메뉴 건너뛰기




Volumn 83, Issue 2, 2009, Pages 337-405

Comparison geometry for the bakry-emery ricci tensor

Author keywords

[No Author keywords available]

Indexed keywords


EID: 73349083915     PISSN: 0022040X     EISSN: 1945743X     Source Type: Journal    
DOI: 10.4310/jdg/1261495336     Document Type: Article
Times cited : (551)

References (54)
  • 1
    • 84968508874 scopus 로고
    • On complete manifolds with nonnegative Ricci curvature
    • U. Abresch & D. Gromoll. On complete manifolds with nonnegative Ricci curvature. J. Amer. Math Soc., 3(2), 355-374, 1990.
    • (1990) J. Amer. Math Soc. , vol.3 , Issue.2 , pp. 355-374
    • Abresch, U.1    Gromoll, D.2
  • 2
    • 84972506465 scopus 로고
    • Short geodesics and gravitational instantons
    • M. T. Anderson. Short geodesics and gravitational instantons. J. Differential Geom., 31(1), 265-275, 1990.
    • (1990) J. Differential Geom. , vol.31 , Issue.1 , pp. 265-275
    • Anderson, M.T.1
  • 3
    • 0002408575 scopus 로고
    • On the topology of complete manifolds of nonnegative Ricci curvature
    • M. T. Anderson. On the topology of complete manifolds of nonnegative Ricci curvature. Topology, 91, 41-55, 1990.
    • (1990) Topology , vol.91 , pp. 41-55
    • Anderson, M.T.1
  • 4
    • 0001119849 scopus 로고
    • Diffusions hypercontractives
    • 1983/84, volume 1123 of Lecture Notes in Math, Springer, Berlin
    • D. Bakry & M. Emery. Diffusions hypercontractives. In: Seminaire de proba-bilites, XIX, 1983/84, volume 1123 of Lecture Notes in Math., pages 177-206. Springer, Berlin, 1985.
    • (1985) Seminaire De Proba-Bilites , vol.19 , pp. 177-206
    • Bakry, D.1    Emery, M.2
  • 5
    • 0034666932 scopus 로고    scopus 로고
    • Some new Results on Eigenvectors via dimension, diameter, and Ricci curvature
    • D. Bakry & Z. Qian. Some new Results on Eigenvectors via dimension, diameter, and Ricci curvature. Advances in Mathematics, 155, 98-153(2000).
    • (2000) Advances in Mathematics , vol.155 , pp. 98-153
    • Bakry, D.1    Qian, Z.2
  • 6
    • 67149123755 scopus 로고    scopus 로고
    • Volume Comparison Theorems without Jacobi fields
    • Current trends in potential theory, 2005, volume 4 of Theta Ser, Theta, Bucharest
    • D. Bakry & Z. Qian. Volume Comparison Theorems without Jacobi fields. In: Current trends in potential theory, 2005, volume 4 of Theta Ser. Adv. Math., pages 115-122. Theta, Bucharest, 2005.
    • (2005) Adv. Math. , pp. 115-122
    • Bakry, D.1    Qian, Z.2
  • 7
    • 0008989693 scopus 로고    scopus 로고
    • Existence of gradient KShler-Ricci solitons
    • (Minneapolis, MN, 1994), A K Peters, Wellesley, MA
    • H.-D. Cao. Existence of gradient KShler-Ricci solitons. In: Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994), pages 1-16. A K Peters, Wellesley, MA, 1996.
    • (1996) Elliptic and Parabolic Methods in Geometry , pp. 1-16
    • Cao, H.-D.1
  • 9
    • 33644554483 scopus 로고    scopus 로고
    • Conformed invariants associated to a measure
    • electronic
    • S.-Y. A. Chang, M. J. Gursky & P. Yang. Conformed invariants associated to a measure. Proc. Natl. Acad. Sci. USA, 103(8), 2535-2540 (electronic), 2006.
    • (2006) Proc. Natl. Acad. Sci. USA , vol.103 , Issue.8 , pp. 2535-2540
    • Chang, S.-Y.A.1    Gursky, M.J.2    Yang, P.3
  • 10
    • 84972500359 scopus 로고
    • The splitting theorem for manifolds of nonnegative Ricci curvature
    • J. Cheeger & D. Gromoll. The splitting theorem for manifolds of nonnegative Ricci curvature. J. Differential Geometry, 6, 119-128, 1971/72.
    • (1971) J. Differential Geometry , vol.6 , pp. 119-128
    • Cheeger, J.1    Gromoll, D.2
  • 13
    • 3042672574 scopus 로고    scopus 로고
    • Rotationally symmetric shrinking and, expanding gradient KShler-Ricci solitons
    • M. Feldman, T. Ilmanen & D. Knopf. Rotationally symmetric shrinking and, expanding gradient KShler-Ricci solitons. J. Differential Geom., 65(2), 169-209, 2003.
    • (2003) J. Differential Geom. , vol.65 , Issue.2 , pp. 169-209
    • Feldman, M.1    Ilmanen, T.2    Knopf, D.3
  • 14
    • 38549153690 scopus 로고    scopus 로고
    • A remark on compact Ricci solitons
    • M. Fernández-López & E. García-Ríio. A remark on compact Ricci solitons. Math. Ann., 340(4), 893-896, 2008.
    • (2008) Math. Ann. , vol.340 , Issue.4 , pp. 893-896
    • Fernández-López, M.1    García-Ríio, E.2
  • 15
    • 51649160427 scopus 로고
    • Groups of polynomial growth and expanding maps
    • Inst. Hautes Etudes Sci
    • M. Gromov. Groups of polynomial growth and expanding maps. Inst. Hautes Etudes Sci. Publ. Math., 53, 53-73, 1981.
    • (1981) Publ. Math. , vol.53 , pp. 53-73
    • Gromov, M.1
  • 18
    • 0000307628 scopus 로고
    • On rotationally symmetric Hamilton’s equation for KShler-Einstein metrics
    • Recent topics in differential and analytic geometry, volume 18 of Adv, Academic Press, Boston, MA
    • N. Koiso. On rotationally symmetric Hamilton’s equation for KShler-Einstein metrics. In: Recent topics in differential and analytic geometry, volume 18 of Adv. Stud. Pure Math., pages 327-337. Academic Press, Boston, MA, 1990.
    • (1990) Stud. Pure Math. , pp. 327-337
    • Koiso, N.1
  • 20
    • 25444447987 scopus 로고    scopus 로고
    • Liouville theorems for symmetric diffusion operators on complete Rie-mannian manifolds
    • X.-D. Li. Liouville theorems for symmetric diffusion operators on complete Rie-mannian manifolds. J. Math Pures Appl., 84, 1295-1361, 2005.
    • (2005) J. Math Pures Appl. , vol.84 , pp. 1295-1361
    • Li, X.-D.1
  • 21
    • 0013555422 scopus 로고
    • On extensions of Myers’ theorem
    • X.-M. Li. On extensions of Myers’ theorem. Bull. London Math. Soc., 27(4), 392-396, 1995.
    • (1995) Bull. London Math. Soc. , vol.27 , Issue.4 , pp. 392-396
    • Li, X.-M.1
  • 22
    • 60849120915 scopus 로고
    • Variétés riemanniennes à tenseur C non négatif
    • A. Lichnerowicz. Variétés riemanniennes à tenseur C non négatif. C.R. Acad. Sc. Paris Serie A, 271, A650-A653, 1970.
    • (1970) C.R. Acad. Sc. Paris Serie A , vol.271 , pp. A650-A653
    • Lichnerowicz, A.1
  • 23
    • 0010139041 scopus 로고
    • Variétés kahlériennes a premiere classe de Chern non negative et variétés riemanniennes a courbure de Ricci généralisée non negative
    • A. Lichnerowicz. Variétés kahlériennes a premiere classe de Chern non negative et variétés riemanniennes a courbure de Ricci généralisée non negative. J. Differential Geom., 6, 47-94, 1971/72.
    • (1971) J. Differential Geom , vol.6 , pp. 47-94
    • Lichnerowicz, A.1
  • 24
    • 62049083002 scopus 로고    scopus 로고
    • Ricci curvature for metric-measure spaces via optimal transport
    • J. Lott & C. Villani. Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math., 169(3), 903-991, 2009.
    • (2009) Ann. of Math. , vol.169 , Issue.3 , pp. 903-991
    • Lott, J.1    Villani, C.2
  • 25
    • 0242350976 scopus 로고    scopus 로고
    • Some geometric properties of the Bakry-Emery-Ricci tensor. Comment
    • J. Lott. Some geometric properties of the Bakry-Emery-Ricci tensor. Comment. Math. Helv., 78(4), 865-883, 2003.
    • (2003) Math. Helv. , vol.78 , Issue.4 , pp. 865-883
    • Lott, J.1
  • 26
    • 84972530010 scopus 로고
    • A note on curvature and fundamental group
    • J. Milnor. A note on curvature and fundamental group. J. Differential Geometry, 2, 1-7, 1968.
    • (1968) J. Differential Geometry , vol.2 , pp. 1-7
    • Milnor, J.1
  • 27
    • 32644448806 scopus 로고    scopus 로고
    • Manifolds with Density. Notices of the Amer
    • F. Morgan. Manifolds with Density. Notices of the Amer. Math. Soc., 52(8), 853-858, 2005.
    • (2005) Math. Soc. , vol.52 , Issue.8 , pp. 853-858
    • Morgan, F.1
  • 28
    • 55249113353 scopus 로고    scopus 로고
    • Myers’ theorem with density
    • F. Morgan. Myers’ theorem with density. Kodai Math Journal, 29(3), 454-460, 2006.
    • (2006) Kodai Math Journal , vol.29 , Issue.3 , pp. 454-460
    • Morgan, F.1
  • 31
    • 0036058289 scopus 로고    scopus 로고
    • Category bounds for nonnegative Ricci curvature manifolds with infinite fundamental group
    • J. Oprea. Category bounds for nonnegative Ricci curvature manifolds with infinite fundamental group. Proc. Amer. Math Soc., 130(3), 833-839. 2002.
    • (2002) Proc. Amer. Math Soc. , vol.130 , Issue.3 , pp. 833-839
    • Oprea, J.1
  • 34
    • 0001136902 scopus 로고    scopus 로고
    • Integral curvature bounds, distance estimates, and, applications
    • P. Petersen & C. Sprouse. Integral curvature bounds, distance estimates, and, applications. Jour. Differential Geom., 50(2), 269-298, 1998.
    • (1998) Jour. Differential Geom. , vol.50 , Issue.2 , pp. 269-298
    • Petersen, P.1    Sprouse, C.2
  • 35
    • 0031283631 scopus 로고    scopus 로고
    • Relative volume comparison with integral curvature bounds
    • P. Petersen, & G. Wei. Relative volume comparison with integral curvature bounds. GAFA, 7, 1031-1045, 1997.
    • (1997) GAFA , vol.7 , pp. 1031-1045
    • Petersen, P.1    Wei, G.2
  • 36
    • 0037333980 scopus 로고    scopus 로고
    • On Frankel’s Theorem. Canad
    • P. Petersen & F. Wilhelm. On Frankel’s Theorem. Canad. Math. Bull., 46(1), 130-139, 2003.
    • (2003) Math. Bull. , vol.46 , Issue.1 , pp. 130-139
    • Petersen, P.1    Wilhelm, F.2
  • 38
    • 70350159705 scopus 로고    scopus 로고
    • On gradient Ricci solitons with symmetry
    • P. Petersen & W. Wylie. On gradient Ricci solitons with symmetry. Proc. Amer. Math. Soc., 137, 2085-2092, 2009.
    • (2009) Proc. Amer. Math. Soc. , vol.137 , pp. 2085-2092
    • Petersen, P.1    Wylie, W.2
  • 40
    • 0031285629 scopus 로고    scopus 로고
    • Estimates for weighted volumes and applications
    • Z. Qian. Estimates for weighted volumes and applications. Quart. J. Math. Oxford Ser. (2), 48(190), 235-242, 1997.
    • (1997) Quart. J. Math. Oxford Ser. , vol.2 , Issue.48-190 , pp. 235-242
    • Qian, Z.1
  • 41
    • 0040558001 scopus 로고    scopus 로고
    • Nonnegative Ricci curvature, small linear diameter growth and finite generation of fundamental groups
    • C. Sormani. Nonnegative Ricci curvature, small linear diameter growth and finite generation of fundamental groups. J. Differential Geom., 54(3), 547-559, 2000.
    • (2000) J. Differential Geom. , vol.54 , Issue.3 , pp. 547-559
    • Sormani, C.1
  • 42
    • 0041027402 scopus 로고    scopus 로고
    • On loops representing elements of the fundamental group of a complete manifold with nonnegative Ricci curvature
    • C. Sormani. On loops representing elements of the fundamental group of a complete manifold with nonnegative Ricci curvature. Indiana Univ. Math. J., 50(4), 1867-1883, 2001.
    • (2001) Indiana Univ. Math. J. , vol.50 , Issue.4 , pp. 1867-1883
    • Sormani, C.1
  • 43
    • 33750193237 scopus 로고    scopus 로고
    • On the geometry of metric measure spaces. I
    • K.-T. Sturm. On the geometry of metric measure spaces. I. Acta Math., 196(1), 65-131, 2006.
    • (2006) Acta Math. , vol.196 , Issue.1 , pp. 65-131
    • Sturm, K.-T.1
  • 44
    • 33750180350 scopus 로고    scopus 로고
    • On the geometry of metric measure spaces. II
    • K.-T. Sturm. On the geometry of metric measure spaces. II. Acta Math., 196(1), 133-177, 2006.
    • (2006) Acta Math. , vol.196 , Issue.1 , pp. 133-177
    • Sturm, K.-T.1
  • 45
    • 84968508677 scopus 로고
    • Examples of complete manifolds of positive Ricci curvature with nilpotent isometry groups
    • G. Wei. Examples of complete manifolds of positive Ricci curvature with nilpotent isometry groups. Bull. Amer. Math. Soci., 19(1), 311-313, 1988.
    • (1988) Bull. Amer. Math. Soci. , vol.19 , Issue.1 , pp. 311-313
    • Wei, G.1
  • 47
    • 0034259709 scopus 로고    scopus 로고
    • On fundamental groups of manifolds of nonnegative curvature
    • B. Wilking. On fundamental groups of manifolds of nonnegative curvature. Differential Geom. Appl., 13(2), 129-165, 2000.
    • (2000) Differential Geom. Appl. , vol.13 , Issue.2 , pp. 129-165
    • Wilking, B.1
  • 48
    • 33748295882 scopus 로고    scopus 로고
    • Noncompact manifolds with nonnegative Ricci curvature
    • W. Wylie. Noncompact manifolds with nonnegative Ricci curvature. J. Geom. Anal., 16(3), 535-550, 2006.
    • (2006) J. Geom. Anal. , vol.16 , Issue.3 , pp. 535-550
    • Wylie, W.1
  • 49
    • 44449141156 scopus 로고    scopus 로고
    • Complete shrinking Ricci solitons have finite fundamental group
    • W. Wylie. Complete shrinking Ricci solitons have finite fundamental group. Proc. American Math. Sci., 136(5), 1803-1806, 2008.
    • (2008) Proc. American Math. Sci. , vol.136 , Issue.5 , pp. 1803-1806
    • Wylie, W.1
  • 50
    • 0345759779 scopus 로고    scopus 로고
    • On the fundamental group of open manifolds with nonnegative Ricci curvature
    • S. Xu, Z. Wang, & F. Yang. On the fundamental group of open manifolds with nonnegative Ricci curvature. Chinese Ann. Math. Ser. B., 24(4), 469-474, 2003.
    • (2003) Chinese Ann. Math. Ser. B. , vol.24 , Issue.4 , pp. 469-474
    • Xu, S.1    Wang, Z.2    Yang, F.3
  • 51
    • 85037181182 scopus 로고    scopus 로고
    • A note on nonnegative Bakry-Emery Ricci Curvature
    • N. Yang. A note on nonnegative Bakry-Emery Ricci Curvature. preprint.
    • Preprint
    • Yang, N.1
  • 52
    • 0000567770 scopus 로고
    • Some function-theoretic properties of complete Riemannian manifold, and their applications to geometry
    • S. T. Yau. Some function-theoretic properties of complete Riemannian manifold, and their applications to geometry. Indiana Univ. Math. J., 25(7), 659-670, 1976.
    • (1976) Indiana Univ. Math. J. , vol.25 , Issue.7 , pp. 659-670
    • Yau, S.T.1
  • 53
    • 84961609970 scopus 로고    scopus 로고
    • On the finiteness of the fundamental group of a compact shrinking Ricci soliton
    • Z. Zhang. On the finiteness of the fundamental group of a compact shrinking Ricci soliton. Colloq. Math., 107(2), 297-299, 2007.
    • (2007) Colloq. Math. , vol.107 , Issue.2 , pp. 297-299
    • Zhang, Z.1
  • 54
    • 0040251391 scopus 로고    scopus 로고
    • The comparison geometry of Ricci curvature
    • Comparison geometry (Berkeley, CA, 1993-94), Cambridge Univ. Press, Cambridge
    • S.-H. Zhu. The comparison geometry of Ricci curvature. In: Comparison geometry (Berkeley, CA, 1993-94), volume 30 of Math. Sci. Res. Inst. Publ, pages 221-262. Cambridge Univ. Press, Cambridge, 1997.
    • (1997) Volume 30 of Math. Sci. Res. Inst. Publ , pp. 221-262
    • Zhu, S.-H.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.