \`a courbure positive, C. R. Acad. Sci. Paris, 247, 1165-1168 (1958) · Zbl 0093.35101 |
| [3] | Berger, M., Les vari\'et\'es Riemanniennes \((1/4)\)-pinc\'ees, Ann. Scuola Norm. Sup. Pisa (3), 14, 161-170 (1960) · Zbl 0096.15502 |
| [4] | Brendle, Simon; Schoen, Richard, Manifolds with \(1/4\)-pinched curvature are space forms, J. Amer. Math. Soc., 22, 1, 287-307 (2009) · Zbl 1251.53021 · doi:10.1090/S0894-0347-08-00613-9 |
| [5] | Cao, Huai-Dong, Existence of gradient K\"ahler-Ricci solitons. Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994), 1-16 (1996), A K Peters, Wellesley, MA · Zbl 0868.58047 |
| [6] | Cao, Huai-Dong; Zhou, Detang, On complete gradient shrinking Ricci solitons, J. Differential Geom., 85, 2, 175-185 (2010) · Zbl 1246.53051 |
| [7] | Chow, Bennett; Lu, Peng; Ni, Lei, Hamilton’s Ricci flow, Graduate Studies in Mathematics 77, xxxvi+608 pp. (2006), American Mathematical Society, Providence, RI; Science Press, New York · Zbl 1118.53001 · doi:10.1090/gsm/077 |
| [8] | do Carmo, Manfredo Perdig{\~a}o, Riemannian geometry, Mathematics: Theory & Applications, xiv+300 pp. (1992), Birkh\"auser Boston, Inc., Boston, MA · Zbl 0752.53001 · doi:10.1007/978-1-4757-2201-7 |
| [9] | Fang, Fu-quan; Man, Jian-wen; Zhang, Zhen-lei, Complete gradient shrinking Ricci solitons have finite topological type, C. R. Math. Acad. Sci. Paris, 346, 11-12, 653-656 (2008) · Zbl 1144.53083 · doi:10.1016/j.crma.2008.03.021 |
| [10] | Feldman, Mikhail; Ilmanen, Tom; Knopf, Dan, Rotationally symmetric shrinking and expanding gradient K\"ahler-Ricci solitons, J. Differential Geom., 65, 2, 169-209 (2003) · Zbl 1069.53036 |
| [11] | Futaki, Akito; Sano, Yuji, Lower diameter bounds for compact shrinking Ricci solitons, Asian J. Math., 17, 1, 17-31 (2013) · Zbl 1281.53046 · doi:10.4310/AJM.2013.v17.n1.a2 |
| [12] | Gromoll, Detlef; Meyer, Wolfgang, On complete open manifolds of positive curvature, Ann. of Math. (2), 90, 75-90 (1969) · Zbl 0191.19904 |
| [13] | Grove, Karsten; Shiohama, Katsuhiro, A generalized sphere theorem, Ann. of Math. (2), 106, 2, 201-211 (1977) · Zbl 0341.53029 |
| [14] | Hamilton, Richard S., The formation of singularities in the Ricci flow. Surveys in differential geometry, Vol.II, Cambridge, MA, 1993, 7-136 (1995), Int. Press, Cambridge, MA · Zbl 0867.53030 |
| [15] | Hatcher, Allen, Algebraic topology, xii+544 pp. (2002), Cambridge University Press, Cambridge · Zbl 1044.55001 |
| [16] | Kennard, Lee; Wylie, William, Positive weighted sectional curvature · Zbl 1371.53032 |
| [17] | Klingenberg, Wilhelm, \`“Uber Riemannsche Mannigfaltigkeiten mit positiver Kr\'”ummung, Comment. Math. Helv., 35, 47-54 (1961) · Zbl 0133.15005 |
| [18] | Klingenberg, Wilhelm, Riemannian geometry, de Gruyter Studies in Mathematics 1, x+396 pp. (1982), Walter de Gruyter & Co., Berlin-New York · Zbl 0495.53036 |
| [19] | Koiso, Norihito, On rotationally symmetric Hamilton’s equation for K\"ahler-Einstein metrics. Recent topics in differential and analytic geometry, Adv. Stud. Pure Math. 18, 327-337 (1990), Academic Press, Boston, MA · Zbl 0739.53052 |
| [20] | Lichnerowicz, Andr{\'e}, Vari\'et\'es riemanniennes \`“a tenseur C non n\'”egatif, C. R. Acad. Sci. Paris S\'er. A-B, 271, A650-A653 (1970) · Zbl 0208.50003 |
| [21] | Lichnerowicz, Andr{\'e}, Vari\'et\'es k\`“ahl\'”eriennes \`“a premi\`ere classe de Chern non negative et vari\'”et\'es riemanniennes \`“a courbure de Ricci g\'”en\'eralis\'ee non negative, J. Differential Geometry, 6, 47-94 (1971/72) · Zbl 0231.53063 |
| [22] | Lott, John, Some geometric properties of the Bakry-\'Emery-Ricci tensor, Comment. Math. Helv., 78, 4, 865-883 (2003) · Zbl 1038.53041 · doi:10.1007/s00014-003-0775-8 |
| [23] | Lott, John; Villani, C{\'e}dric, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2), 169, 3, 903-991 (2009) · Zbl 1178.53038 · doi:10.4007/annals.2009.169.903 |
| [24] | Munteanu, Ovidiu; Wang, Jiaping, Analysis of weighted Laplacian and applications to Ricci solitons, Comm. Anal. Geom., 20, 1, 55-94 (2012) · Zbl 1245.53039 · doi:10.4310/CAG.2012.v20.n1.a3 |
| [25] | Munteanu, Ovidiu; Wang, Jiaping, Geometry of shrinking Ricci solitons · Zbl 1339.53036 |
| [26] | Morgan, Frank, Manifolds with density, Notices Amer. Math. Soc., 52, 8, 853-858 (2005) · Zbl 1118.53022 |
| [27] | Morgan, Frank, Geometric measure theory, viii+249 pp. (2009), Elsevier/Academic Press, Amsterdam · Zbl 1179.49050 |
| [28] | Morgan, Frank, Manifolds with density and Perelman’s proof of the Poincar\'e conjecture, Amer. Math. Monthly, 116, 2, 134-142 (2009) · Zbl 1170.53001 · doi:10.4169/193009709X469896 |
| [29] | Perelman, G., Ricci flow with surgery on three-manifolds. · Zbl 1130.53002 |
| [30] | Perelman, G., The entropy formula for the {R}icci flow and its geometric applications. · Zbl 1130.53001 |
| [31] | Petersen, Peter, Riemannian geometry, Graduate Texts in Mathematics 171, xvi+401 pp. (2006), Springer, New York · Zbl 1220.53002 |
| [32] | Sturm, Karl-Theodor, On the geometry of metric measure spaces. I, Acta Math., 196, 1, 65-131 (2006) · Zbl 1105.53035 · doi:10.1007/s11511-006-0002-8 |
| [33] | Sturm, Karl-Theodor, On the geometry of metric measure spaces. II, Acta Math., 196, 1, 133-177 (2006) · Zbl 1106.53032 · doi:10.1007/s11511-006-0003-7 |
| [34] | Wang, Xu-Jia; Zhu, Xiaohua, K\"ahler-Ricci solitons on toric manifolds with positive first Chern class, Adv. Math., 188, 1, 87-103 (2004) · Zbl 1086.53067 · doi:10.1016/j.aim.2003.09.009 |
| [35] | Weber, Brian, Convergence of compact Ricci solitons, Int. Math. Res. Not. IMRN, 1, 96-118 (2011) · Zbl 1211.53087 · doi:10.1093/imrn/rnq055 |
| [36] | Wei, Guofang; Wylie, Will, Comparison geometry for the Bakry-Emery Ricci tensor, J. Differential Geom., 83, 2, 377-405 (2009) · Zbl 1189.53036 |
| [37] | Wylie, William, Complete shrinking Ricci solitons have finite fundamental group, Proc. Amer. Math. Soc., 136, 5, 1803-1806 (2008) · Zbl 1152.53057 · doi:10.1090/S0002-9939-07-09174-5 |
| [38] | Wylie, William, Sectional curvature for Riemannian manifolds with density · Zbl 1325.53047 |
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