We show that a complete Riemannian manifold has finite topological type
(i.e., homeomorphic to the interior of a compact manifold with boundary),
provided its Bakry-\'{E}mery Ricci tensor has a positive lower bound, and
either of the following conditions:
  (i) the Ricci curvature is bounded from above; (ii) the Ricci curvature is
bounded from below and injectivity radius is bounded away from zero.
  Moreover, a complete shrinking Ricci soliton has finite topological type if
its scalar curvature is bounded

Fang, Fuquan

man, Jianwen

Zhang, Zhenlei

English

arXiv

Fang, Fu-quan

Man, Jian-wen

Zhang, Zhen-lei

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 346 (2008) 653–656http://france.elsevier.com/direct/CRASS1/Differential GeometryComplete gradient shrinking Ricci solitons havefinite topological typeFu-quan Fang a, Jian-wen Man b, Zhen-lei Zhang ba Department of Mathematics, Capital Normal University, Beijing, PR Chinab Chern Institute of Mathematics, Weijin Road 94, Tianjin 300071, PR ChinaReceived 3 January 2008; accepted after revision 18 March 2008Available online 29 April 2008Presented by Étienne GhysAbstractWe show that a complete Riemannian manifold has finite topological type (i.e., homeomorphic to the interior of a compactmanifold with boundary), provided its Bakry–Émery Ricci tensor has a positive lower bound, and either of the following conditions:(i) the Ricci curvature is bounded from above;(ii) the Ricci curvature is bounded from below and injectivity radius is bounded away from zero.Moreover, a complete shrinking Ricci soliton has finite topological type if its scalar curvature is bounded. To cite this article:F.-q. Fang et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméLes solitons de Ricci contractants complets sont de type fini. Dans cette Note nous montrons qu’une variété riemaniennecomplète est de type topologique fini – c’est-à-dire qu’elle est homéomorphe à une variété compacte à bord – si son tenseur deBakry–Emery–Ricci est bornée inférieurement par une constante positive et vérifie l’une des conditions suivantes :(i) la courbure de Ricci est bornée supérieurement.(ii) la courbure de Ricci est bornée inférieurement et le rayon d’injectivité est positif.De plus, un soliton de Ricci contractant complet est de type topologique fini si sa coubure scalaire est bornée. Pour citer cetarticle : F.-q. Fang et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionIn 1968, J. Milnor [8] conjectured that a complete non-compact Riemannian manifold with non-negative Riccicurvature has a finitely generated fundamental group. However, such a manifold may not have finite topologicaltype. Examples of complete non-compact manifold with positive Ricci curvature without finite topological type wasconstructed by Gromoll–Meyer [5]. It has been an interesting topic in Riemannian geometry to study the topology ofcomplete manifolds with positive (non-negative) Ricci curvature.E-mail addresses: fuquan_fang@yahoo.com (F.-q. Fang), zhleigo@yahoo.com.cn (Z.-l. Zhang).1631-073X/$ – see front matter © 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2008.03.021654 F.-q. Fang et al. / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 653–656In this Note we are concerned with complete Riemannian manifold (M,g) satisfying that Ric+Hess(f )  λgfor some constant λ > 0 and f ∈ C∞(M), i.e., whose Bakry–Émery Ricci tensor is bounded below by λ in the senseof [7]. When the equality holds, the manifold is a shrinking Ricci soliton, i.e., a self-similar solution of the well-knownRicci flow equation. If f is constant, Bakry–Émery Ricci tensor reduces to the Ricci tensor, and so the classical Myers’theorem implies that M is compact with finite fundamental group. In general, M may not be compact, but from thework of [1,3,4,6,7,10,9,11] etc., M still has finite fundamental group. Under some additional conditions, the manifoldmay be of finite topological type:Theorem 1.1. Suppose (M,g) is a complete Riemannian manifold satisfying Ric+Hess(f )  λg for some constantλ > 0 and f ∈ C∞(M). Then M is of finite topological type, if either of the following alternative conditions holds:(i) Ric  Cg for some constant C < ∞;(ii) Ric  −δ−1g and the injectivity radius inj(M,g)  δ > 0 for some δ > 0.If (M,g) is a shrinking Ricci soliton, then the Ricci curvature bounds can be relaxed by scalar curvature.Theorem 1.2. Suppose (M,g) is a complete shrinking Ricci soliton Ric+Hess(f ) = g2 , where f ∈ C∞(M). If thescalar curvature R is bounded, then M has finite topological type.In view of Theorem 1.2 it is natural to pose the following:Conjecture 1.3. Any shrinking Ricci soliton has finite topological type.We prove Theorem 1.1 in Section 2 and Theorem 1.2 in Section 3.2. Proof of Theorem 1.1Let (M,g) be such a manifold satisfying that Ric+Hess(f )  λg for some λ > 0 and f ∈ C∞(M). By thedeformation lemma of Morse theory, to prove Theorem 1.1, it suffices to show that the function f is proper and hasno critical points outside of a compact set.First fix one point p ∈ M as a base point. For any q ∈ M with d(p,q) = L, choose a shortest geodesic γ from pto q parametrized by arc length. Then〈∇f, γ̇ 〉(q) = 〈∇f, γ̇ 〉(p) +L∫0d2dt2f(γ (t))dt  〈∇f, γ̇ 〉(p) +L∫0(λ − Ric(γ̇ , γ̇ ))dt λL − |∇f |(p) −L∫0Ric(γ̇ , γ̇ )dt.If the integral∫ L0 Ric(γ̇ , γ̇ )dt  Λ for some constant Λ independent of q and the choice of γ , then |∇f |(q) 〈∇f, γ̇ 〉(q)  λd(p,q) − |∇f |(p) − Λ, which implies that |∇f |(q) has a linear growth in d(p,q) and so f is aproper function without critical points outside of a compact set. In the remainder of this section, we will focus onproving that∫ L0 Ric(γ̇ , γ̇ ) has an upper bound under the assumptions of Theorem 1.1.Case (i): Ric  Cg for some constant C < ∞. By Lemma 2.2 of [10], the integral bound is given by Λ = 2(n − 1) +2C.Case (ii): Ric  −δ−1g and inj(M,g)  δ > 0 for some δ > 0. Suppose d(p,q) = L  δ. Let ϕ(t) : [0,L] → [0,1] bean arcwise smooth function such that ϕ(0) = ϕ(L) = 0. By the second variation formula, as proposed in [10] or [11],we have the estimate:∫ L0 ϕ2(t)Ric(γ̇ , γ̇ )dt  (n − 1) ∫ L0 |ϕ̇|2 dt . Now define ϕ byϕ(t) = 3 t, t ∈[0,δ]; ϕ(t) = 1, t ∈[δ,L − δ]; ϕ(t) = 3 (L − t), t ∈[L − δ ,L],δ 3 3 3 δ 3F.-q. Fang et al. / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 653–656 655then we have the estimateL∫0Ric(γ̇ , γ̇ )dt  (n − 1)L∫0|ϕ̇|2 dt +δ/3∫0(1 − ϕ2)Ric(γ̇ , γ̇ )dt +L∫L−δ/3(1 − ϕ2)Ric(γ̇ , γ̇ )dt 6δ(n − 1) + 23+δ/3∫0Ric(γ̇ , γ̇ )dt +L∫L−δ/3Ric(γ̇ , γ̇ )dt,where in the second inequality, we used the fact that−δ/3∫0ϕ2 Ric(γ̇ , γ̇ )dt  1δδ/3∫0ϕ2 dt = 19, and similarly, −L∫L−δ/3ϕ2 Ric(γ̇ , γ̇ )dt  19.We next prove that∫ δ/30 Ric(γ̇ , γ̇ )dt and∫ LL−δ/3 Ric(γ̇ , γ̇ )dt are bounded from above and so finish the proof ofTheorem 1.1. This is given by the following lemma:Lemma 2.1. If Ric  −δ−1g and inj(M,g)  δ > 0 for some δ > 0, thenδ/3∫0Ric(γ̇ , γ̇ )dt  6δ(n − 1) + 23for any minimal arc length parametrized geodesic γ : [0, δ3 ] → M .Proof. Firstly, by inj(M,g)  δ, we can extend the geodesic γ to a shortest geodesic σ : [0, δ] → M , such that γ (t) =σ(t + δ3 ), t ∈ [0, δ3 ]. Set L = δ in the arguments above, we have∫ δ0 ϕ2 Ric(σ̇ , σ̇ )dt  (n − 1) ∫ δ0 |ϕ̇|2 dt = 6δ (n − 1),then using Ric  −δ−1g, we get the estimateδ/3∫0Ric(γ̇ , γ̇ )dt =2δ/3∫δ/3Ric(σ̇ , σ̇ )dt  6δ(n − 1) −δ/3∫0ϕ2 Ric(σ̇ , σ̇ )dt −δ∫2δ/3ϕ2 Ric(σ̇ , σ̇ )dt 6δ(n − 1) + 23. 3. Proof of Theorem 1.2As before, we will prove that the potential function f to the Ricci soliton is proper and has no critical points outsideof B(p,ρ) for large ρ. Suppose (M,g) is a complete shrinking Ricci soliton which satisfies Ric+Hess(f ) = g2for some potential function f . Suppose further that the scalar curvature |R|  C for some constant C < ∞. It’swell-known that the following analytic equality holds for the soliton (after modifying f by a translation, see [2] forexample):R + |∇f |2 = f. (1)Let q ∈ M be one point and denote by ρ = d(p,q) the distance from p to q . Let γ be a shortest arc lengthparametrized geodesic from p to q . First we have:Lemma 3.1. ρ2 − |∇f |(p) − |∇f |(q) ∫ ρ0 Ric(γ̇ , γ̇ )dt .Proof. It follows from a direct computation,|∇f |(q)  〈∇f, γ̇ 〉(q) = 〈∇f, γ̇ 〉(p) +ρ∫d2dt2f(γ (t))dt  −|∇f |(p) +ρ∫ (12− Ric(γ̇ , γ̇ ))dt. 0 0656 F.-q. Fang et al. / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 653–656On the other hand, by second variation formula as did in above section, we can get an upper bound of∫ ρ0 Ric(γ̇ , γ̇ )dt . Precisely, for the function ψ defined byψ(t) = t, t ∈ [0,1]; ψ(t) ≡ 1, t ∈ [1, ρ − 1]; ψ(t) = ρ − t, t ∈ [ρ − 1, ρ],we have the estimateρ∫0Ric(γ̇ , γ̇ )dt ρ∫0(n − 1)|ψ̇ |2 dt +1∫0(1 − ψ2)Ric(γ̇ , γ̇ )dt +ρ∫ρ−1(1 − ψ2)Ric(γ̇ , γ̇ )dt 2(n − 1) + supB(p,1)|Ric | +ρ∫ρ−1(1 − ψ2)(12− d2dt2f (γ (t)))dt.By integration by parts, we have the estimate for the last termρ∫ρ−1(1 − ψ2)(12− d2dt2f(γ (t)))dt  1 + 2ρ∫ρ−1ψddtf(γ (t))dt = 1 − 2f (γ (ρ − 1)) + 2ρ∫ρ−1f(γ (t))dt.Substituting this equality into above estimate, we obtain:ρ∫0Ric(γ̇ , γ̇ )dt  2n + supB(p,1)|Ric | + 2ρ∫ρ−1f(γ (t))dt − 2f (γ (ρ − 1)) 2n + supB(p,1)|Ric | + supx,y∈B(q,1)2∣∣f (x) − f (y)∣∣. (2)Now we are ready to show f is proper. By assumption |R|  C and then using Eq. (1), we see that |∇f | √f − R  √f + C. It follows that |√f (x) + C − √f (q) + C|  12 for all x ∈ B(q,1), and so∣∣f (x) − f (y)∣∣ = ∣∣√f (x) + C + √f (y) + C∣∣ · ∣∣√f (x) + C − √f (y) + C∣∣  2√f (q) + C + 1,∀x, y ∈ B(q,1).The combination of Lemma 3.1, Eq. (2) and above estimate givesρ2 2n + |∇f |(p) + |∇f |(q) + supB(p,1)|Ric | + 4√f (q) + C + 2 2(n + 1) + |∇f |(p) + supB(p,1)|Ric | + 5√f (q) + C.It concludes that f is a proper function and then by boundedness of R and Eq. (1), f has no critical points outside ofa compact domain. This finishes the proof of the theorem.References[1] W. Ambrose, A theorem of Myers, Duke Math. J. 24 (1957) 345–348.[2] H.D. Cao, R. Hamilton, T. Ilmanen, Gauss densities and stability for some Ricci solitons, arXiv: math.DG/0404165.[3] A. Derdzinski, A Myers-type theorem and compact Ricci solitons, Proc. Amer. Math. Soc. 134 (2006) 3645–3648.[4] M. Fernández-López, E. García-Río, A remark on compact Ricci solitons, Math. Ann. 340 (2008) 893–896.[5] D. Gromoll, W.T. Meyer, Examples of complete manifolds with positive Ricci curvature, J. Differential Geom. 21 (2) (1985) 195–211.[6] X.M. Li, On extensions of Myers’ theorem, Bull. London Math. Soc. 27 (1995) 392–396.[7] J. Lott, Some geometric properties of the Bakry–Émery–Ricci tensor, Comment. Math. Helv. 78 (2003) 865–883.[8] J. Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968) 1–7.[9] G.F. Wei, W. Wylie, Comparison geometry for the Bakry–Émery Ricci tensor, arXiv: math.DG/0706.1120v1.[10] W. Wylie, Complete shrinking Ricci solitons have finite fundamental group, Proc. Amer. Math. Soc. 136 (2008) 1803–1806.[11] Z.L. Zhang, On the finiteness of the fundamental group of a compact shrinking Ricci soliton, Colloq. Math. 107 (2007) 297–299.

Complete gradient shrinking Ricci solitons have finite topological type

Fu-quan Fang

Jian-wen Man

Zhen-lei Zhang

Comptes Rendus Mathématique

http://www.numdam.org/item/10.1016/j.crma.2008.03.021.pdf

Complete gradient shrinking Ricci solitons have finite topological type

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