aa r X i v : . [ m a t h . DG ] J un Manifolds with Pointwise Ricci PinchedCurvature
Hui-Ling Gu
Department of MathematicsSun Yat-Sen UniversityGuangzhou, P.R.China
Abstract
In this paper, we proved a compactness result about Riemannian man-ifolds with an arbitrary pointwisely pinched Ricci curvature tensor.
1. Introduction
Let M n be an n -dimensional complete Riemannian manifold with n ≥
3. Oneof the basic problems is under which condition on its curvature the Riemannianmanifold is compact. The classical Bonnet-Myers’ theorem states that a completeRiemannian manifold with positive lower bound for its Ricci curvature is compact.In [11], Hamilton proved that:
Any convex hypersurface with dimension ≥ in Euclidean space with secondfundamental form h ij ≥ δ · tr ( h ) n must be compact. In [5], Chen-Zhu proved an intrinsic analogue of the Hamilton’s result by usingthe Ricci flow which was introduced by Hamilton in 1982. They proved that:1 f M n is a complete n -dimensional ( n ≥ Riemannian manifold with posi-tive and bounded scalar curvature and satisfies the following pointwisely pinchingcondition | W | + | V | ≤ δ n (1 − ε ) | U | , for ε > , δ = , δ = and δ n = n − n +1) , ( n ≥ , where W, V, U denote theWeyl conformal curvature tensor, traceless Ricci part and the scalar curvature partof the curvature operator respectively. Then M n is compact. For the 3-dimensional case, they weaken the curvature operator pinching con-dition to an arbitrary Ricci curvature pinching condition:
Let M be a complete 3-dimensional Riemannian manifold with bounded andnonnegative sectional curvature. If M satisfies the positive Ricci pinching condi-tion: R ij ≥ ε · scal · g ij > for some ε > . Then M must be compact. Recently, by the Ricci flow and the new invariant cone construction introducedby B¨ o hm-Wilking [1], Ni-Wu [13] proved the following compactness result in termsof curvature operator: If M n is a complete n -dimensional ( n ≥ Riemannian manifold with boundedcurvature and satisfies Rm ≥ δU > for δ > , where Rm, U denote the curvature operator and its scalar curvaturepart. Then M n must be compact. Naturally, from the above results, one expects that: any complete Riemannianmanifold with dimension ≥ Theorem 1.1
Let n ≥ . Suppose M n is a smooth complete locally conformallyflat n -dimensional manifold with bounded and positive scalar curvature. Suppose M n has nonnegative sectional curvature and satisfies the following Ricci curvaturepinching condition R ij ≥ ε · scal · g ij (1 . or some ε > . Then M n is compact. We briefly describe the proof of the theorem. Our proof of Theorem 1.1 dependson the Yamabe flow and the limit solution of Yamabe flow. Suppose there existssuch a noncompact Riemannian manifold satisfying the Ricci pinching condition(1.1), we evolve it by the Yamabe flow. By the short-time existence result [6]and the Ricci pinching condition, we can obtain a long-time existence result. Insection 2, we will study the asymptotic behaviors of the solution to the Yamabeflow. Finally in section 3, we will complete the proof of the main theorem by usingthe results obtained in section 2.
2. The Asymptotic Behaviors of the Yamabe Flow
In the geometric flows, in order to know the initial manifold well, we usuallyneed to study the asymptotic behaviors of the solution of the flow. In this section,we study the asymptotic behaviors of the Yamabe flow. First we recall the Li-Yau-Hamilton inequality of Chow [8] on locally conformally flat manifolds.
Theorem 2.1 (Chow [8])
Suppose ( M n , g ij ) is a smooth n -dimensional ( n ≥ complete locally conformally flat manifold with bounded and nonnegative Riccicurvature. Let R ( x, t ) be the scalar curvature of the solution of the Yamabe flowwith g ij as initial metric. Then we have ∂R∂t + h∇ R, X i + 12( n − R ij X i X j + Rt ≥ for any vector X on M . In his paper [8], Chow proved the above theorem for compact locally confor-mally flat manifolds with positive Ricci curvature. However, by a perturbationargument as in [9], it is clear that the Li-Yau-Hamilton inequality actually holdsfor complete locally conformally flat manifolds with nonnegative Ricci curvature.
Lemma 2.2
Let g ij ( t ) be a locally conformally flat complete solution to the Yamabeflow for t > which has bounded and positive Ricci curvature. If the Harnackquantity Z = ∂R∂t + h∇ R, X i + 12( n − R ij X i X j + Rt s positive for all X ∈ T x M n at some point x = x and t = t > , then it ispositive for all X ∈ T x M n at every point x ∈ M n for any t > t . Proof.
By the calculation in [8], we know( ∂∂t − ( n − △ ) Z ≥ ( R − t ) Z ≥ − t Z. (2 . Z is positive for all X ∈ T x M n at t = t >
0, we can find a nonnegativefunction F on M n with support in a neighborhood of x so that F ( x ) > Z ≥ Ft for all X everywhere at t = t . Let F evolve by the heat equation ∂F∂t = ( n − △ F. (2 . F > t > t . We only need to prove that Z ≥ Ft , for all t ≥ t . By (2.1) and (2.2) we know( ∂∂t − ( n − △ )( Z − Ft ) ≥ − t ( Z − Ft ) , for t ≥ t . By the maximum principle we get Z ≥ Ft .This completes the proof of the Lemma 2.2. t → + ∞ . Definition 2.3 (i)
A complete solution to the Yamabe flow is called a Type I limitsolution if the solution has nonnegative Ricci curvature and exists for −∞ < t < Ω for some constant Ω with < Ω < + ∞ and R ≤ ΩΩ − t everywhere with equalitysomewhere at t = 0.(ii) A complete solution to the Yamabe flow is called a Type II limit solutionif the solution has nonnegative Ricci curvature and exists for −∞ < t < + ∞ and R ≤ everywhere with equality somewhere at t = 0.4iii) A complete solution to the Yamabe flow is called a Type III limit solution ifthe solution has nonnegative Ricci curvature and exists for − A < t < + ∞ for someconstant A with < A < + ∞ and R ≤ AA + t everywhere with equality somewhereat t = 0. Definition 2.4 (i)
We call a solution to the Yamabe flow a steady soliton, if itsatisfies Rg ij = g jk ∇ i X k , where X i is a vector field on the manifold. (ii) We call a solution to the Yamabe flow a shrinking soliton, if it satisfies ( R − λ ) g ij = g jk ∇ i X k , where X i is a vector field on the manifold and λ is a positive constant. (iii) We call a solution to the Yamabe flow an expanding soliton, if it satisfies ( R + λ ) g ij = g jk ∇ i X k , where X i is a vector field on the manifold and λ is a positive constant.Moreover, if the vector field X is the gradient of some function f , then wewill call the corresponding soliton a steady, shrinking, expanding gradient solitonrespectively. We now follow Hamilton [10] and Chen-Zhu [5] (or also Cao [2]) to give aclassification for Type II and Type III limit solutions.
Theorem 2.5
Let M n be a smooth n -dimensional locally conformally flat andsimply connected Riemannian manifold. Then: (i) any Type II limit solution with positive Ricci curvature to the Yamabe flowon M n is necessarily a homothetically steady gradient soliton; (ii) any Type III limit solution with positive Ricci curvature to the Yamabe flowon M n is necessarily a homothetically expanding gradient soliton. Proof.
The following arguments are adapted from Hamilton [10] and Chen-Zhu[5] (or also Cao [2]), where the classification for the limit solutions of the Ricciflow were given. We only give the complete proof of (ii), since the proof of (i) is5imilar and easier. At the end of the proof we point the difference between (i) and(ii), and then it is easy to see that the rest of the arguments are the same.By the definition of the Type III limit solution, after a shift of the time variable,we may assume the Type III limit solution g ij ( t ) is defined for 0 < t < + ∞ withuniformly bounded curvature and positive Ricci curvature where tR assumes itsmaximum in space-time.Suppose tR assumes its maximum at a point ( x , t ) in space-time, then t > Z = ∂R∂t + h∇ R, X i + 12( n − R ij X i X j + Rt , (2 . X = 0 at ( x , t ). By Lemma 2.2 we know that at anyearlier time t < t and at every point x ∈ M n , there is a vector X ∈ T x M n suchthat Z = 0.By the first variation of Z in X ∇ i R + 1 n − R ij X j = 0 , (2 . X is unique at each point and varies smoothlyin space-time.Combining (2.3) and (2.4) we obtain that ∂R∂t + Rt + 12 ∇ i R · X i = 0 . (2 . X i ( ∂∂t − ( n − △ )( ∇ i R ) + n − X i X j ( ∂∂t − ( n − △ ) R ij −∇ k R ij ∇ k X j X i − ( n − ∇ k ∇ i R · ∇ k X i +( ∂∂t − ( n − △ )( ∂R∂t + Rt ) = 0 , (2 . ∂∂t − ( n − △ )( ∇ i R ) = ∇ i [( ∂∂t − ( n − △ ) R ] − ( n − R il ∇ l R = ∇ i ( R ) − ( n − R il ∇ l R, (2 . ∂∂t − ( n − △ )( ∂R∂t + Rt ) = 3( n − R △ R + ( n − − n ) |∇ R | +2 R + R t − Rt , (2 . ∂∂t − ( n − △ ) R ij = 1 n − B ij , (2 . B ij = ( n − | Ric | g ij + nRR ij − n ( n − R ij − R g ij . The combination of(2.6)-(2.9) gives − R ( R + t ) + n − n − B ij X i X j − n − RR ij X i X j + n n − R il R jl X i X j + R ij ∇ k X i ∇ k X j = 0 . (2 . ∇ k ∇ i R = − n − X j · ∇ k R ij + R ij · ∇ k X j ) , (2 . R ij (( R + 1 t ) g ij − ∇ i X j ) = 0 . (2 . R ij ( ∇ k X i − ( R + 1 t ) g ik )( ∇ k X j − ( R + 1 t ) g jk ) + A ij X i X j = 0 , (2 . A ij = n − n − B ij + n − ( nR il R jl − RR ij ) . In local coordinate { x i } where g ij = δ ij and the Ricci tensor is diagonal, i.e., Ric = diag ( λ , λ , · · · , λ n ), with λ ≤ λ ≤ · · · ≤ λ n , and e i , (1 ≤ i ≤ n ) is thedirection corresponding to the eigenvalue λ i of the Ricci tensor, we have X i λ i ( ∇ k X i − ( R + 1 t ) g ik ) + A ij X i X j = 0and A ij = diag ( ν , ν , · · · , ν n ) , where ν i = 12( n − n − X k,l = i,k>l ( λ k − λ l ) ≥ . ∇ j X i = ( R + 1 t ) g ij , and A ij X i X j = 0 . Thus ∇ j X i is symmetric and by the simply connectedness of M n , there exists afunction f such that ∇ i X j = ∇ i ∇ j f. Hence ( R + 1 t ) g ij = ∇ i ∇ j f. This means that g ij ( t ) is a homothetically expanding gradient soliton.So we have proved that if the solution exists on 0 < t < + ∞ , and the Harnackquantity Z = ∂R∂t + h∇ R, X i + 12( n − R ij X i X j + Rt vanishes, then it must be an expanding gradient soliton. If we have a solution on α < t < + ∞ , we can replace t by t − α in the Harnack quantity. Then if α → −∞ ,the expression t − α → Z = ∂R∂t + h∇ R, X i + 12( n − R ij X i X j . Then the rest of the arguments for the proof of (i) follows.Hence we complete our proof of Theorem 2.5.
Proposition 2.6
There exists no noncompact locally conformally flat Type IIIlimit solution of the Yamabe flow which satisfies the Ricci pinching condition: R ij ≥ ε · scal · g ij > , for some ε > . roof. We argue by contradiction. Suppose there is a noncompact locally con-formally flat Type III limit solution g ij ( t ) on M which satisfies the above Riccipinching condition. By Theorem 2.5, we know that the solution must be a homo-thetically expanding gradient soliton. This means that for any fixed time t = t ,we have : ( R + ρ ) g ij = ∇ i ∇ j f (2 . ρ and some function f on M .Differentiating the equation (2.14) and switching the order of differentiationsand then taking trace, we have − ( n − ∇ i R = R ij ∇ j f. (2 . t = t and consider a long shortest geodesic γ ( s ), 0 ≤ s ≤ ¯ s . Let x = γ (0) and X ( s ) = ˙ γ ( s ). Following by the same arguments as in the proof ofLemma 1.2 of Perelman [14] (or see the proof of Lemma 6.4.1 of [3] for the details)and using the Ricci pinching condition, we can obtain that | dfds − ρs | ≤ const. (2 . | f − ρs | ≤ const · ( s + 1) (2 . s large enough. From (2.16) and (2.17) we obtain that |∇ f | ( x ) ≥ cρf ( x ) ≥ c ρ s = c ρ d ( x, x )for some constant c >
0. Then by the same argument as in Theorem I in [5], wecan obtain a contradiction!Hence we complete the proof of Proposition 2.6.
Proposition 2.7
Suppose ( M n , g ij ( t )) is an n -dimensional ( n ≥ complete non-compact locally conformally flat steady gradient soliton with bounded and positive icci curvature. Assume the scalar curvature assumes its maximum at a point p ∈ M , then the asymptotic scalar curvature ratio is infinite, i.e., A = lim sup s → + ∞ Rs = + ∞ where s is the distance to the point p . Proof.
We argue by contradiction. Suppose R ≤ Cs , for some constant C > Rg ij = ∇ i ∇ j f, (2 . f on M .Consider the integral curve γ ( s ) , ≤ s ≤ ¯ s , of ∇ f with γ (0) = p and X ( s ) =˙ γ ( s ). We first claim that M is diffeomorphic to R n . Indeed, by differentiating theequation (2.18) and switching the order of differentiations and then taking trace,we have − ( n − ∇ i R = R ij ∇ j f. (2 . n − ∇ X R + CR ∇ X f ≥ , for some positive constant C depends only on n . This is equivalent to ∇ X (( n −
1) log R + Cf ) ≥ . That is the function ( n −
1) log R + Cf is nondecreasing along γ ( s ).But by the assumption R ≤ Cs , we have log R → −∞ as s → + ∞ . So f ( γ ( s )) → + ∞ as s → + ∞ . That is f is a exhaustion function on M . By(2.18) we know that f is a strictly convex function, so any two level sets of f arediffeomorphic via the gradient curves of f . Combining these and f is a exhaustionfunction, we know that M is diffeomorphic to R n . So we have proved the claim.(We can have another proof by using the main result of [4].)10ext, we follow the argument of Hamilton [12] to prove that we can take alimit on M − { p } of g ij ( x, t ) as t → −∞ and the limit is flat.By (2.18) we have ∇ X ∇ X f = R. Integrating it we obtain X ( f ( γ ( s ))) − X ( f ( γ (0))) = Z s Rds ≥ C > C > . So we have |∇ f | ≥ C >
0. Then we can evolve thefunction f backward with time along the gradient of f . When we go backwardin time, this is equivalent to following outwards along the gradient of f , and thespeed |∇ f | ≥ C >
0. So we have s | t | ≥ C as | t | large. Then R ≤ Cs ≤ CC | t | as | t | large. (2 . ≥ ∂∂t g ij = − Rg ij ≥ − CC | t | g ij . Then by the same argument as in [12], we can take a limit on M − { p } of g ij ( x, t )as t → −∞ and the limit is flat.Since M is diffeomorphic to R n , we know that M − { p } is diffeomorphic to S n − × R , but for n ≥
3, there exists no flat metric on it. So we obtain a contra-diction.Hence we complete the proof of the Proposition 2.7.
3. The Proof of the Main Theorem
Proof of the Main Theorem 1.1.
We will argue by contradiction to prove ourTheorem. Let M n be a noncompact conformally flat manifold with nonnegative11ectional curvature. Suppose M n has positive and bounded scalar curvature andsatisfies the Ricci pinching condition: R ij ≥ ε · scal · g ij for some ε >
0. We evolve the metric by the Yamabe flow: ∂g ij ( x,t ) ∂t = − Rg ij ( x, t ) ,g ij ( x,
0) = g ij ( x ) , (3 . , T ) with T > T = + ∞ or the evolvingmetric contracts to a point at a finite time T .Moreover, for locally conformally flat manifolds, we have R ijkl = 1 n − R ik g jl + R jl g ik − R il g jk − R jk g il ) − R ( n − n −
2) ( g ik g jl − g il g jk ) . Then by direct computation, we have the following evolution equation: ∂∂t R ijkl = ( n − △ R ijkl − R · R ijkl + n − n − [( R imkn R mn − R ik ) g jl +( R jmln R mn − R jl ) g ik − ( R jmkn R mn − R jk ) g il − ( R imln R mn − R il ) g jk ]= ( n − △ R ijkl − R · R ijkl + n − ( B ik g jl + B jl g ik − B il g jk − B jk g il ) , where B ij = ( n − | Ric | g ij + nRR ij − n ( n − R ij − R g ij . In a moving frame,we have: ∂∂t R abcd = ( n − △ R abcd − R · R abcd + n − ( B ac g bd + B bd g ac − B ad g bc − B bc g ad )+ Rn − · ( R ac g bd + R bd g ac − R ad g bc − R bc g ad ) . At a point where g ab = δ ab and the Ricci tensor is diagonal: Ric = diag ( λ , λ , · · · , λ n ) , with λ ≤ λ ≤ · · · ≤ λ n , we also have B ab is diagonal and the sectional curvature R abab = 1 n − λ a + λ b ) − R ( n − n − .
12f at some point, the sectional curvature R = 0, then λ + λ = Rn − . Hence if n ≥
4, we have : n − ( B aa + B bb ) + Rn − ( λ a + λ b )= n − [2( n − | Ric | + nR ( λ a + λ b ) − n ( n − λ a + λ b ) − R ] + R ( n − n − ≥ n − [ n − n R + nR n − − n ( n − R ( n − − R ] + R ( n − n − = n − n +2 n ( n − n − R > , if n = 3, by direct calculation, we have: n − ( B + B ) + Rn − ( λ + λ )= B + B + R = 4 | Ric | + 3 R ( λ + λ ) − λ + λ ) − R + R = 4( λ + λ + λ ) − λ + λ )= 4 λ − λ + λ )= R − λ + λ )= λ + λ + λ + 2 λ λ + 2( λ + λ ) λ − λ + λ )= ( λ − λ ) + ( λ − λ ) + 2 λ λ > . So we obtain that the nonnegative sectional curvature is preserved under the Yam-abe flow.Next we claim that under our assumption, the solution g ij ( t ) has a long-timeexistence. Otherwise, using the same argument as in Theorem 1.2 in [8], we knowthat the Ricci pinching condition is preserved under the Yamabe flow. Then bya scaling argument as in Ricci flow, we can take a limit to obtain a noncompactsolution to the Yamabe flow with constant positive Ricci curvature, which is acontradiction with Bonnet-Myers’ Theorem. So we have the long-time existenceresult. 13y a standard rescaling argument similarly as in Ricci flow, we know that thereexists a sequence of dilations of the solution which converges to a noncompact limitsolution, which we also denote by g ij ( t ), of Type II or Type III with positive scalarcurvature and it still satisfies the Ricci pinching condition.Now we consider its universal covering space, then we also have a solution onits universal cover which is of Type II or Type III. So in the following we considerthe limit solution is defined on its universal cover.If the limit solution is of Type III, then by Theorem 2.5, we know that it isa homothetically expanding gradient soliton, but from Proposition 2.6, we knowthat there exists no such limit solution of Type III satisfies the Ricci pinchingcondition. So the limit must be of Type II.Suppose the limit solution is of Type II, then by Theorem 2.5, we know thatit is a homothetically steady gradient soliton. From Proposition 2.7, we also knowthat lim sup s → + ∞ Rs = + ∞ , where s is the distance function from the point p where the scalar curvature R assumes its maximum. Then by the result of Hamilton [12], we can take a sequenceof points x k divergent to infinity and a sequence of r k , such that r k R ( x k ) → + ∞ and d ( p,x k ) r k → + ∞ and R ( x ) ≤ R ( x k )for all points x ∈ B ( x k , r k ). Then by a same argument as in Ricci flow, we obtainthat ( M, R ( x k ) g ij , x k ) converge to a limit manifold ( f M , f g ij , e x ) with nonnegativesectional curvature. By Proposition 2.3 in [7], we know that the limit manifoldwill split a line. Since the Ricci pinching condition is preserved under dilations, weconclude that the limit must be also satisfies the Ricci pinching condition. Andthis is a contradiction.Therefore the proof of the main theorem 1.1 is completed. eferences [1] C. B¨ o hm, and B. Wilking, Manifolds with positive curvature operators arespace forms , arXiv:math.DG/0606187 June 2006.[2] Cao, H. D.,
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