Abstract
Abstract. In this note, we prove that a 3-dimensional steady Ricci soliton is rotationally symmetric if its scalar curvature R(x) satisfies
Abstract
for some constant C0 > 0, where ρ(x) denotes the distance from a fixed point x0. Our result doesn't assume that the soliton is κ-noncollapsed.
Introduction
In his celebrated paper [13] , Perelman conjectured that all 3-dimensional κ-noncollapsed steady (gradient) Ricci solitons must be rotationally symmetric. The conjecture is solved by Brendle in 2012 [1] . For a general dimension n ≥ 3, under an extra condition that the soliton is asymptotically cylindrical, Brendle also proves that any κ-noncollapsed steady Ricci soliton with positive sectional curvature must be rotationally symmetric [2] . In general, it is still open whether an n-dimensional κ-noncollapsed steady Ricci soliton with positive curvature operator is rotationally symmetric for n ≥ 4. For κ-noncollapsed steady Kähler-Ricci solitons with nonnegative bisectional curvature, the authors have recently proved that they must be flat [10] , [11] .
Introduction
Recall from [2] , Definition 1.1. An n-dimensional steady Ricci soliton (M, g, f ) is called asymptotically cylindrical if the following holds: (i) Scalar curvature R(x) of g satisfies
Introduction
, ∀ ρ(x) ≥ r 0 , * where C 0 > 0 is a constant and ρ(x) denotes the distance of x from a fixed point x 0 .
Introduction
(ii) Let p m be an arbitrary sequence of marked points going to infinity. Consider rescaled metrics g m (t) = r −1 m φ * rmt g, where r m R(p m ) = n−1 2 + o(1) and φ t is a one-parameter subgroup generated by X = −∇f . As m → ∞, flows (M, g m (t), p m ) converge in the Cheeger-Gromov sense to a family of shrinking cylinders (R × S n−1 (1), g(t)), t ∈ (0, 1). The metric g(t) is given by
Introduction
where S n−1 (1) is the unit sphere of euclidean space.
Introduction
In this note, we discuss 3-dimensional steady (gradient) Ricci solitons without assuming the κ-noncollapsed conditon. 1 We prove
Introduction
be a 3-dimensional steady Ricci soliton. Then, it is rotationally symmetric if the scalar curvature R(x) of (M, g, f ) satisfies
Introduction
for some constant C > 0, where ρ(x) denotes the distance from a fixed point x 0 .
Introduction
Under the condition (1.2), we need to check the property (ii) in Definition 1.1 to prove Theorem 1.2. Actually, we show that for any sequence p i → ∞, there exists a subsequence
Introduction
where
Introduction
) is a shrinking cylinders flow, i.e.
Introduction
As in [10] , we study the geometry of neighborhood
Introduction
for any p ∈ M . We are able to give a uniform injective radius estimate for (M, R p i g) at each sequence of p i . Then we can still get a limit flow for rescaled flows (M, g p i (t)), which will split off a line. By using a classification result of Daskalopoulos-Hamilton-Sesum for ancient flows on a compact surface [8] , we finish the proof of Theorem 1.2. We remark that the curvature condition in Theorem 1.2 cannot be removed, since there does exist a 3-dimensional non-flat steady Ricci soliton with exponential curvature decay. For example, (R 2 ×S 1 , g cigar +ds 2 ), where (R 2 , g cigar ) is a cigar soliton. Also, Theorem 1.2 is not true for dimension n ≥ 4 by Cao's examples of steady Kähler-Ricci solitons with positive sectional curvature [3] .
Introduction
At last, we remark that it is still open wether there exists a 3-dimensional collapsed steady Ricci soliton with positive curvature. Hamilton has conjectured that there should exist a family of collapsed 3-dimensional complete gradient steady Ricci solitons with positive curvature and S 1 -symmetry (cf. [5] Proof. We need to show that (M, g) has positive Ricci curvature. On the contrary, (M, g) locally splits off a flat piece of line by Shi's splitting theorem [14] . Then, the universal cover ( M , g) of (M, g) is isometric to a product Riemannian manifold of a real line and the cigar soliton. Namely, ( M , g) = (R 2 × R, g cigar + ds 2 ). Let π : M → M be a universal covering. We fix x 0 ∈ M and x 0 ∈ M such that π( x 0 ) = x 0 . For any x ∈ M and x ∈ M such that π( x) = x, one sees
Introduction
where ρ and ρ are the distance functions w.r.t g and g respectively. Let { x i } i≥1 be a sequence of points so that x i = (p i , 0) ∈ R 2 × R tend to infinity. Then, one may check that
Introduction
Again by (1.2) and (2.2), we get
Introduction
This is a contradiction to (2.3). Hence, the lemma is proved. Corollary 2.2. (M, g, f ) in Theorem 1.2 has a unique equilibrium point o, i.e., ∇f (o) = 0. As a consequence, Σ r = {f (x) = r} is diffeomorphic to S 2 , for any r > f (o).
Introduction
Proof. Note that
Introduction
By taking covariant derivatives on both sides of (2.5), it follows 2Ric(∇f, ∇f ) = − ∇R, ∇f . By the Morse theorem, Σ r = {f (x) = r > f (o)} is diffeomorphic to S 2 (cf. [10] , Lemma 2.1).
Geometry of M p,k
For any p ∈ M and number k > 0, we set
Geometry of M p,k
Let g p = R(p)g be a rescaled metric and denote B(p, r; g p ) a r-geodesic ball centered at p with respect to g p . Then by Corollary 2.2, we have (cf. [10] , Lemma 3.3)
Geometry of M p,k
By Lemma 3.1, we prove Lemma 3.2. Under (1.2), there exists a constant C such that
Geometry of M p,k
It is known by [4] ,
Geometry of M p,k
Thus by (1.2), (3.3) and (3.4), we get
Geometry of M p,k
On the other hand, by (3.1), we have
Geometry of M p,k
Let φ t be generated by −∇f . Then g(t) = φ * t g satisfies the Ricci flow,
Geometry of M p,k
Also rescaled flow g p (t) = R(p)g(R −1 (p)t) satisfies (3.7). Since the Ricci curvature is positive,
Geometry of M p,k
Combining with (3.6), we get
Geometry of M p,k
Thus, by Shi's higher order estimates, we obtain
Geometry of M p,k
It follows
Geometry of M p,k
In particular, we have
Geometry of M p,k
The lemma is proved. * Remark 3.3. Under (1.2), by the same argument as in the proof of Lemma 3.2, for each k ∈ N, there exists a constant C(k) such that
Geometry of M p,k
Next, we want to show that M p,k is bounded by a finite ball B(p, Ck; g p ), where C is a uniform constant. We need to use the Gauss formula,
Geometry of M p,k
We choose a normal basis {e 1 , e 2 } on (Σ r ,ḡ) with the induced metricḡ. Then {e 1 , e 2 , ∇f |∇f | } spans a normal basis of (M, g). Thus
Geometry of M p,k
Since (Σ r ,ḡ) is a surface, K = R 11 = R 22 . Hence, we get Lemma 3.4. The Gauss curvature of (Σ r ,ḡ) is given by
Geometry of M p,k
Lemma 3.5. Under (1.2), there exists a uniform B > 0 such that the following is true: for any k ∈ N, there existsr 0 =r 0 (k) such that
Geometry of M p,k
Proof. By (1.2) and (3.4), we have
Geometry of M p,k
as long as ρ(p) ≥ r 0 >> 1. Then by Lemma 3.4 and Lemma 3.2, we get
Geometry of M p,k
It follows
Geometry of M p,k
On the other hand, by (1.2), (3.3) and (3.4), we see
Geometry of M p,k
as long as ρ(p) ≥ r 0 . Then similar to (3.5), there exists ar 0 ≥ r 0 such that
Geometry of M p,k
Thus by (3.13), we get
Geometry of M p,k
where B > 0 is a uniform constant. By the Myer's theorem, the diameter of Σ f (p) is bounded by
Geometry of M p,k
.
Geometry of M p,k
As a consequence,
Geometry of M p,k
For any q ∈ M p,k , there exists q ′ ∈ Σ f (p) such that φ s (q) = q ′ for some s ∈ R. Then by (3.16) and (3.12), we have *
Geometry of M p,k
The lemma is proved.
Geometry of M p,k
By Lemma 3.5, we get the following volume estimate of B(p, s; g p ).
Geometry of M p,k
Proposition 3.6. Under (1.2) of Theorem 1.2, there exists s 0 and c > 0 such that
Geometry of M p,k
Moreover, the injective radius of (M, g p ) at p has a uniform lower bound δ > 0, i.e.,
Geometry of M p,k
Proof. By Lemma 3.5, we have
Geometry of M p,k
In the following, we give an estimate of Vol(Σ l ,ḡ) for any l with
Geometry of M p,k
.
3D STEADY GRADIENT RICCI SOLITONS WITH LINEAR CURVATURE DECAY 9
By (3.5) and (3.15), we see
3D STEADY GRADIENT RICCI SOLITONS WITH LINEAR CURVATURE DECAY 9
By (3.13), it follows that the Gauss curvature K l of (Σ l , g p | Σ l ) satisfies
3D STEADY GRADIENT RICCI SOLITONS WITH LINEAR CURVATURE DECAY 9
By the Co-Area formula, we get
3D STEADY GRADIENT RICCI SOLITONS WITH LINEAR CURVATURE DECAY 9
By the volume comparison theorem, we derive from (3.19),
3D STEADY GRADIENT RICCI SOLITONS WITH LINEAR CURVATURE DECAY 9
. This proves (3.17). By (3.17), we can apply a result of Cheeger-Gromov-Taylor for Riemannian manifolds with bounded curvature to get the injective radius estimate (3.18) immediately [6] .
Proof of Theorem 1.2
First we prove the following convergence of rescaled flows. (N, g N (t) ) is an ancient solution of Ricci flow on N . * Proof. For a fixed r, as in (3.5), it is easy to see that there exists a uniform C 1 independent of r such that
Proof of Theorem 1.2
as long as i is large enough. By Lemma 3.1, it follows
Proof of Theorem 1.2
where g p i = g p i (0). Since the scalar curvature is increasing along the flow, we get
Proof of Theorem 1.2
Thus together with the injective radius estimate in Proposition 3.6, we can apply the Hamilton compactness theorem to see that g p i (t) converges subsequently to a limit flow ( M ,g(t); p ∞ ) on t ∈ (−∞, 0] [12] . Moreover, the limit flow has uniformly bounded curvature. It remains to prove the splitting property. By Remark 3.3, we have
Proof of Theorem 1.2
It follows from (4.1),
Proof of Theorem 1.2
On the other hand, by Remark 3.3, we also have
Proof of Theorem 1.2
Thus X (i) converges subsequently to a parallel vector field X (∞) on ( M ,g (0)). Moreover,
Proof of Theorem 1.2
as long as f (p i ) is large enough. This implies that X (∞) is non-trivial. Hence, ( M , g(t)) locally splits off a piece of line along X (∞) . It remains to show that X (∞) generates a line through p ∞ . By Lemma 3.5,
Proof of Theorem 1.2
and so,
Proof of Theorem 1.2
Thus X (∞) generates a line γ ∞ (s) through p ∞ as k → ∞. As a consequence, ( M , g(0)) splits off a line and so does the flow ( M , g(t); p ∞ ). The lemma is proved.
Proof of Theorem 1.2
Next we estimate the curvature of (N, g N (t)).
Proof of Theorem 1.2
Lemma 4.2. Under (1.2), there exists a constant C independent of t such that the scalar curvature R N (t) of (N, g N (t)) satisfies
Proof of Theorem 1.2
Proof. Let R(x, t) be the scalar curvature of (R × N, g(t)). It suffices to prove the following is true:
Proof of Theorem 1.2
for some constant C. For any x, y ∈ R × N , we choose r > 0 such that x, y ∈ B(p ∞ , r; g(0)). By the convergence of g p i (t), there are sequences {x i } and {y i } in B(p i , r; g p i (0)) such that x i and y i converge to x and y in the Cheeger-Gromov sense, respectively. By Lemma 3.1, we have By (4.4) and the fact
Proof of Theorem 1.2
It follows
Proof of Theorem 1.2
Hence we obtain
Proof of Theorem 1.2
This proves (4.3).
Proof of Theorem 1.2
The proof of Theorem 1.2 is completed by the following lemma. Proof. By Lemma 4.2, the Gauss curvature of (N, g N (0)) has a uniform positive lower bound. Then N is compact by Myer's Theorem. On the other hand, by a classification theorem of Daskalopoulos-Hamilton-Sesum [8] , an ancient solution on a compact surface N is either a shrinking spheres flow or a Rosenau solution. The Rosenau solution is obtained by compactifying (R× S 1 (2), h(x, θ, t) = u(x, t)(dx 2 + dθ 2 )) by adding two points, where u(x, t) = sinh(−t)
Proof of Theorem 1.2
cosh(x)+cosh(t) and t ∈ (−∞, 0). By a direct computation, R h(t) = cosh(t) cosh(x) + 1 sinh(−t)(cosh(x) + cosh(t)) . (4.5)

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