On Locally Conformally Flat Gradient Steady Ricci Solitons

In this paper, we classify n-dimensional (n>2) complete noncompact locally conformally flat gradient steady solitons. In particular, we prove that a complete noncompact non-flat conformally flat gradient steady Ricci soliton is, up to scaling, the Bryant soliton.


The results
A complete Riemannian metric g ij on a smooth manifold M n is called a gradient steady Ricci soliton if there exists a smooth function F on M n such that the Ricci tensor R ij of the metric g ij is given by the Hessian of F : The function F is called a potential function of the gradient steady soliton. Clearly, when F is a constant the gradient steady Ricci soliton is simply a Ricci flat manifold. Thus Ricci solitons are natural extensions of Einstein metrics. Gradient steady solitons play an important role in Hamilton's Ricci flow as they correspond to translating solutions, and often arise as Type II singularity models. Thus one is interested in classifying them and understand their geometry. It turns out that compact gradient steady solitons must be Ricci flat. In dimension n = 2, Hamilton [14] discovered the first example of a complete noncompact gradient steady soliton on R 2 , called the cigar soliton, where the metric is given by ds 2 = dx 2 + dy 2 1 + x 2 + y 2 . The cigar soliton has positive curvature and is asymptotic to a cylinder of finite circumference at infinity. Furthermore, Hamilton [14] showed that the only complete steady soliton on a two-dimensional manifold with bounded (scalar) curvature R which assumes its maximum at an origin is, up to scaling, the cigar soliton. For n ≥ 3, Bryant [2] proved that there exists, up to scaling, a unique complete rotationally symmetric gradient Ricci soliton on R n , see, e.g., Chow et al. [8] for details. The Bryant soliton has positive sectional curvature, linear curvature decay and volume growth on the order of r (n+1)/2 . In the Kähler case, the first author [3] constructed a complete gradient steady Kähler-Ricci soliton on C m , for m ≥ 2, with positive sectional curvature and U (m) symmetry.
A well-known conjecture is that, in dimension n = 3, the Bryant soliton is the only complete noncompact (κ-noncollapsed) gradient steady soliton with positive 0 2000 Mathematics Subject Classication. Primary 53C21, 53C25. The first author was partially supported by NSF Grants DMS-0506084 and DMS-0909581; the second author was partially supported by NSF Grant DMS-0354621 and a Dean's Fellowship of the School of Arts and Sciences at Lehigh University. 1 sectional curvature 1 . We remark that S.-C. Chu [9] and H. Guo [12] have studied the geometry of 3-dimensional gradient steady solitons with positive sectional curvature and the scalar curvature R attaining its maximum at some origin. For n ≥ 4, it is also natural to ask if the Bryant soliton is the only complete noncompact, positively curved, locally conformally flat gradient steady soliton. In this paper, we classify ndimensional (n ≥ 3) complete noncompact locally conformally flat gradient steady solitons, and give an affirmative answer to the latter question. Our main results: Theorem 1.1. Let (M n , g ij , F ), n ≥ 3, be a n-dimensional complete noncompact locally conformally flat gradient steady Ricci soliton with positive sectional curvature. Then, (M n , g ij , F ) is isometric to the Bryant soliton. Theorem 1.2. Let (M n , g ij , F ), n ≥ 3, be a n-dimensional complete noncompact locally conformally flat gradient steady Ricci soliton. Then, (M n , g ij , F ) is either flat or isometric to the Bryant soliton.
Our work was motivated in part by the works of physicists Israel [16] and Robinson [21] concerning the uniqueness of the Schwarzschild black hole among all static, asymptotically flat vacuum space-times. In their setting, the Einstein field equations take the form 2) and ∆V = 0 for certain positive potential function V on a three-dimensional space-like hypersurface (N 3 , g ij ). They proved that such (N 3 , g ij , V ) must be rotationally symmetric without assuming locally conformal flatness. In fact, they were able to prove such (N 3 , g ij , V ) is locally conformally flat. However, it remains a challenge to do the same for 3-dimensional gradient steady Ricci solitons.
In this section, we recall some basic facts and collect several known results about gradient steady solitons.
Let (M n , g ij , F ) be a gradient steady Ricci soliton so that the Ricci tensor R ij of the metric g ij is given by the Hessian of the potential function F : Taking the covariant derivatives and using the commutating formula for covariant derivatives, we obtain (2.1) Taking the trace on j and k, and using the contracted second Bianchi identity 1 Perelman ([20], 11.9) claimed that the conjecture is true but didn't give any detail, or sketch, of a proof.
we get [15]) Let (M n , g ij , F ) be a complete gradient stead soliton satisfying Eq. (1.1). Then we have

2)
and R + |∇F | 2 = C 0 (2.3) for some constant C 0 . Here R denotes the scalar curvature.  [14] discovered the first example of a complete noncompact steady soliton on R 2 , called the cigar soliton, where the metric is given by . The (scalar) curvature of the cigar soliton is given by Hence it is positive, attains its maximum at the origin, and decays to zero exponentially fast (in terms of the geodesic distance) at space infinity. Furthermore, the cigar soliton has linear volume growth, and is asymptotic to a cylinder of finite circumference at ∞.

Now a Ricci flat metric is clearly a stationary solution of Hamilton's Ricci flow
This happens, for example, on a flat torus or on any K3-surface with a Calabi-Yau metric.
On the other hand, suppose that we have a complete steady Ricci soliton g ij on a smooth manifold M n with potential function F . As observed by Z.-H. Zhang [22], the gradient vector field V = ∇F is a complete vector field on M . Let ϕ t denote the one-parameter group of diffeomorphisms of M n generated by −V . Then it easily follows thatg Next we present a useful result, which was implicitly proved by B.-L. Chen [7]. For the reader's convenience, we include a proof here (see also Proposition 5.5 in [4]).
Proposition 2.2. Let g ij (t) be a complete ancient solution to the Ricci flow on a noncompact manifold M n . Then the scalar curvature R of g ij (t) is nonnegative for all t.
Proof. Suppose g ij (t) is defined for −∞ < t ≤ T for some T > 0. We divide the argument into two steps: Step 1: Consider any complete solution g ij (t) defined on [0, T ]. For any fixed point x 0 ∈ M , pick r 0 > 0 sufficiently small so that for all t ∈ [0, T ]. Then for any positive number A > 2, pick K A > 0 such that R ≥ −K A on B 0 (x 0 , Ar 0 ) at t = 0. We claim that there exists a universal constant C > 0 (depending on the dimension n) such that Indeed, take a smooth nonnegative decreasing function φ on R such that φ = 1 on (−∞, 7/8], and φ = 0 on [1, ∞). Consider the function On the other hand, by Lemma 8.3(a) of Perelman [20] (see also Lemma 3.4.1 (i) of [5]), we know that the liminf of all forward difference quotients, then Hence, and the inequality (2.6) in our claim follows.
Step 2: Now if our solution g ij (t) is ancient, we can replace t by t − α in (2.6) and get Letting A → ∞ and then α → −∞, we see that R(·, t) ≥ 0 for all t. This completes the proof of Proposition 2.2.
As an immediate corollary, we have Lemma 2.2. Let (M n , g ij , F ) be a complete gradient steady soliton. Then it has nonnegative scalar curvature R ≥ 0.
We remark that Munteanu-Sesum [19] recently proved a Liouville type theorem for gradient steady solitons, namely a gradient steady soliton does not admit any non-trivial harmonic function with finite Dirichlet energy. As a consequence, a gradient steady soliton has at most one nonparabolic end.
For steady solitons with nonnegative Ricci curvature, we have the following Lemma 2.3. (Hamilton [15]) Let (M n , g ij , F ) be a complete gradient steady soliton with nonnegative Ricci curvature Rc ≥ 0 and assume the scalar curvature R attains its maximum at some point x 0 . Then the potential function F is weakly convex and attains its minimum at x 0 .
Moreover, if the Ricci curvature of (M n , g ij , F ) is assumed to be positive, then Lemma 2.3 can be strengthened to the following 2 Proposition 2.3. Let (M n , g ij , F ) be a complete noncompact gradient steady soliton with positive Ricci curvature Rc > 0. Assume the scalar curvature R attains its maximum at some origin x 0 . Then, there exist some constants 0 < c 1 ≤ √ C 0 and c 2 > 0 such that the potential function F satisfies the estimates where r(x) = d(x 0 , x) is the distance function from x 0 , and C 0 = R max is the constant in (2.2). In particular, F is a strictly convex exhaustion function achieving its minimum at the only critical point x 0 , and the underlying manifold M n is diffeomorphic to R n .
Proof. It is clear that the upper estimate in (2.7) in fact holds for complete gradient steady solitons in general, because |∇F | 2 ≤ C 0 by (2.3) and Lemma 2.2.
To prove the lower estimate, we consider any minimizing normal geodesic γ(s), 0 ≤ s ≤ s 0 for large s 0 > 0, starting from the origin x 0 = γ(0). Denote by X(s) =γ(s), the unit tangent vector along γ, andḞ = ∇ X F (γ(s)). By (1.1), we have ∇ XḞ = ∇ X ∇ X F = Rc(X, X). where c 1 > 0 is the least eigenvalue of Rc on the unit geodesic ball B x0 (1). Thus, (1)). Now we turn our attention to locally conformally flat steady Ricci solitons. For any Riemannian manifold (M n , g), let be the Weyl tensor, and denote by the Cotton tensor. It is well known that, for n = 3, W ijkl vanishes identically while C ijk = 0 if and only if (M 3 , g ij ) is locally conformally flat; for n ≥ 4, W ijkl = 0 if and only if (M n , g ij ) is locally conformally flat. Moreover, for n ≥ 4, the vanishing of Weyl tensor W ijkl implies the vanishing of the Cotton tensor C ijk , while C ijk = 0 corresponds to the Weyl tensor being harmonic.
In proving Theorem 1.2, we need the following important facts due to B.-L. Chen  [23]. In fact, the same arguments in [23] imply that a complete ancient solution g ij (t) to the Ricci flow with vanishing Weyl tensor for each time t is necessarily of nonnegative curvature operator.

Combining Proposition 2.4 with (2.3) in Lemma 2.1, we have
Proposition 2.5. Let (M n , g ij , F ) be a complete gradient steady soliton such that either n = 3, or n ≥ 4 and (M n , g ij , F ) is locally conformally flat. Then (M n , g ij , F ) has bounded and nonnegative curvature operator 0 ≤ Rm ≤ C.

The proofs of Theorem 1.1 and Theorem 1.2
Throughout this section, we assume that (M n , g ij , F ) (n ≥ 3) is a complete noncompact locally conformally flat gradient steady soliton. We are going to prove Theorem 1.1, which is the same as Proposition 3.1 below, and Theorem 1.2.
Proposition 3.1. If (M n , g ij , F ), n ≥ 3, is a n-dimensional complete noncompact locally conformally flat gradient steady soliton with positive sectional curvature, then (M n , g ij , F ) is a rotationally symmetric gradient staedy soliton on R n , hence isometric to the Bryant soliton.
Proof. First of all, since the sectional curvature of (M n , g ij , F ) is positive we know that M n is diffeomorphic to R n by Gromoll-Meyer [11] (or by Proposition 2.3 if R attains its maximum at some origin x 0 ). Moreover, since the Ricci curvature is positive, the potential function F is strictly convex, thus having at most one critical point. Secondly, if we denote by G = |∇F | 2 , then in any neighborhood, where G = 0, of the equipotential hypersurface Σ c =: {x ∈ M : F (x) = c} of a regular value c of F , we can express the metric ds 2 = g ij (x)dx i dx j as where θ = (θ 2 , · · · , θ n ) denotes intrinsic coordinates for Σ c . It is clear that the key step in proving Proposition 3.1 is to show that in (3.1) we have G = G(F ), g ab = g ab (F ), and that (Σ c , g ab ) is a space form of positive curvature. Note that an n-dimensional rotationally symmetric metric is of the form whereḡ is the standard metric on the unit sphere S n−1 , and the Ricci tensor of such a metric is given by In particular, it has at most two distinct eigenvalues depending only on t.
We shall first derive a useful formula for the norm square of the Cotton tensor for steady solitons with vanishing Weyl tensor. This formula plays an important role in our derivation of the desired property of the Ricic tensor for the steady solton metric. Moreover, as W ijkl = 0 always when n = 3, this formula is of particular interest in the three-dimensional case.
Lemma 3.1. For any n-dimensional gradient steady soliton (n ≥ 3) with vanishing Weyl tensor W ijkl = 0, we have Here C ijk is the Cotton tensor defined by (2.9).
Proof. Notice that, by (1.1) and Lemma 2.1, we have ∇G = 2Rc(∇F, ·) = −∇R. Using (2.1), (3.3), and the assumption W ijkl = 0, we can express Hence, by direct computations, we have On the other hand, by (3.3), and Rc(∇F, ∇F ) = 1 2 ∇G · ∇F. Thus, Next, using Lemma 3.1, we show that the Ricci tensor of our (M n , g ij , F ) has, at least pointwisely, the desired property that it has at most two distinct eigenvalues.
Proof. For any regular value c of F , pick an orthonormal basis {E 1 , · · · , E n } of the tangent space T p M at p ∈ Σ c = {x ∈ M : F (x) = c} so that Rc(E i , E j ) = λ i δ ij . Then, we have Plugging the above two identities into (3.2) and noticing that C ijk = 0 by assumption, it follows that Since c is a regular value of F and p ∈ Σ c , ∇F (p) = 0. On the other hand, from (3.4) it is easy to see that if ∇F (p) has two or more nonzero components with respect to {E i } n i=1 , then λ 1 = · · · = λ n so the Ricci tensor has a unique eigenvalue. Otherwise, say ∇ 1 F = 0 and ∇ i F = 0 for i = 2, · · · , n, then ∇F = |∇F |E 1 , with E 1 = ∇F/|∇F |, is an eigenvector of Rc, and λ 2 = · · · = λ n . In either case, we conclude that ∇F is an eigenvector, and the Ricci tensor has the desired properties.
Remark 3.1. Fernández-López and García-Río [10] showed, by a different argument, that for shrinking solitons with harmonic Weyl tensor the gradient of any potential function is an eigenvector of the Ricci tensor. (e) Σ c , with the induced metric g ab , is of constant sectional curvature.
Hence G is constant on Σ c , and so is R by (2.3). This proves (a) and (b). Next, since R ab = R aa g ab and R 22 = · · · = R nn by Lemma 3.1, we have where Moreover, the Codazzi equation says that, for a, b, c = 2, · · · , n, R 1cab = ∇ Σc a h bc − ∇ Σc b h ac . Tracing over b and c, we obtain

proving (c) and (d).
Finally, by the Gauss equation, the sectional curvatures of (Σ c , g ab ) are given by On the other hand, since W ijkl = 0, But we already showed that G, H, R are constant on Σ c . Therefore (Σ c , g ab ) has constant sectional curvature proving (e).
With Lemma 3.3 in our hands, we now conclude the proof of Proposition 3.1: Recall that in any neighborhood, where G = 0, of the equipotential hypersurface Σ c =: {x ∈ M : F (x) = c} of a regular value c of F , we can express the metric ds 2 = g ij (x)dx i dx j as where θ = (θ 2 , · · · , θ n ) denotes intrinsic coordinates for Σ c . Then Lemma 3.3 tells us that G = G(F ), g ab = g ab (F ), and that (Σ c , g ab ) is a space form, with positive curvature (this follows from (3.5) and the assumption that (M n , g ij , F ) has positive sectional curvature). Also, F has exactly one minima at some origin x 0 , otherwise (M n , g ij , F ) would split out a flat factor which is impossible. Hence, on M \ {x 0 } we have whereḡ denotes the standard metric on the unit sphere S n−1 , and φ is some smooth function on M n depending only on F and vanishing only at x 0 . Thus, (M n , g ij , F ) is a rotationally symmetric gradient steady soliton on R n . Therefore, it is the Bryant soliton.
Proof of Theorem 1.2 Proof. Now, by assumption, (M n , g ij , F ) is a complete noncompact locally conformally flat gradient steady soliton. By Proposition 2.5, we know that (M n , g ij , F ) has bounded and nonnegative curvature operator 0 ≤ Rm ≤ C. From Hamilton's strong maximum principle (see [13] and [15]), (M n , g ij , F ) is either of positive curvature operator, or its holonomy group reduces. If (M n , g ij , F ) has positive curvature operator Rm > 0, then by Proposition 3.1/Theorem 1.1, (M n , g ij , F ) must be the Bryant soliton.
On the other hand, if (M n , g ij , F ) has reduced holonomy, then (M n , g ij , F ) is either a Riemannian product, or a locally symmetric space, or irreducible but not locally symmetric.
• Case (a): (M n , g ij , F ) is a Riemannian product. It is known (cf. p.61 in [1]) that the only conformally flat Riemannian products are either the product of a space form N n−1 with S 1 or R 1 , or the product of two Riemannian manifolds, one with constant sectional curvature −1 and the other with constant sectional curvature 1. However, since our (M n , g ij ) is a steady Ricci soliton with nonnegative sectional curvature, this implies that the latter case cannot occur and that in the former case N n−1 must be flat. Hence (M n , g ij , F ) is flat in case (a); • Case (b): (M n , g ij , F ) is a locally symmetric space. In case (b), (M n , g ij , F ) is necessarily Einstein. However, (M n , g ij , F ) is noncompact and of nonnegative sectional curvature. Therefore it follows that it must be Ricci flat and hence flat; • Case (c): (M n , g ij , F ) is irreducible and not locally symmetric.
In this case the flatness of (M n , g ij , F ) follows from the holonomy classification theorem of Berger and Simons (cf. [1]), the fact that for Kähler manifolds of complex dimension m > 1 (cf. Proposition 2.68 in [1]), and that our (M n , g ij , F ) is noncompact, conformally flat and of nonnegative sectional curvature.
Thus we have shown that in all above three cases (M n , g ij , F ) are flat. This completes the proof of Theorem 1.2.
Remark 3.2. Shortly after our work appeared on the arXiv, Catino and Mantegazza [6] obtained a similar result to our Theorem 1.1 for dimension n ≥ 4 by studying the evolution of Weyl tensor under the Ricci flow. However, as pointed out in [6], their argument does not work for n = 3.

Further Remarks
In this section, we point out that several lemmas in the proof of Theorem 1.1 for gradient steady Ricci solitons also hold for gradient shrinking and expanding Ricci solitons satisfying R ij = ∇ i ∇ j F + ρg ij , (4.1) with ρ = 1/2 for shrinkers and ρ = −1/2 for expanders respectively. In particular, our method, when combined with a result of Kotschwar [18], yields another proof of the classification theorem for n-dimensional (n ≥ 4) complete locally conformally flat shrinking gradient solitons (see Proposition 4.1).  Therefore, by replacing G = |∇F | 2 by G ρ = G + 2ρF and carrying out the same arguments as in the proof of Lemma 3.1, we obtain Lemma 4.1. For any n-dimensional gradient shrinking or expanding soliton (n ≥ 3) with vanishing Weyl tensor W ijkl = 0, the Cotton tensor C ijk has the property that Consequently, Lemma 3.2 and Lemma 3.3 hold for locally conformally flat gradient shrinkers and expanders as well.
mean curvature H are given respectively by n-dimensional (n ≥ 4) complete locally conformally flat shrinking gradient solitons (which is first due to the combined works of Z.-H. Zhang and Ni-Wallach, and also the combination of more recent work of Munteanu-Sesum and the earlier works of either Petersen-Wylie, or X. Cao, B. Wang and Z. Zhang).
Proposition 4.1. Any n-dimensional, n ≥ 4, complete locally conformally flat gradient shrinking Ricci soliton is a finite quotient of R n , or S n , or S n−1 × R.