The Fundamental Groups of m-Quasi-Einstein Manifolds

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]):𝜕𝑔𝜕𝑡(𝑔0)==−𝑔2₀Ric,.(1.1) Recently, Perelman supplemented Hamilton’s result and solved the Poincaré 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+𝐿_𝑋𝑔=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+∇∇𝑓=𝑐𝑔 and says 𝑔 is a gradient Ricci soliton. In 2008, Lōpez and Río have shown that if (𝑀,𝑔) is a complete manifold with Ric+𝐿_𝑋𝑔≥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 𝜋₁(𝑀) is finite. Furthermore, Wylie [2] has shown that under these conditions if 𝑀 is complete, then 𝜋₁(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 𝑚-quasi-Einstein with 1≤𝑚<∞ and 𝑐>0 has a positive scalar curvature. Let me introduce the definition of 𝑚-quasi-Einstein.

Definition 1.1. The triple (𝑀,𝑔,𝑓) is an 𝑚-quasi-Einstein manifold if it satisfies the equationRic+Hess𝑓−1𝑚𝑑𝑓⊗𝑑𝑓=𝑐𝑔(1.3) for some 𝑐∈𝑅.
Here 𝑚-Bakry-Emery Ricci tensor Ric^𝑚𝑓≐Ric+Hess𝑓−(1/𝑚)𝑑𝑓⊗𝑑𝑓 for 0<𝑚≤∞ is a natural extension of the Ricci tensor to smooth metric measure spaces (cf. [6, Section  1 ]). Note that if 𝑚=∞, then it reduces to a gradient Ricci soliton. In this paper, we will prove the finiteness of the fundamental group of an 𝑚-quasi-Einstein with 𝑐>0.

Theorem 1.2. Let (𝑀,𝑔,𝑓) be a complete manifold with 𝑐>0 and Ric+Hess𝑓−(1/𝑚)𝑑𝑓⊗𝑑𝑓≥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. ‖∇𝑓‖is bounded. We claim that the bounded ‖∇𝑓‖ implies the compactness of 𝑀. Let 𝑞 be a point in 𝑀, and consider any geodesic 𝛾∶[0,∞)→𝑀 emanating from 𝑞 and parametrized by arc length 𝑡. Then we have ^𝑇0Ric(𝛾̇,𝛾̇)≥𝑐𝑇+1𝑚^𝑇0(𝑑𝑓(𝛾̇))²−^𝑇0𝛾̇(𝑔(∇𝑓,𝛾̇))≥𝑐𝑇−𝑔(∇𝑓,𝛾̇)|^𝑇0.(2.1)
Since 𝑔(∇𝑓,𝛾̇)|^𝑇0 is bounded we have that ∫^∞0Ric(𝛾̇,𝛾̇)=∞. Hence, the claim is followed by the proof of [4, Theorem  1]. Let (𝑀,𝑔̃) be the Riemannian universal cover of (𝑀,𝑔), let 𝑝∶(𝑀,𝑔̃)→(𝑀,𝑔) be a projection map, and let 𝑓 be a map 𝑓∘𝑝. Since 𝑝 is a local isometry, then the same inequality holds, that is, Ric(𝑔̃)+Hesse_̃𝑔𝑓−(1/𝑚)𝑑𝑓⊗𝑑𝑓≥𝑐𝑔̃. Now, since ‖∇𝑓‖ is bounded, it is followed from the above argument that 𝑀 is compact. So 𝜋₁(𝑀) is finite.Case 2. ‖∇𝑓‖ is unbounded. We will prove this case by following the proof of [2]. By Case 1, 𝑀 is noncompact. For any 𝑝∈𝑀, define 𝐻_𝑝≐max0,supRic_𝑦(𝑣,𝑣)∶𝑦∈𝐵(𝑝,1),‖𝑣‖=1.(2.2) Note that by [7, Lemma  2.2] we have ^𝑟0Ric(𝛾̇,𝛾̇)𝑑𝑠≤2(𝑛−1)+𝐻_𝑝+𝐻_𝑞.(2.3) Assume that 𝑑(𝑝,𝑞)>1. On the other hand, from the inequality of Theorem 1.2, we have ^𝑟0Ric(𝛾̇,𝛾̇)𝑑𝑠≥𝑐𝑑(𝑝,𝑞)+1𝑚^𝑟0≥𝑐𝑑(𝑝,𝑞)−‖∇𝑓‖(_𝑝𝑑𝑓(𝛾̇))²−^𝑟0𝛾̇(𝑔(∇𝑓,𝛾̇))−‖∇𝑓‖_𝑞,(2.4) since 𝑔(∇𝑓,𝛾̇)≤‖∇𝑓‖‖𝛾̇‖. Hence, we have that for any 𝑝, 𝑞∈𝑀𝑑(𝑝,𝑞)≤max1,1𝑐2(𝑛−1)+𝐻^𝑝+𝐻_𝑞+‖∇𝑓‖_𝑝+‖∇𝑓‖_𝑞.(2.5) Now we will apply a similar argument like Case 1. Fix 𝑝̃∈𝑀, and let ℎ∈𝜋₁(𝑀) identified as a deck transformation on 𝑀. Note that 𝐵(𝑝,̃1) and 𝐵(ℎ(𝑝)̃,1) are isometric, and thus 𝐻_̃𝑝=𝐻_ℎ(𝑝₎̃. Also ‖∇𝑓‖_̃𝑝=‖∇𝑓‖_ℎ(𝑝₎̃. So we conclude that 𝑑(𝑝,̃ℎ(𝑝)̃)≤max1,2𝑐n−1+𝐻^̃𝑝+‖‖∇𝑓‖‖_̃𝑝(2.6) for any ℎ∈𝜋₁(𝑀). Since the right-hand side is independent of ℎ, this proves this case.