The Splitting Theorem and topology of noncompact spaces with nonnegative N-Bakry Émery Ricci curvature

Lim, Alice

doi:10.1090/proc/15240

The Splitting Theorem and topology of noncompact spaces with nonnegative N-Bakry Émery Ricci curvature
HTML articles powered by AMS MathViewer

by Alice Lim PDF

Proc. Amer. Math. Soc. 149 (2021), 3515-3529 Request permission

Abstract:

In this paper, we generalize topological results known for noncompact manifolds with nonnegative Ricci curvature to spaces with nonnegative $N$-Bakry Émery Ricci curvature. We study the Splitting Theorem and a property called the geodesic loops to infinity property in relation to spaces with nonnegative $N$-Bakry Émery Ricci curvature. In addition, we show that if $M^n$ is a complete, noncompact Riemannian manifold with nonnegative $N$-Bakry Émery Ricci curvature where $N>n$, then $H_{n-1}(M,\mathbb {Z})$ is $0$.

References

D. Bakry and Michel Émery, Diffusions hypercontractives, Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 177–206 (French). MR 889476, DOI 10.1007/BFb0075847
Gilles Carron and Emmanuel Pedon, On the differential form spectrum of hyperbolic manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004), no. 4, 705–747. MR 2124586
Jeff Cheeger and Detlef Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry 6 (1971/72), 119–128. MR 303460
Fuquan Fang, Xiang-Dong Li, and Zhenlei Zhang, Two generalizations of Cheeger-Gromoll splitting theorem via Bakry-Emery Ricci curvature, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 2, 563–573 (English, with English and French summaries). MR 2521428, DOI 10.5802/aif.2440
M. Fernández-López and E. García-Río, A remark on compact Ricci solitons, Math. Ann. 340 (2008), no. 4, 893–896. MR 2372742, DOI 10.1007/s00208-007-0173-4
Shin-ichi Ohta, $(K,N)$-convexity and the curvature-dimension condition for negative $N$, J. Geom. Anal. 26 (2016), no. 3, 2067–2096. MR 3511469, DOI 10.1007/s12220-015-9619-1
Marcus Khuri, Eric Woolgar, and William Wylie, New restrictions on the topology of extreme black holes, Lett. Math. Phys. 109 (2019), no. 3, 661–673. MR 3910139, DOI 10.1007/s11005-018-1121-9
André Lichnerowicz, Variétés kählériennes à première classe de Chern non negative et variétés riemanniennes à courbure de Ricci généralisée non negative, J. Differential Geometry 6 (1971/72), 47–94 (French). MR 300228
John Lott, Some geometric properties of the Bakry-Émery-Ricci tensor, Comment. Math. Helv. 78 (2003), no. 4, 865–883. MR 2016700, DOI 10.1007/s00014-003-0775-8
Emanuel Milman, Beyond traditional curvature-dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension, Trans. Amer. Math. Soc. 369 (2017), no. 5, 3605–3637. MR 3605981, DOI 10.1090/tran/6796
James R. Munkres, Elements of algebraic topology, Addison-Wesley Publishing Company, Menlo Park, CA, 1984. MR 755006
Ovidiu Munteanu and Jiaping Wang, Geometry of manifolds with densities, Adv. Math. 259 (2014), 269–305. MR 3197658, DOI 10.1016/j.aim.2014.03.023
Barrett O’Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With applications to relativity. MR 719023
Peter Petersen, Riemannian geometry, 2nd ed., Graduate Texts in Mathematics, vol. 171, Springer, New York, 2006. MR 2243772
Zhongmin Qian, Estimates for weighted volumes and applications, Quart. J. Math. Oxford Ser. (2) 48 (1997), no. 190, 235–242. MR 1458581, DOI 10.1093/qmath/48.2.235
Zhongmin Shen and Christina Sormani, The codimension one homology of a complete manifold with nonnegative Ricci curvature, Amer. J. Math. 123 (2001), no. 3, 515–524. MR 1833151, DOI 10.1353/ajm.2001.0020
C. Sormani, On loops representing elements of the fundamental group of a complete manifold with nonnegative Ricci curvature, Indiana Univ. Math. J. 50 (2001), no. 4, 1867–1883. MR 1889085, DOI 10.1512/iumj.2001.50.2048
Guofang Wei and Will Wylie, Comparison geometry for the Bakry-Emery Ricci tensor, J. Differential Geom. 83 (2009), no. 2, 377–405. MR 2577473, DOI 10.4310/jdg/1261495336
Eric Woolgar and William Wylie, Curvature-dimension bounds for Lorentzian splitting theorems, J. Geom. Phys. 132 (2018), 131–145. MR 3836773, DOI 10.1016/j.geomphys.2018.06.001
William Wylie, A warped product version of the Cheeger-Gromoll splitting theorem, Trans. Amer. Math. Soc. 369 (2017), no. 9, 6661–6681. MR 3660237, DOI 10.1090/tran/7003
Shing Tung Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976), no. 7, 659–670. MR 417452, DOI 10.1512/iumj.1976.25.25051