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On a dichotomy of the curvature decay of steady Ricci soliton

We establish a dichotomy on the curvature decay for four dimensional complete noncompact non Ricci flat steady gradient Ricci soliton with linear curvature decay and proper potential function. A similar dichotomy is also shown in higher dimensions under the additional assumption that the Ricci curvature is nonnegative outside a compact subset.


Introduction
Let (M n , g) be a smooth connected Riemannian manifold and X a smooth vector field on M . The triple (M, g, X) is said to be a Ricci soliton if there is a constant λ ∈ R such that (1) Ric where Ric and L X denote the Ricci curvature and Lie derivative with respect to X respectively. A Ricci soliton is called steady (shrinking, expanding) if λ = 0 (> 0, < 0 resp.). Upon scaling the metric by a constant, we may assume λ ∈ {− 1 2 , 0, 1 2 }. The soliton is called complete if (M, g) is complete as a Riemannian manifold. It is said to be gradient if X can be chosen as X = ∇f for some smooth function f on M . In this case, f is called a potential function and (1) can be rewritten as (2) Ric + ∇ 2 f = λg.
Ricci soliton is of great importance as it sometimes arises as a rescaled limit of the Ricci flow near its singularities. When f in (2) is a constant, the metric becomes Einstein. Hence Ricci soliton can also be viewed as a natural generalization of the Einstein manifold as well.
In view of different examples of steady gradient Ricci solitons, the exponential and linear curvature decays seem to be two generic decays for noncompact steady solitons (see [25,11] and ref. therein). Munteanu-Sung-Wang [52] raised the following conjecture on the curvature decay of steady solitons: Conjecture 1.1. [52] If (M, g, f ) is a complete non Ricci flat steady gradient Ricci soliton with |Rm| → 0 as x → ∞, then either one of the following estimates holds outside a compact set of M : (3) C −1 r −1 ≤ |Rm| ≤ Cr −1 ; (4) where C is a positive constant and r is the distance function.
Recently, Lai [49] has resolved a conjecture of Hamilton and constructed a family of 3 dimensional gradient steady solitons that are flying wings. It is natural to examine their curvature properties to see if they fulfill the expectation from Conjecture 1.1. As shown in [49], |Rm| does not decay uniformly to 0 at infinity on these 3 dimensional examples. Analogous steady solitons were also constructed in dimension n ≥ 4 [49]. Nonetheless, It is unclear at this point if the curvature |Rm| → 0 at infinity on these higher dimensional solitons.
It follows from the Hamilton-Ivey pinching estimate that any 3 dimensional complete steady gradient Ricci soliton must have nonnegative sectional curvature (see [20] and ref. therein). However, this significant feature does not hold in higher dimensions. In particular, the examples on some line bundles over the complex projective space constructed by Cao [10] and Appleton [1] don't have nonnegative sectional curvature on the entire manifold. Hence, a reasonable next step is to consider if the dichotomy holds under linear curvature decay condition in higher dimensions. Under the assumption that the soliton is κ noncollapsed with dimension n ≥ 4, Deng-Zhu [29] obtained the estimate (7) if |Rm| ≤ C(r + 1) −1 and Ric ≥ 0 outside a compact subset (see also [31,22,5]). However, there does exist collapsed steady solitons (see [10,28,32]). Munteanu-Sung-Wang [52] and the first named author [16] proved that the upper bound of |Rm| in (4) holds, i.e. |Rm| ≤ Ce −r , if r|Rm| is uniformly small near infinity and f is bounded from above. The lower bound in (4) is a result by Chow-Lu-Yang [24] (see also [52]).
In this note, we are aimed at studying the curvature decay without any assumption of noncollapsedness and uniform smallness of r|Rm| at infinity. In particular, in real dimension 4, if the scalar curvature R decays at least linearly and the potential function f is proper, then Conjecture 1.1 will hold. Our main theorem Theorem 1.3. Let (M 4 , g, f ) be a 4 dimensional, complete, non Ricci flat steady gradient Ricci soliton with proper potential function lim x→∞ f = −∞ and linear scalar curvature decay, i.e. R ≤ C ′ (r + 1) −1 . Then there exists a positive constant C such that either one of the following estimates holds near infinity where r is the distance function.
If the Ricci curvature is nonnegative and goes to 0 uniformly at infinity, then the potential function f is proper [12]. Hence Theorem 1.3 also generalizes Theorem 1.2 and the related results in [29,52]. Instead of using Gromov's almost flatness theorem [40] or looking at the rescaled flow at infinity as in [29]. Here, we apply the parabolic maximum principle to the level set flow of f as in [8,55] to obtain the dichotomy estimates. One advantages of this approach is that it doesn't require any volume noncollapsing condition. Similar argument works as well in higher dimensions under some assumptions on the Ricci curvature. Theorem 1.4. Let (M n , g, f ) be an n ≥ 5 dimensional, complete, non Ricci flat steady gradient Ricci soliton with Ric ≥ 0 outside a compact subset and linear Riemann curvature decay, i.e. |Rm| ≤ C ′′ (r + 1) −1 . Then there exists positive constant C such that either one of the following estimates holds near infinity where c n is some dimensional constant.
Remark 1.5. Under the linear curvature decay condition, we only need to assume that |Ric| ≤ A 0 R for some constant A 0 > 0 and f is proper, instead of Ric ≥ 0 near infinity, then the same conclusion in Theorem 1.4 will also hold. It can be seen from the volume estimate in [33] that when estimate (8) is true, the soliton is collapsed. Hence Theorem 1.4 also recovers an estimate in [29]. On the other hand, Theorem 1.4 tells us that See [14] and references therein for the case of n = 4. Ancient solution to the Ricci flow and steady soliton satisfying estimate (9) have been recently studied by Ma-Zhang [50] and the first named author-Ma-Zhang [17].
For the level set of potential function under the dichotomy, we have the following two topological classifications of level set for large τ .
• If we assume that (5) or (7) holds, then by the Gauss equation and Shi's estimate (28), the level set Σ τ := {−f = τ } must have positive scalar curvature for large τ ; • If we assume that (6) or (8) holds, then it follows from Proposition 4.1 and Shoen-Yau's trick on minimal surface that the level set is diffeomorphic to a compact flat manifold (see also [33]).
Hence, as a corollary, we obtain that Corollary 1.6. Under the same assumptions as in Theorem 1.3 or Theorem 1.4, there exists a large constant τ 0 such that either one of the following is true for the compact connected hypersurface Σ τ := {−f = τ }: (a) Σ τ is diffeomorphic to a finite quotient of torus for all τ ≥ τ 0 ; (b) Σ τ has positive scalar curvature w.r.t. the intrinsic metric induced by (M, g) for all τ ≥ τ 0 . Remark 1.7. Under the same conditions as in Theorem 1.4 (with n ≥ 4 instead) and additionally the soliton is κ noncollapsed, Deng-Zhu [29] proved that that the level sets of f at infinity are diffeomorphic to a compact gradient Ricci shrinker with nonnegative Ricci curvature. In particular for as n = 4, the level sets at infinity are diffeomorphic to a spherical 3 manifolds (see also [22]). However, generally it may not be the case if noncollapsed condition is not assumed. For instance, let B 3 be the three dimensional Bryant steady soliton, the level sets of f at infinity of the product soliton B 3 × S 1 are diffeomorphic to S 2 × S 1 which has infinite fundamental group π 1 and admits no shrinker structure [38]. However, it does support a metric of positive scalar curvature as in Corollary 1.6 (b). Here, in the case of positive scalar curvature • As n = 4, by the resolutions of the Poincáre Conjecture and the Thurston's Geometrization Conjecture [56,57,58], Σ τ is a connected sum of spherical 3-manifolds and some copies of S 1 × S 2 ; • As n ≥ 5, by the Ric ≥ 0 near infinity condition, the level sets Σ τ in Corollary 1.6(b) have almost nonnegative Ricci curvature and hence have first Betti number b 1 (Σ τ ) ≤ n−2 by a result of Cheeger-Colding [19,Theorem A.1.13], where n is the real dimension of the ambient manifold M .
Dimension reduction serves as an important tool in the studies of Ricci flow and has led to a lot of successes in the classification of Ricci solitons (for example, see [5,7,8,22,23,28,29,30,31,42,55] and ref. therein). Under some volume noncollapsing conditions, one may extract different geometric information by looking at the blow down profiles of the soliton at infinity, which usually split like R × X, where X is some ancient solution to the Ricci flow of lower dimension. On 4 dimensional complete noncompact, noncollapsed, non Ricci flat steady gradient soliton with R → 0 at ∞ and Ric ≥ 0 outside compact set, Chow-Deng-Ma [22] studied the possible rescaled limits and showed that the limits at ∞ split off a line. In general, existence of smooth limit at infinity is not expected without the noncollapsedness condition. In such a case, Gromov-Hausdorff topology is one of the most natural notions to work on for the convergence of manifolds. Under the conditions of Theorems 1.3 and 1.4, we apply the level set method in [29,30,31] to show that the Gromov-Hausdorff limit of the blow down metrics based at any sequence p i → ∞ also splits isometrically. Very recently, Bamler [2,3,4] has developed new compactness and partial regularity theories of Ricci flow under relatively weak conditions. Bamler-Chow-Deng-Ma-Zhang [5] also classified the tangent flows at infinity (see [3] for the definition) of 4 dimensional steady soliton singularity models.
Let (N, h) be a complete Riemannian manifold and L the cylinder R × N with product metric g L = dt 2 + h. For any λ ∈ R, the translation ρ λ : L −→ L is given by ρ λ (s, ω) := (s + λ, ω). A Riemannian manifold (M, g) is said to be smoothly asymptotic to the cylinder L if there exist a compact set K of M and a diffeomorphism Φ : [55]). The asymptotic convergence is at exponential rate if in addition for any integer k ≥ 0, there is a positive constant C k such that for all s > 0, (10) sup In this case, (M, g) is smoothly asymptotic to the flat cylinder R × T n−1 ∼ at exponential rate, where T n−1 ∼ is diffeomorphic to the quotient of torus in Corollary 1.6(a).
Since we assume Ric ≥ 0 outside a compact subset when n ≥ 5, so in this case, the result in Corollary 1.8(a) also follows from the splitting theorem by Cheeger-Colding [18]. Instead, we shall adopt the level set approach by Deng-Zhu [29,30,31] which works for all n ≥ 4 and provides more geometric information of the metric space (Y, d Y ) (see also Proposition 6.1). In particular, it can be seen from the proof that (Y, d Y ) in Corollary 1.8(a) is the Gromov Hausdorff limit of a sequence of level sets as in Corollary 1.6(b) with uniformly bounded curvature and diameter after scaling. In general, due to volume collapsing, the limit R × Y in Corollary 1.8(a) can be of lower dimension compared to M . On the positively curved Cao steady Kähler soliton on C 2 [10], the corresponding metric space (Y, d Y ) is CP 1 endorsed with a scalar multiple of the Fubini-Study metric (see [28]).
Steady Ricci solitons with fast curvature decay were extensively studied in [13,15,27,29,33,52]. As an application of Theorems 1.3 and 1.4, by exploiting the real analyticity of the soliton metric, we prove some classification results on steady gradient Ricci soliton with fast curvature decay under milder conditions. Here, we do not require any global non-negative curvature condition and uniform smallness of the quantity rR at infinity. Further assume that the following conditions are satisfied, Then M is holomorphically isometric to a quotient of Σ × C m−1 and the curvature Rm decays exponentially in r, where Σ is the Hamilton Cigar soliton.
Theorem 1.10. Let (M n , g, f ) be a complete noncompact and nonflat steady gradient Ricci soliton with nonnegative sectional curvature outside a compact subset of M and linear scalar curvature decay, that is, R ≤ C(r + 1) −1 . If in addition that then M is isometric to a quotient of Σ × R n−2 , where Σ is the Hamilton Cigar soliton.
Remark 1.11. If we further assume sectional curvature is nonnegative everywhere on M , then Theorem 1.10 also follows from [29] (see also Theorem 1.2 and [15,52]).
We wrap up the introduction by looking at the analogous dichotomy in other types of Ricci solitons. For complete noncompact and nonflat gradient Ricci shrinker, Munteanu-Wang [54] showed that If |Ric| is sufficiently small at infinity, then the shrinker is smoothly asymptotic to a cone and hence |Rm| ∼ r −2 at infinity by a result of Chow-Lu-Yang [24] (see also [46]). In real dimension 4, it remains an open problem whether shrinker with bounded scalar curvature R must have either R ≥ c for some positive constant c or R → 0 at infinity (this holds in the Kähler case [55]). The dichotomy will be extremely useful toward the classification of 4 dimensional shrinker in view of different studies on the asymptotic geometry of the solitons (see [46,54,55] and ref. therein).
Before we move on, let us recall the notion of asymptotically conical expanding soliton [35]. Let X be a smooth n − 1 dimensional closed manifold with Riemannian metric g X , C(X) is defined to be the cone over X, i.e. {(t, ω) : t > 0, ω ∈ X}. g C and ∇ C denote the metric dt 2 + t 2 g X on C(X) and its Levi-Civita connection respectively. For any positive constant S, Just like the steady case, there are two generic curvature decays at infinity for nonflat asymptotically conical expander [35]: The curvature decays (15) and (16) can be found in the Bryant expanding soliton and the Kähler expander constructed by Feldman-Ilmanen-Knopf [37] respectively. It will be interesting to see whether or not (15) and (16) are the only possible curvature decays of expanding soliton. The upper bound in (15) is always satisfied on conical expander. If lim x→∞ r 2 |Rm| = 0, then the upper and lower estimates of Rm in (16) are due to Deruelle [35] and the first named author [16] respectively. Dichotomy result in the expanding case similar to Theorems 1.3 and 1.4 seems to be quite promising. However using the existence and compactness results for Ricci expander by Deruelle [34,35], one can prove that the above dichotomy fails in 3 dimensional expanding case: where v = n 2 − f and lim x→∞ 4r −2 v = 1.
Though Theorem 1.13 is not explicitly stated in [34,35], it is essentially due to Deruelle and is a direct consequence of the results in [34,35]. We shall include its proof in the appendix for the sake of completeness.
The paper is organized as follows. In Section 2, we include the basic preliminaries of steady soliton. In Section 3, we show the exponential curvature decay and prove Theorems 1.3 and 1.4. Sections 4 and 5 will then be devoted to the proofs of Theorems 1.9 and 1.10. In Section 6, we will prove Corollary 1.8. Finally, the proof of Theorem 1.13 will be presented in the appendix.
Acknowledgement: The authors are indebted to Jiaping Wang for helpful discussions and suggestions. The authors would also like to thank Bennett Chow, Zilu Ma, Ovidiu Munteanu and Yongjia Zhang for valuable comments on this work. The first named author was partially supported by an AMS-Simons Travel Grant and would like to express his gratitude to Lei Ni for constant encouragement.

Preliminaries
Let (M, g) be an n dimensional connected smooth Riemannian manifold and f be a smooth function on M . ∇ is the Levi Civita connection of g. Rm, Ric and R denote the Riemann, Ricci and scalar curvatures respectively. (M, g, f ) is said to be a gradient steady Ricci soliton with the potential function f if (18) Ric The steady soliton is complete if (M, g) is a complete Riemannian manifold. (M, g, f ) is a gradient steady Kähler Ricci soliton if M satisfies (18) and is a complex manifold with complex structure J and g is a Kähler metric compatible with J on M . Ricci soliton is a self similar solution to the Ricci flow. Given a complete steady gradient Ricci soliton, we consider g(t) := ϕ * t g, where t ∈ R and ϕ t is the flow of ∇f with ϕ 0 = id. Then g(t) is an ancient solution to the Ricci flow: ∂g(t) ∂t = −2Ric(g(t)), It was shown by Chen [20] that any complete ancient solution to the Ricci flow must have nonnegative scalar curvature (see also [61]). Using the strong maximum principle, we see that any complete steady gradient Ricci soliton must have positive scalar curvature R > 0 unless it is Ricci flat. It is also known that compact steady soliton must be Ricci flat [25]. Upon scaling the metric by a constant if necessary, Hamilton [42] showed that for a complete steady gradient Ricci soliton, (20) |∇f | 2 + R = 1 on M, we adopt the above scaling convention throughout this article. By virtue of (18) and the Ricci identity, for any vectors X, Y and Z Hereinafter, we fix a point p 0 in M , then for any x in M , r, r(x) and d(x, p 0 ) will be used interchangeably and refer to the distance between x and p 0 . Then, for any smooth function ω, we define the weighted Laplacian w.r.t. ω by ∆ ω := ∆ − ∇ω · ∇, where ∆ is Beltrami Laplacian operator. The following identities are well known for steady gradient Ricci soliton (see [25]) where by adding a constant to f if necessary, we may assume that v = −f ≥ 10 on M . Furthermore by [14,Lemma 2], f and v must be distance-like and satisfy (26) lim Since we assume that the scalar curvature R −→ 0 as x → ∞, lim x→∞ |∇f | 2 = lim x→∞ 1−R = 1 and the level sets Σ τ := {−f = τ } are smooth compact hypersurfaces for all large τ . Moreover, Σ τ are connected by a result of Munteanu-Wang [53].

Curvature decay
The goal of this section is to prove the following key proposition Proposition 3.1. Let (M n , g, f ) be a complete noncompact non Ricci flat steady gradient Ricci soliton of real dimension n with proper potential function. Further suppose that the following conditions are satisfied: We will make use of the diameter and curvature estimates by Deng-Zhu [29,31]. By viewing the soliton as a solution to the Ricci flow and our assumption |Rm| ≤ C(r + 1) −1 implies, by the Shi's estimate [23], that for any nonnegative integer k, there exists positive constant C k such that on M . Using (28), we may argue in the same way as in [29,31] to get the following lemma With the diameter estimate on the level sets, we may proceed as in [29] to show the following lemma: Let (M n , g, f ) be a complete noncompact steady gradient Ricci soliton of real dimension n with proper potential function f . Further suppose that the curvature tensor Rm decays at least linearly i.e.
If in addition, r(p i )R(p i ) → 0 for a sequence of points p i → ∞, then there exists a positive constant ε(n, A 1 ) such that Proof. The argument is due to Deng-Zhu [29] and we include the proof for the sake of completeness. Let r i = r(p i ) and g i = r −1 i g. We consider the Ricci flow associated to the steady soliton, i.e. g(t) := φ * t g, φ t is the flow of the vector field ∇f with φ 0 = id and t ∈ R. For any z Let g i (t) := 1 r i g(r i t), then g i (0) = g i and by (30) Thank to the Shi's estimates (28), for all positive integer k We are about to take subsequential limit of g i (t). Let η i : (R n , g e ) −→ (T p i M, g i (p i )) be a linear isometry with η(0) = 0. By the Rauch Comparison theorem, the map F i := exp is the exponential map at p i w.r.t. g i (0). By the Hamilton compactness theorem, (34) and (35) (see also [43], [25], [29,Lemma 4.4] and [16,Theorem 12]), there exist positive constant δ = δ(n, A 1 ) < min 1 2 , π √ 2A 1 and a subsequence i l such that as l → ∞ where B δ (0) := {x ∈ R n : |x| < δ} and g ∞ (t) is a solution to the Ricci flow. Note that δ is independent on the subsequence taken. F * i l g i l (t) has nonnegative scalar curvature, so does g ∞ (t). Furthermore by the assumption and smooth convergence Due to the strong minimum principle, R g∞(t) ≡ 0 on B δ (0) × (− 1 10 , 0]. For the above δ, we want to show that the original sequence p i satisfies (38) lim Suppose on the contrary, by passing to a subsequence if necessary, we may assume there is constant ǫ 0 > 0 such that for all i By the locally uniform convergence (36) and distance estimate (32) which is impossible. This completes the proof of the lemma.
Using Lemma 3.3, Deng-Zhu [29] can show a better convergence result. sup Proof. The proof is due to Deng-Zhu [29] and we provide a sketch of the argument for reader's convenience. We argue by contradiction, by passing to subsequence if necessary, there exists constant ǫ 0 > 0 such that for all i, we can find q i ∈ Σ −f (p i ) such that Let γ i be a normalized minimizing intrinsic geodesic joining p i to q i on Σ −f (p i ) with respect to induced metric g i = r(p i ) −1 g, l(γ i ) the length of γ i with respect to g i .
for some unique nonnegative integer N i , where ε is the constant as in Lemma 3.3. Moreover by (26), for all large i and a ∈ Σ −f (p i ) In view of Lemma 3.2, we have the following upper bound for N i Hence by taking further subsequence, we may assume N i ≡ N 1 for all i. For each 0 ≤ j ≤ N 1 , we define the sequence p j i := γ i jǫ 8 . Using (43), We see that p 0 j = p i and the distance function on M satisfies and similarly By (45) and Similarly by (46) and repeating the same procedure finitely many times, we conclude that which is absurd. Result follows.
To apply the parabolic maximum principle as in [8,55], we need the evolution equation of vR along the level set flow: where v := −f and ∆ Στ denotes the intrinsic Laplacian of the level set Σ τ := {−f = τ } with respect to the induced metric by g.
Proof. By (20), be an orthonormal frame near infinity such that e n = ∇f |∇f | . It is known that the Laplace operators on M and Σ τ are related by the following formula: for any smooth function ω, (49) ∆ω = ∇ 2 ω(e n , e n ) + H τ e n ω + ∆ Στ ω.
where ∆ and ∆ Στ are the Laplacian operators on M and Σ τ respectively, H τ is the mean curvature of Σ τ . From the soliton equation, we see that e i , ∇ e i e n = −Ric(e i , e i )|∇f | −1 for 1 ≤ i ≤ n − 1 and H τ = n−1 i=1 ∇ e i e n , e i . Hence We substitute ω = vR in (50) and estimate the terms on the R.H.S. of (50) one by one. Using (28), (26) and For the Hessian term in the normal direction in (50), direct calculation and (51) yield By (24) ∇R = 2Ric(∇f ), the second Bianchi identity and Shi's estimate (28), Hence by (22) and (25), there exists a constant C 0 > 0 such that near infinity.
As an intermediate step toward the estimate on the curvature tensor |Rm|, we use the maximum principle to establish the exponential decay of the norm of Ricci tensor |Ric|. Lemma 3.6. Under the same assumptions as Proposition 3.1, the Ricci curvature decays exponentially near infinity, (56) |Ric| ≤ Ce −r .
Proof. For all small ε > 0, we can find a large a 1 > 1 + a 0 such that where a 0 , C 0 and A 0 are the constants in Lemma 3.5 and Proposition 3.1 respectively. Claim: For any y in the set of {v ≥ a 1 }, We first assume the claim and prove the lemma. By (26) (see also [14,Lemma 2]), lim x→∞ r −1 v = 1 and hence from (58) lim x→∞ rR = 0. Thanks to the assumption |Ric| ≤ A 0 R, We may then apply Proposition 1 and Lemma 3 in [14] to conclude that Now it remains to justify the claim, i.e. (58). Suppose the claim does not hold. Then there exists y 0 in the set of {v ≥ a 1 } such that v(y 0 )R(y 0 ) > ε. By lim inf x→∞ rR = 0 and Lemma 3.4, we can find sequences of p i → ∞ with τ i := v(p i ) > v(y 0 ) and ε i := sup Στ i vR → 0 as i → ∞. Moreover for all large i, ε i < ε. Therefore, we can always choose a z 0 ∈ {τ i ≥ v ≥ a 1 } with largest possible v(z 0 ) such that v(z 0 )R(z 0 ) = ε. By the choices of ε i and z 0 , v(z 0 ) < τ i and vR < ε on {τ i ≥ v > v(z 0 )}. We may invoke the parabolic maximum principle to conclude that at z 0 (61) ∇f, ∇(vR) ≥ 0; (62) ∆ Σ v(z 0 ) (vR) ≤ 0.
By |Ric| ≤ A 0 R and Lemma 3.5, From the above inequality, we have two possible cases: which is impossible for all small ε. We also used v(z 0 ) ≥ a 1 and (57) in the last inequality. Case 2: It follows from (57) that We again arrive at a contradiction. This justifies our claim and thus completes the proof. As pointed out by him in [35], the same argument also works for steady gradient Ricci soliton. Hence we apply his result with T = Ric and λ = 0 to get Let ψ s be the flow of the vector field − ∇f |∇f | 2 with ψ 0 = id. For all q near infinity, ψ s (q) is well defined for all s ≥ 0 and Hence ψ s (q) → ∞ as s → ∞. It follows from the Ricci identity and the soliton equation (see also (21)) that R kl,i − R ki,l = R ilkj f j .
By the second Bianchi identity and (68), we differentiate the quantity |Rm| 2 along the flow We may integrate the above inequality and get (71) ln 1 + |Rm| 2 (ψ s (q)) 1 + |Rm| 2 (q) 14 Letting s → ∞, together with the fact that lim x→∞ |Rm| = 0, one has We also used f ≤ −10 on M in the last inequality. Thank to (26) and (72), we see that lim x→∞ r|Rm| = 0. We then apply [14,Theorem 2] (see also [52]) and conclude that Proof of Theorem 1.3: By our assumption R → 0 as x → ∞. It then follows from [14] that • If lim inf x→∞ rR > 0, then R ∼ r −1 near infinity and estimate (5) on Rm is a consequence of (74). • If lim inf x→∞ rR = 0, we apply Proposition 3.1 to see that By the lower bound for R established by Chow-Lu-Yang [24] (see also [52]), |Rm| is bounded below by a constant multiple of e −r .
Proof of Theorem 1.4: Since Ric ≥ 0 outside a compact subset, there exists a large positive constant A 0 such that Using |Rm| → 0 as x → ∞ and an estimate on the potential function f in [15, Proposition 1] (see also [22]), we have lim x→∞ f = −∞. Theorem 1.4 then follows similarly from Proposition 3.1 as in the proof of Theorem 1.3.

Proof of Theorem 1.9
We first investigate the topology of the level sets of f and show that they are diffeomorphic to a quotient of torus at infinity if the curvature decays sufficiently fast. • |Ric| ≤ A 0 R on M for some constant A 0 > 0; • |Rm| ≤ A 1 (r + 1) −1 on M for some constant A 1 > 0; • lim inf r→∞ rR = 0. Then for all sufficiently large τ , the level sets Σ τ := {x : −f (x) = τ } are diffeomorphic to a finite quotient of torus.
Proof. By Proposition 3.1 and a volume estimate by Munteanu-Sesum [51], we see that |Rm| is integrable, i.e. in L 1 (M ). Thanks to the properness of f and a result of Munteanu-Wang [53], M is connected at infinity and hence the level sets {−f = τ } are connected smooth compact hypersurfaces in M for all large τ . Moreover, there exists a large τ 0 such that ∇f = 0 on {−f ≥ τ 0 } and Σ τ = {−f = τ } are diffeomorphic to each other for all τ ≥ τ 0 . The same argument used by Deruelle [33,Proposition 2.3] shows that the level sets are diffeomorphic to a compact flat manifold. Indeed. if ψ s is the flow of − ∇f |∇f | 2 with ψ 0 = id. Then ψ s : Σ τ 0 −→ Σ τ 0 +s are diffeomorphisms for all s ≥ 0 and when restricted on the tangent space of the level set T Σ τ 0 , the pull back metric satisfies Proposition 3.1 was used in the last equation. Hence ψ * s g are uniformly equivalent to each other for all s ≥ 0 and we can find a constant C > 0 such that for all τ ≥ τ 0 Similar argument by Deruelle [33,Lemma 2.5] shows that for all nonnegative integer k, there exists C k independent of τ such that (79) |∇ k Στ Rm Στ | ≤ C k , where Rm Στ and ∇ Στ denote the curvature tensor and the Riemannian connection of Σ τ with respect to the induced metric by g. It follows from lim τ →∞ sup Στ |Rm Στ | = 0 and the Hamilton Compactness Theorem [43] that for any sequence τ i → ∞, by passing to subsequence if necessary, Σ τ i converges smoothly to a compact flat manifold as i → ∞. Hence, for large τ , Σ τ is diffeomorphic to a finite quotient of the torus T n−1 by the Bieberbach's Theorem (see [33] and reference therein).
With the topological restriction on the level sets in Proposition 4.1, we will prove that the level sets Σ τ are flat with respect to the induced metric for all large τ . The key observation is that torus does not admit any nontrivial metric with nonnegative scalar curvature. This approach was used by Deruelle [33] to study steady soliton with integrable curvature. Proof. By Remark 1.5, |Rm| ≤ cR for some positive constant c > 0 (when m = 2, the estimate follows from [14], where m is the complex dimension). It can be seen from [15,Proposition 1] (see also [22]) that f is proper. We may then invoke Proposition 4.1 to see that the level sets Σ τ := {x : −f (x) = τ } are diffeomorphic to a quotient of torus.
We are going to show that they have nonnegative scalar curvature. Using the properness of f , we may choose larger τ such that Ric ≥ 0 on {x : −f (x) ≥ τ }. The second fundamental form of Σ τ (w.r.t. the normal − ∇f |∇f | ) is given by − ∇ 2 f |∇f | = Ric |∇f | ≥ 0. Let 0 ≤ µ 1 ≤ µ 2 ≤ · · · ≤ µ 2m−1 be the eigenvalues of the second fundamental form of Σ τ . Then we must have where H Στ and A Στ refer to the mean curvature and the second fundamental form of Σ τ respectively. By the Gauss equation, the intrinsic scalar curvature of Σ τ , denoted by R Στ satisfies we also used the Kählerity of ambient metric g in the last equality. Thus, the metric on Σ τ is of nonnegative scalar curvature and hence is flat since there exists no nonflat metric on torus with nonnegative scalar curvature (See [41,60]). Now, we are in a position to prove Theorem 1.9.
Proof of Theorem 1.9: By Lemma 4.2, we have Σ τ are flat for all large τ . Hence equalities hold in (81) and we have the following identities outside a compact subset of M R|∇f | 2 = 2Ric(∇f, ∇f ) (82) Using a result of [45] (see also [36,44]): Ricci solitons are real-analytic. Hence, the soliton metric g ij is real analytic in its geodesic normal coordinates. Moreover f satisfies (84) ∆f = −R and by the elliptic regularity theory (see [39, p.110] and ref. therein), f is also real analytic in the geodesic normal coordinates. Thanks to the analytic continuation, we see that (82) and (83) indeed hold globally on M . Since the set of critical points {|∇f | = 0} is nowhere dense in M , it follows from (82) and (83) that Ric ≥ 0 on M and the kernel of Ric is a smooth subbundle of the tangent bundle of M with real rank 2m − 2. By the Kählerity of g, Ric only has two distinct eigenvalues, namely 0 with real multiplicity 2m − 2 and R 2 with real multiplicity 2. Moreover, ∇f and J∇f span the eigenspace of the eigenvalue R 2 wherever ∇f = 0. To prove the splitting of M , we proceed as in [15] to show that the kernel of Ric is invariant under parallel translation. Let E be the kernel of Ric, it is a smooth subbundle of tangent bundle of real rank 2m − 2 (see also [52]). Suppose at p, ∇f = 0, by the orthogonal decomposition, the tangent space at p can be splitted into T p M = E p ⊕ ⊥ span{∇f, J∇f }. Let X be a smooth section of E defined locally near p and Y be any smooth vector field defined near p, then JX is also a smooth section of E. At p Similarly, ∇ Y JX ⊥ ∇f , thus ∇ Y X(p) is in E p . If ∇f = 0 at p, by the real analyticity of g (see [36,44,45]), {∇f = 0} = {R = 1} has no interior point in M , we may find a sequence p k → p as k → ∞ with ∇f (p k ) = 0, Hence, we conclude that E is invariant under parallel translation. By the De Rham Splitting Theorem [47,48], the universal covering space of M splits isometrically and holomorphically as M 1 × M 2 , where M 1 , M 2 are complex m − 1 and 1 dimensional Kähler manifolds respectively. Moreover, the tangent bundle of M 1 is given by the kernel of Ric and the tangent bundle of M 2 coincides with the nonzero eigenspace of Ric. From this we conclude that M 1 is Ricci flat and M 2 is nonflat. M also induces a steady Kähler gradient soliton structure on M 2 and by the classification of complex 1 dimensional nontrivial complete steady Kähler gradient Ricci soliton [23], M 2 is holomorphically isometric to the Cigar soliton Σ. When m = 2, M 1 is flat. If m ≥ 3, since |Rm| ≤ cR (see (9)) also holds on M 1 × Σ and M 1 is Ricci flat, for any (a, b) ∈ M 1 × Σ, we used the fact that the curvature of Σ decays at infinity. Hence M 1 is also flat for m ≥ 3. The exponential curvature decay is a consequence of the conditions lim inf x→∞ rR = 0, Theorems 1.3 and 1.4. This completes the proof of the theorem.

Proof of Theorem 1.10
We begin with an elementary lemma on the regularity of the component functions of a parallel vector field on real analytic manifolds.
Lemma 5.1. Let (M n , g) be a complete Riemannian manifold such that in any geodesic normal coordinates, the corresponding metric coefficients g ij are real analytic functions. Suppose that γ : [a, b] −→ M is a geodesic and V (t) is a parallel vector field along γ(t). For any t 0 ∈ (a, b) and {x i } a geodesic normal coordinate centered at γ(t 0 ), the component functions V i (t) of V are real analytic functions in t.
Proof. Let t 0 and {x i } be the number and geodesic coordinate as in the statement of the lemma. In the local coordinate, γ(t) = (t − t 0 )a for some a ∈ R n , V (t) = V i (t) ∂ ∂x i (γ(t)) and DV dt = 0 can be rewritten as where Γ i jk are the Christoffel symbols. Since g ij (x) are real analytic, Γ i jk (x) are also real analytic in x and hence Γ i jk ((t − t 0 )a) are real analytic in t. By [21,Theorem 1.3], V i (t), as the solution to the above ODE, is real analytic on (t 0 − δ, t 0 + δ) for some δ > 0.
With the above preparations, we shall prove Theorem 1.10.
Proof of Theorem 1.10: Since the sectional curvature of M is nonnegative near infinity and R → 0 as x → ∞, f is proper ( [15,22]) and |Ric| ≤ A 0 R on M for some positive constant A 0 . Using Proposition 4.1, the level sets Σ τ = {−f = τ } are diffeomorphic to a quotient of torus T n−1 for all sufficiently large τ . If M has nonnegative sectional curvature at p, then for all unit tangent vector v ∈ T p M , Therefore, we can apply the Gauss equation as in (81) to see that the intrinsic scalar curvature R Στ is nonnegative for all large τ . Since it is well-known that on T n−1 , n ≥ 2, any metric with nonnegative scalar curvature is flat (See [41,60]). Hence, the induced metric on Σ τ is flat and the equality in (80) holds and It is thanks to the above equation and (86) that ∇f is an eigenvector of Ric and Ric(∇f, v) = 0 for all v ∈ T Σ τ . Hence where a = ∇f |∇f | . We also used (87) in the last equality. It can be seen from (88) and an argument by Munteanu-Sung-Wang [52,Proposition 5.4] that wherever M has nonnegative sectional curvature, Ric has two distinct eigenvalues, 0 with multiplicity n − 2 and R/2 with multiplicity 2.

Gromov-Hausdorff Limit at infinity
In this section, we prove the following proposition which implies Corollary 1.8. Moreover, Ric ≥ 0 near infinity is not needed in the proposition.
We will separate the argument into two cases, namely linear and exponential curvature decays.
Case (a): Linear curvature decay We shall apply the level set method by Deng-Zhu [29,30,31] to show the convergence. By the Gauss equation, (5) or (7) and (96), we see that By Gromov Compactness theorem [9, Theorem 10.7.2] (see also [59]), after passing to a subsequence, converges in pointed Gromov-Hausdorff topology as i → ∞, where (Y, d Y , p ∞ ) is a compact Alexandrov space. Next we consider a type of sets introduced by Deng-Zhu [29,30,31], for any p ∈ M and k > 0 Moreover, they [30,Lemma 3.1] showed that for all k > 0, there is a large I such that for all i ≥ I Let ψ s be the flow of the vector field − ∇gf |∇gf | 2 g with ψ 0 = id . Then for all large i, ψ s : ]. Hence we can define the following We compare the pull back metrics as in [31, Lemma 4.2]. When restricting on T Σ τ i , Similarly, By integrating the above differential inequalities, we have for all large i and s ∈ [−k, k]

Hence by
In view of (98), for any ε > 0, we can consider a sequence of Gromov Hausdorff Case (b): Exponential curvature decay Again by Lemma 3.2 and Hence we have the following convergence in Gromov-Hausdorff sense (without taking subsequence) where d 0 is the discrete metric on the singleton {0}.
To proceed, we define another type of sets similar to M p,k in (99), namely We first show an analog to (100): for any k > 0 Suppose on the contrary, we can find a point z ∈ B h i (p i , k) \ N p i ,k and a distance minimizing geodesic γ : [0, T ] −→ M with respect to h i joining p i to z. By restricting γ on a smaller interval if necessary, we may further assume that for all t, −f (γ(t)) ≤ τ i +kR −1/2 i , f (γ(0)) = f (p i ) = −τ i and f (γ(T )) = −τ i − kR −1/2 i . Hence by |∇ g f | g ≤ 1, which is impossible. Therefore, we must have (111). To show the pointed Gromov-Hausdorff convergence, for any ε > 0, we construct an ε isometry from F i : N p i ,k −→ [0, ∞) for all large i, Obviously, F i (p i ) = 0. As before, ψ s denote the flow of the vector field − ∇gf |∇gf | 2 g with ψ 0 = id. For any a, b ∈ N p i ,k satisfying −τ i ≥ f (a) ≥ f (b), let β := f (a) − f (b) ≥ 0, by the proof of (78) in Proposition 3.1, Using a similar argument as in the proof of (111), we also have Hence by (108), (112) and fixing k > 0, we may take i to be sufficiently large such that When f (a) > −τ i ≥ f (b), it follows from (96) that there is a positive constant c 0 such that for all large i and thus for all y, z ∈ {x : f (x) ≥ −τ i }, By (108) and (114), we see that It remains to verify that F i is almost surjective, i.e. [0, k − ε) ⊆ F i (B h i (p i , k)). From the construction of F i , we have for any s ∈ [0, k − ε), s = F i Σ τ i +s/ √ R i . Thanks to (112), for all large i (fixing k), Hence F i is an ε isometry and we get the pointed Gromov-Hausdorff convergence to the ray. For the smooth convergence to a cylinder, by the local Shi derivative estimates [35, Lemma 2.6], ∇R = 2Ric(∇f ) and (9), we have for all integer k ≥ 0, there is a positive constant C k such that (116) |∇ k Rm(g)| ≤ C k R ≤ C ′ k e −r on M. When restricted on Σ τ 0 , ψ s : Σ τ 0 −→ Σ τ 0 +s are diffeomorphisms for all s ≥ 0 and we denote the pull back metric on Σ τ 0 = {−f = τ 0 } by g s := ψ * s g. Since the second fundamental form of Σ τ 0 +s is − Ric |∇gf |g , we may apply (116) and the computation in (77) to conclude that (117) ∇ k gs ∂ ∂s g s gs ≤ C k e −τ 0 −s 24 for all s ≥ 0 and integer k ≥ 0. By [6, Proposition A.5], g s converges in C ∞ sense to a smooth metric g ∞ on Σ τ 0 as s → ∞. The limit g ∞ agrees with the subsequential limit in the proof of Proposition 4.1 and thus is flat. Moreover, C −1 g s ≤ g ∞ ≤ Cg s for all s ≥ 0. By the compactness of Σ τ 0 and the smooth convergence, for all k ≥ 0, ∇ k g∞ g s g∞ ≤ C ′ k for all s ≥ 0. One may then argue by induction as in [6,Lemma A.4] to see that Hence by integrating the above estimates, we have for all k ≥ 0, We define the asymptotic cylinder L for the steady soliton (M, g) as follows. Let L := R × Σ τ 0 with the product metric g L := ds 2 + g ∞ . Φ : (0, ∞) × Σ τ 0 −→ {−f > τ 0 } is the diffeomorphism given by Φ(s, ω) := ψ s (ω). It can be seen that (120) Φ * g = |∇ g f | −2 g ds 2 + g s . Then by (116), (119) and a direct (though tedious) induction argument, we have for any integers k, p ≥ 0, and vectors w i ∈ T Σ τ 0 with |w i | g∞ = 1, where C k,p is some positive constant. Estimate (121), together with the fact that ∇ g L ∂ ∂s ≡ 0 on L, implies the asymptotic convergence to (L, g L ) at exponential rate. This completes the proof of Proposition 6.1.