International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 812541 | https://doi.org/10.5402/2011/812541

Hee Kwon Lee, "The Fundamental Groups of m-Quasi-Einstein Manifolds", International Scholarly Research Notices, vol. 2011, Article ID 812541, 4 pages, 2011. https://doi.org/10.5402/2011/812541

The Fundamental Groups of m-Quasi-Einstein Manifolds

Academic Editor: A. Borowiec
Received21 Nov 2011
Accepted08 Dec 2011
Published10 Apr 2012

Abstract

In Ricci flow theory, the topology of Ricci soliton is important. We call a metric quasi-Einstein if the m-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of gradient shrinking Ricci soliton. In this paper, we will prove the finiteness of the fundamental group of m-quasi-Einstein with a positive constant multiple.

1. Introduction and Main Results

Ricci flow is introduced in 1982 and developed by Hamilton (cf. [1]):πœ•πœ•π‘‘π‘”=βˆ’2Ric,𝑔(0)=𝑔0.(1.1) Recently, Perelman supplemented Hamilton’s result and solved the PoincarΓ© Conjecture and the Geometrization Conjecture by using a Ricci flow theory. But in higher dimension greater than 4 classification using Ricci flow is still far-off. Most above all the classification of Ricci solitons, which are singularity models, is not completed. But there exist many properties of Ricci solitons. Here we say 𝑔 is a Ricci soliton if (𝑀,𝑔) is a Riemannian manifold such that the identityRic+𝐿𝑋𝑔=𝑐𝑔(1.2) holds for some constant 𝑐 and some complete vector field 𝑋 on 𝑀. If 𝑐>0, 𝑐=0, or 𝑐<0, then we call it shrinking, steady, or expanding. Moreover, if the vector field 𝑋 appearing in (1.2) is the gradient field of a potential function (1/2)𝑓, one has Ric+βˆ‡βˆ‡π‘“=𝑐𝑔 and says 𝑔 is a gradient Ricci soliton. In 2008, Lōpez and RΓ­o have shown that if (𝑀,𝑔) is a complete manifold with Ric+𝐿𝑋𝑔β‰₯𝑐𝑔 and some positive constant 𝑐, then 𝑀 is compact if and only if ‖𝑋‖ is bounded. Moreover, under these assumptions if 𝑀 is compact, then πœ‹1(𝑀) is finite. Furthermore, Wylie [2] has shown that under these conditions if 𝑀 is complete, then πœ‹1(𝑀) is finite. Moreover, in 2008, Fang et al. (cf. [3]) have shown that a gradient shrinking Ricci soliton with a bounded scalar curvature has finite topological type. By [4, Proposition 1.5.6], Cao and Zhu have shown that compact steady or expanding Ricci solitons are Einstein manifolds. In addition by [4, Corollary 1.5.9 (ii)] note that compact shrinking Ricci solitons are gradient Ricci solitons. So we are interested in shrinking gradient Ricci solitons. In [6, page 354], Eminenti et al. have shown that compact shrinking Ricci solitons have positive scalar curvatures. In [6] Case et al. have shown that an π‘š-quasi-Einstein with 1β‰€π‘š<∞ and 𝑐>0 has a positive scalar curvature. Let me introduce the definition of π‘š-quasi-Einstein.

Definition 1.1. The triple (𝑀,𝑔,𝑓) is an π‘š-quasi-Einstein manifold if it satisfies the equation1Ric+Hessπ‘“βˆ’π‘šπ‘‘π‘“βŠ—π‘‘π‘“=𝑐𝑔(1.3) for some π‘βˆˆπ‘….
Here π‘š-Bakry-Emery Ricci tensor Ricπ‘šπ‘“β‰Ric+Hessπ‘“βˆ’(1/π‘š)π‘‘π‘“βŠ—π‘‘π‘“ for 0<π‘šβ‰€βˆž is a natural extension of the Ricci tensor to smooth metric measure spaces (cf. [6, Section  1 ]). Note that if π‘š=∞, then it reduces to a gradient Ricci soliton. In this paper, we will prove the finiteness of the fundamental group of an π‘š-quasi-Einstein with 𝑐>0.

Theorem 1.2. Let (𝑀,𝑔,𝑓) be a complete manifold with 𝑐>0 and Ric+Hessπ‘“βˆ’(1/π‘š)π‘‘π‘“βŠ—π‘‘π‘“β‰₯𝑐𝑔. Then it has a finite fundamental group.

2. The Proof of Theorem 1.2

The proof of Theorem 1.2 is similar to the proofs of [2, 7].

Proof. We will prove it by dividing into two cases.Case 1. β€–βˆ‡π‘“β€–is bounded. We claim that the bounded β€–βˆ‡π‘“β€– implies the compactness of 𝑀. Let π‘ž be a point in 𝑀, and consider any geodesic π›ΎβˆΆ[0,∞)→𝑀 emanating from π‘ž and parametrized by arc length 𝑑. Then we have ξ€œπ‘‡01Ric(̇𝛾,̇𝛾)β‰₯𝑐𝑇+π‘šξ€œπ‘‡0(𝑑𝑓(̇𝛾))2βˆ’ξ€œπ‘‡0̇𝛾(𝑔(βˆ‡π‘“,̇𝛾))β‰₯π‘π‘‡βˆ’π‘”(βˆ‡π‘“,̇𝛾)|𝑇0.(2.1)
Since 𝑔(βˆ‡π‘“,̇𝛾)|𝑇0 is bounded we have that ∫∞0Ric(̇𝛾,̇𝛾)=∞. Hence, the claim is followed by the proof of [4, Theorem  1]. Let (𝑀,̃𝑔) be the Riemannian universal cover of (𝑀,𝑔), let ξ‚‹π‘βˆΆ(𝑀,̃𝑔)β†’(𝑀,𝑔) be a projection map, and let 𝑓 be a map π‘“βˆ˜π‘. Since 𝑝 is a local isometry, then the same inequality holds, that is, Ric(̃𝑔)+HesseΜƒπ‘”ξ‚ξ‚ξ‚π‘“βˆ’(1/π‘š)π‘‘π‘“βŠ—π‘‘π‘“β‰₯𝑐̃𝑔. Now, since β€–ξ‚βˆ‡ξ‚π‘“β€– is bounded, it is followed from the above argument that 𝑀 is compact. So πœ‹1(𝑀) is finite.
Case 2. β€–βˆ‡π‘“β€– is unbounded. We will prove this case by following the proof of [2]. By Case 1, 𝑀 is noncompact. For any π‘βˆˆπ‘€, define 𝐻𝑝≐max0,supRic𝑦(𝑣,𝑣)βˆΆπ‘¦βˆˆπ΅(𝑝,1),‖𝑣‖=1ξ€Ύξ€Ύ.(2.2) Note that by [7, Lemma  2.2] we have ξ€œπ‘Ÿ0Ric(̇𝛾,̇𝛾)𝑑𝑠≀2(π‘›βˆ’1)+𝐻𝑝+π»π‘ž.(2.3) Assume that 𝑑(𝑝,π‘ž)>1. On the other hand, from the inequality of Theorem 1.2, we have ξ€œπ‘Ÿ01Ric(̇𝛾,̇𝛾)𝑑𝑠β‰₯𝑐𝑑(𝑝,π‘ž)+π‘šξ€œπ‘Ÿ0(𝑑𝑓(̇𝛾))2βˆ’ξ€œπ‘Ÿ0̇𝛾(𝑔(βˆ‡π‘“,̇𝛾))β‰₯𝑐𝑑(𝑝,π‘ž)βˆ’β€–βˆ‡π‘“β€–π‘βˆ’β€–βˆ‡π‘“β€–π‘ž,(2.4) since 𝑔(βˆ‡π‘“,̇𝛾)β‰€β€–βˆ‡π‘“β€–β€–Μ‡π›Ύβ€–. Hence, we have that for any 𝑝, π‘žβˆˆπ‘€ξ‚†1𝑑(𝑝,π‘ž)≀max1,𝑐2(π‘›βˆ’1)+𝐻𝑝+π»π‘ž+β€–βˆ‡π‘“β€–π‘+β€–βˆ‡π‘“β€–π‘žξ€Έξ‚‡.(2.5) Now we will apply a similar argument like Case 1. Fix ξ‚‹π‘€Μƒπ‘βˆˆ, and let β„Žβˆˆπœ‹1(𝑀) identified as a deck transformation on 𝑀. Note that 𝐡(̃𝑝,1) and 𝐡(β„Ž(̃𝑝),1) are isometric, and thus 𝐻̃𝑝=π»β„Ž(̃𝑝). Also β€–ξ‚βˆ‡ξ‚π‘“β€–Μƒπ‘ξ‚βˆ‡ξ‚=β€–π‘“β€–β„Ž(̃𝑝). So we conclude that 2𝑑(̃𝑝,β„Ž(̃𝑝))≀max1,𝑐nβˆ’1+𝐻̃𝑝+β€–β€–ξ‚βˆ‡ξ‚π‘“β€–β€–Μƒπ‘ξ‚ξ‚‡(2.6) for any β„Žβˆˆπœ‹1(𝑀). Since the right-hand side is independent of β„Ž, this proves this case.

References

  1. R. S. Hamilton, β€œThree-manifolds with positive Ricci curvature,” Journal of Differential Geometry, vol. 17, no. 2, pp. 255–306, 1982. View at: Google Scholar | Zentralblatt MATH
  2. W. Wylie, β€œComplete shrinking Ricci solitons have finite fundamental group,” Proceedings of the American Mathematical Society, vol. 136, no. 5, pp. 1803–1806, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  3. F.-Q. Fang, J.-W. Man, and Z.-L. Zhang, β€œComplete gradient shrinking Ricci solitons have finite topological type,” Comptes Rendus Mathématique, vol. 346, no. 11-12, pp. 653–656, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  4. H.-D. Cao and X.-P. Zhu, β€œA complete proof of the Poincaré and geometrization conjectures—application of the Hamilton-Perelman theory of the Ricci flow,” Asian Journal of Mathematics, vol. 10, no. 2, pp. 165–492, 2006. View at: Google Scholar | Zentralblatt MATH
  5. M. Eminenti, G. La Nave, and C. Mantegazza, β€œRicci solitons: the equation point of view,” Manuscripta Mathematica, vol. 127, no. 3, pp. 345–367, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  6. J. Case, Y.-J. Shu, and G. Wei, β€œRigidity of quasi-Einstein metrics,” Differential Geometry and its Applications, vol. 29, no. 1, pp. 93–100, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. M. Fernández-López and E. García-Río, β€œA remark on compact Ricci solitons,” Mathematische Annalen, vol. 340, no. 4, pp. 893–896, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH

Copyright © 2011 Hee Kwon Lee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views843
Downloads407
Citations

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.