Ricci curvature integrals, local functionals, and the Ricci flow

Consider a Riemannian manifold $(M^{m}, g)$ whose volume is the same as the standard sphere $(S^{m}, g_{round})$. If $p>\frac{m}{2}$ and $\int_{M} \left\{ Rc-(m-1)g\right\}_{-}^{p} dv$ is sufficiently small, we show that the normalized Ricci flow initiated from $(M^{m}, g)$ will exist immortally and converge to the standard sphere. The choice of $p$ is optimal.


Introduction
The classical Myer's theorem states that if a closed Riemannian manifold (M m , g) satisfies Rc ≥ m − 1, (1.1a) |M | g = (m + 1)ω m+1 , (1.1b) where ω m is the volume of unit ball in the Euclidean space R m , then the inequality (1.1a) must be an equality, and the manifold (M m , g) must be isometric to (S m , g round ).It is a natural question to find a quantitative version of this rigidity theorem.If we replace (1.1) by the following conditions for some small number δ < δ 0 (m), we can still declare many properties of M .For example, by the deep work of Perelman [25], we know that such M must be homeomorphic to S m .The seminal work of Colding [10] [11] proved that (M, g) is Gromov-Hausdorff close to the standard sphere (S m , g round ).Furthermore, it is proved by the foundational work of Cheeger-Colding [9] that M is diffeomorphic to S m , and (M, g) is uniformly bi-Hölder equivalent to (S m , g round ).Their proof relies on the Reifenberg method.In [36], the second named author proved that the normalized Ricci flow initiated from (M, g) exists immortally and converges to a round metric g ∞ , and the identity map from (M, g) to (M, g ∞ ) is uniformly bi-Hölder.Therefore, [36] provides an alternative proof of Cheeger-Colding' result, via Ricci flow smoothing, which seems to be natural and was intensively studied (e.g.See [13] [33] [21] and the references therein.).
The results mentioned above are based on the point-wise Ricci lower bound.It is interesting to investigate whether this point-wise Ricci lower bound can be replaced by an integral Ricci lower bound.Actually, many important results based on integral Ricci lower bound were established by the work of Petersen-Wei [28] [29], D. Yang [41] [42] [43], S. Gallot [16], etc.The Ricci flow behavior with initial data satisfying Ricci or other curvature's integral pinching conditions was also studied by many people, e.g., see [39] [40] [6].Under an appropriate integral Ricci curvature condition (cf.Definition 2.1 for precise definitions), we have Theorem 1.1.where | • | means volume, ω m+1 is the volume of unit ball in R m+1 , and δ ∈ (0, δ 0 ).Then the normalized Ricci flow initiated from (M m , g) exists immortally and converges to a round metric g ∞ exponentially fast.The limit metric (M, g ∞ ) has constant sectional curvature 1.
(1.7) Also, it is not hard to see from the construction in [1] that the topology of M could be far away from S m .Therefore, the coefficient p > m 2 in Theorem 1.1 is optimal.It would be a possible generalization to replace {Rc − (m − 1)g} − in equation (1.7) by {Rc − (m − 1)g}.If such generalization holds, then Corollary 1.2 can be improved to a topological rigidity result in the similar spirit as in Margerin [24], Bour-Carron [3], and Chang-Gursky-Yang [7], etc.
We briefly discuss the proof of Theorem 1.1.The first thing that needed to be done is to obtain uniform existence time of the Ricci flow solution initiated from (M, g), say t ≥ 1.Then we shall show that the volume closeness at t = 0 implies the Gromov-Hausdorff closeness at t = 0, and consequently C 0 -closeness for all t ∈ [0.5,1].By standard regularity improvement, we then obtain C ∞ -closeness between (M, g (1)) and (S m , g round ).Then it is well-known (cf.[22] [2] [4]) that the normalized Ricci flow started from g (1) exists forever and converges to the standard sphere (S m , g round ).The strategy described above is the same as the one used in [36].However, there exist essential technical difficulties to be overcome.
The first difficulty is to obtain uniform existence time for the Ricci flow initiated from (M, g).In [36], this is achieved through the application of the improved pseudo-locality, whose key is the estimate of the local μ-functional.Note that in [36], the μ-functional can be obtained either through the blowup analysis or it can be derived through the isoperimetric constant estimate via the needledecomposition method for RCD space (cf.[5]).However, in the current setting, the Ricci curvature is not bounded from below point-wisely.Thus the RCD theory cannot be applied directly here.Fortunately, the blowup analysis method survives, as it only requires gradient estimates, volume comparison theorems and compactness properties, all of which are available.
The second difficulty is the distance distortion estimate.Note that it is a key ingredient for the distance distortion estimate to obtain the smallness of the space-time integral ˜|R|dvdt, which is the almost Einstein condition.In [36], this condition is obtained through applying maximum principle for the scalar curvature R, since the almost non-negativity of R is preserved under the Ricci flow.In the current paper, we only have an integral estimate of |Rc − | at initial time.Therefore, the maximum principle argument fails.We overcome this difficulty by delicately applying the curvature condition t|Rm|(x, t) ≤ , (1.8) where = (m) is a small number far less than 1.We construct new cutoff functions and develop an ODE system for rough volume ratio upper bound and ´|R − | p .Then it follows from ODE comparison that both of them are well-controlled.A further interpolation argument then implies the almost Einstein condition.
In conclusion, the first difficulty can be overcome if we are able to apply the pseudo-locality property in Theorem 1.2 of [36].Then it suffices to show a delicate μ-functional estimate under the initial metric, which is achieved by Theorem 1.3.Theorem 1.3.Let (M, g) be a closed Riemannian manifold of dimension m ≥ 3 and p > m 2 .For x 0 ∈ M and each pair of positive numbers (η, A), there exists a constant δ = δ(m, p, η, A) with the following properties. Suppose Note that (1.9) is a version of the "almost Euclidean" condition.(1.9b) means the volume ratio is very close to the Euclidean one, (1.9a) means that Ricci curvature is almost non-negative in the integral sense.The coefficient 1  2 ω m in (1.9a) is only for technical convenience and could be replaced by other small constant.The entropy estimate (1.11) assures us to apply the pseudo-locality theorem.In particular, we shall have the curvature estimate (1.8).Therefore, under the help of condition (1.8), we are able to overcome the second difficulty in light of Theorem 1.4.
By Theorem 1.4, we actually obtain the volume ratio upper bound and the L p -integral bound of scalar curvature along the flow, under appropriate initial conditions.If we have the almost Euclidean condition (1.9) at initial time, then conditions (1.12) and (1.13) in Theorem 1.4 are all satisfied automatically.Furthermore, the inequality (1.15) implies the local almost Einstein condition, via another volume comparison argument.Theorem 1.5.Suppose {(M m , g(t)), 0 ≤ t ≤ 1} is a Ricci flow solution, m ≥ 3 and p > m 2 .For x 0 ∈ M and each pair of positive numbers (ξ, A), there exists a constant δ = δ(m, p, ξ, A) with the following property.
If (1.9) is satisfied with respect to the initial metric g(0), then we have In light of the local almost Einstein condition (1.16), we are able to develop the distance distortion estimate.Thus the second difficulty is overcome.Consequently, we can apply the distance distortion estimate and C 0 -continuous dependence of initial data for the Ricci-Deturck flow, as done in [36], to obtain the continuous dependence of the metric with respect to the Gromov-Hausdorff topology.
For each small, there exists η = η(N , p, ) with the following properties.
Suppose (M m , g) is a Riemannian manifold satisfying where p > m 2 .Then the (normalized) Ricci flow initiated from (M, g) exists on [0, T ].Furthermore, there exists a family of diffeomorphisms In Theorem 1.3 of [36], the author proved the continuous dependence when Rc M > −(m − 1)A.In the current paper, this condition is replaced by the integral Therefore, Theorem 1.6 is an improvement of Theorem 1.3 in [36].
It is not hard to see that Theorem 1.1 follows from Theorem 1.6.In fact, if condition (1.3) is satisfied for sufficiently small δ, then it follows from Petersen-Sprouse [27] and Aubry [1] that (1.17) holds with (N m , h) = (S m , g round ).Thus we can apply Theorem 1.6 to obtain that the normalized Ricci flow started from (M, g) exists on [0, 1].Furthermore, the metric g (1) is very close to g round in C 5 -topology.In particular, the curvature operator of g (1) is nearby the standard one.Thus we can apply Huisken's theorem [22] to obtain the convergence of the normalized Ricci flow.The remaining argument is the same as that in [36].
Reviewing the proof of Theorem 1.1, it is clear that whenever the major two difficulties (almost Euclidean entropy estimate and almost Einstein condition) are overcome, most of the theorems in [36] can be naturally extended to the current situation.For example, when the metric is Gromov-Hausdorff close to a stable Einstein manifold, then the normalized Ricci flow has global existence and convergence.The details will be provided at the end of this paper in Section 6.We also remark that Theorem 1.3, Theorem 1.4 and Theorem 1.5 have their independent interests, rather than only being intermediate steps to obtain Theorem 1.1.
The structure of the paper is described as follows.In Section 2, we review some elementary materials needed in the paper.In Section 3, we settle the first difficulty by developing the local μ-functional estimate in Theorem 1.3.In Section 4, we overcome the second difficulty and obtain the almost Einstein condition, by proving Theorem 1.4 and Theorem 1.5.In Section 5, we follow the strategy of [36] to obtain the distance distortion estimate and we improve the rough distance distortion estimate to C ∞ -closeness, i.e., Theorem 1.6.Finally, in Section 6, we study the stability of the Ricci flow and prove Theorem 1.1.

Preliminaries
In this section, we fix notations and overview important results under the integral Ricci curvature condition.
Let (M m , g) be a closed Riemannian manifold of dimension m.Let p be a positive number for studying L p -norm of Ricci curvature.Without further explanation, we always assume by default that m ≥ 3, and p > m 2 .(2.1) We shall use |•| dv g to denote volume of domains in the Riemannian manifold (M, g), and the subscription g will be omitted if g is clear in the context.When no confusion is possible, we also use | • | to denote the area of hypersurfaces in a Riemannian manifold.In the situation we want to highlight the underlying metric to define the volume, we also use vol g (•) to denote volume.We use ω m to denote the volume of unit ball in R m .Therefore, the standard round sphere of sectional curvature 1 has volume (m + 1)ω m+1 .
The following notations are commonly used in the study of comparison geometry in terms of the integral Ricci curvature condition.Definition 2.1.For any x ∈ M , let ζ(x) be the smallest eigenvalue of the Ricci tensor Rc : T x M → T x M .Then we define where R = tr g Rc is the scalar curvature.
By definition, it is clear that Definition 2.2.For p, r > 0, we define For a constant λ, we can similarly define The subscription g will be omitted if there is no ambiguity.
It is clear that κ g (λ, p, r) = κ g (p, r) when λ = 0. Note that κ g (p, r) is scaling invariant.Namely, κ g (p, r) = κ a 2 g (p, ar) for each a > 0. The Sobolev constant estimate and the non-collapsing condition are closely related.The existence of uniform Sobolev constant naturally implies the noncollapsing condition (e.g.Proposition 2.4 of [38]).On the other hand, the noncollapsing condition also implies the estimate of isoperimetric constant and hence the Sobolev constant (e.g.Theorem 3 of [16] and Theorem 7.4 of [41]).If we consider the normalized volume, then non-collapsing always holds true formally and so does the Sobolev constant.The following estimate was part of Corollary 4.6 of [14].
Given z ∈ M , let r(y) = d(y, z) be the distance function and The classical Laplacian comparison states that, if the Ricci curvature Rc M of M satisfies Rc M ≥ 0, then Δr ≤ m−1 r , i.e. ι ≡ 0. In [28], this result is generalized to the case of integral Ricci bounds.
We also need to estimate the area of the sphere under the integral Ricci curvature condition. (2.24) (2.25) The local functionals of Perelman are defined as follows (cf.[36]).
Definition 2.9.For ϕ ∈ W 1,2 0 (Ω), ϕ ≥ 0 and ´Ω ϕ 2 dv = 1, as usual, we define the local entropy functionals (2.29) As we already know (cf.[36]), the estimates of local functionals are crucial for taming the behavior of Ricci flows.The following gradient estimate, obtained recently by Jie Wang and Youde Wang, is important for estimating the local functionals.
Note that equation (2.30) is the Euler-Lagrange equation of the local μ-functional.The gradient estimate (2.31) does not need the non-collapsing condition and has independent interest.
The following weak Harnack inequality, introduced by P. Li in [23], will be very useful in this paper.
Lemma 2.11 ([23,Lemma 11.2]).Let (M, g) be a complete Riemannian manifold of dimension m ≥ 3. Suppose that the geodesic ball B(x, r) satisfies B(x, r) ∩ ∂M = φ.Let u ≥ 0 be a function in the Sobolev space W 1,2 (B r ) and satisfy (2.32) Δu ≤ Au in the weak sense for some constant A ≥ 0 on B r .Suppose the following conditions hold true: (i) Normalized Sobolev inequality: (ii) Poincaré inequalities: where f B r = ffl B r f and C P > 0 is a uniform constant which does not depend on r.
(iii) Volume doubling property: where C V is a uniform constant which does not depend on r.
Then for θ > 0 sufficiently small, there exists a constant C depending only on θ, m, C S , C P , C V and Ar 2 + 1 such that Remark 2.12.In fact, after some technical improvements, (2.37) holds true for any 0 < θ < m m−2 , cf.Theorem 8.18 of [17] and step 2 in Theorem 4.15 of [20].Under the Ricci curvature condition κ(p, D) < (m, p) for some , we obtain (2.33) and (2.36) by Lemma 2.4 and Lemma 2.6 respectively.To obtain the Poincaré inequalities, we need the following result proved by A. Grigor'yan in [18] and L. Saloff-Coste in [31].(i) Poincaré inequality: There exists a uniform constant P 0 such that, for any ball B(x, r), (ii) Volume doubling property: There exists a uniform constant D 0 such that, for any ball B(x, r), 0 < r < D, (

iii) Parabolic Harnack inequality:
There exists a uniform constant H 0 such that, for any ball B(x, r), 0 < r < D, and for any smooth positive solution u to Δu with arbitrary real number s, such that (2.40) sup where As Theorem 5.5 of [14] asserts, by the parabolic gradient estimate obtained in [44] and a scaling argument, (iii) of Theorem 2.13 holds true provided κ(p, D) < (n, p) for some small and H 0 = H 0 (m, p, D), then we obtain the desired Poincaré inequalities in Lemma 2.11 under Ricci curvature bounds at once.
As a consequence, Lemma 2.11 holds true and we have the following estimate analogous to Proposition 2.2 of [34].
Proposition 2.14.Let (M, g) be a complete Riemannian manifold and 2).Then for any 0 < ρ ≤ 1 and θ small enough, there exists a constant (n, p) such that if κ(x, p, 8) < , there holds where θ is the same as in Lemma 2.11.
Fix 0 < ρ ≤ 1 and let g = ρ −2 g.By scaling, we have Let φ = ρ −2 ϕ.Then we have Note that (2.42) still holds true on (M, g) after scaling.Consequently, we obtain which yields that for some , then we obtain (2.41) at once.
The following generalization of Cheeger-Colding theory was achieved by Petersen-Wei [29].

Lemma 2.15 ([29, Theorem 1.3]
). Suppose a sequence of complete Riemannian m-manifolds (M i , g i ) converges to a Riemannian m-manifold (M, g) in the pointed Gromov-Hausdorff topology.Then we can find an (m, p) > 0 such that if for all the manifolds we have κ(p, D) ≤ and the points Consequently, a standard covering argument then implies

Estimate of local functionals
In this section, we shall develop a delicate lower bound of the local μ functional.Our proof is similar to the proof of Proposition 3.1 of [34] (see also Lemma 4.10 of [36]).Since there are extra difficulties caused by the Ricci integral condition, we shall modify and streamline the previous proof, and provide full details.Proposition 3.1.Let (M, g) be a closed Riemannian manifold of dimension m ≥ 3 and p > m 2 .For x 0 ∈ M and each pair of positive numbers (η, A), there exists a constant δ = δ(m, p, η, A) with the following properties: Proof.The proof follows similar strategy as that of Proposition 3.1 of [34].However, we shall encounter new technical difficulties caused for lack of point-wise Ricci lower bound.Furthermore, we now have more systematical notations and estimates of local functionals by the work of [35] and [36].We shall provide more details and intermediate steps than the proof in [34], to make the proof more streamlined.In the proof, if it is not mentioned otherwise, C by default denotes a constant depending only on m, p and A. As usual, the actual value of C may change from line to line.
By monotonicity of local functionals (e.g.See Proposition 2.1 of [35]), we may assume A ≥ 1.By scaling and redefining δ if necessary, it suffices to show (3.2) for τ = 1.Namely, it suffices to prove We shall prove (3.3) by a contradiction argument.For simplicity of notation, in this proof, we denote Therefore, if the Proposition was wrong, we should have no matter how small δ is in (3.1a) and (3.1b).Let ϕ be a minimizer of the functional μ (Ω, g, 1).It satisfies the normalization condition and the Euler-Lagrange equation on Ω: Note that ϕ is continuous on Ω and vanishes on ∂Ω.Therefore, ϕ can be regarded as a function on M by trivial extension.Equation (3.7) can be rewritten as The proof consists of several steps.
Step 1.There is a constant In view of Lemma 2.6, conditions (3.1a) and (3.1b) assure the uniform noncollapsing condition.Then we can apply the local version of Lemma 2.4 to obtain uniform Sobolev constant estimate, which in turn implies that μ is uniformly bounded from below by a constant depending on m and the Sobolev constant of Ω.Since μ (Ω, g, 1) ≤ 0 by (3.5), this lower bound of μ yields (3.10).Combining (3.10) with (3.6) and (3.7), we can apply standard Moser iteration argument to obtain (3.11).For further details, see the proof of Proposition 3.1 in [34].
Step 2. There holds that sup Since ϕ satisfies equation (3.9) in B(z, 0.1) ⊂ Ω = B(x 0 , A) and λ is uniformly bounded, we can apply the gradient estimate of Youde Wang and Jie Wang (cf. the local version of Theorem 2.10).By setting q = m, it is clear that (3.12) follows from (2.31) directly.
Note that (3.13) is the boundary Hölder estimate.We shall follow the argument in section 8.10 of [17] to achieve the proof.Let w be a point in ∂Ω such that d(z, ∂Ω) = d(z, w).For each r ∈ (0, 1), we define Then there holds where we used the fact that both M r and h r are uniformly bounded by (3.11).
Since δ is sufficiently small, we can apply Proposition 2.14 to obtain Note that h r ≥ 0 on B(w, 2r) ∩ Ω; Therefore (3.18) implies that for some constant H = H(m, p, θ) = H(m, p) sufficiently large.The above inequality can be rewritten as (3.20) and induction that (3.21) where we used the uniform boundedness of M 1 2 in terms of (3.11) and (3.15).Consequently, we have By choosing i such that ζ ∈ [2 −i , 2 −i+1 ) and setting Following the route to prove (3.12), we are ready to deduce (3.14) from the above inequality.Actually, since ϕ satisfies equation (3.9) in B(z, ζ) ⊂ Ω = B(x 0 , A) and λ is uniformly bounded, we can apply Theorem 2.10 again.Setting q = m in (2.31), it follows from (3.24) that On the other hand, Green's formula implies that (3.30) From Lemma 2.8, we have By Hölder's inequality and (2.13) in Lemma 2.5, it follows that ˆB(z,2A)\B(z,ζ) In view of the curvature condition (3.1a), we can apply Lemma 2.8 to obtain that for all ρ ≤ 2A.Thus the combination of (3.33) and (3.32) implies that (3.34) < +∞.Thus where ψ = ψ(δ|m, p, A).
Applying integration by parts again, we have ˆΩ We shall estimate the right hand side of (3.37) term by term.Firstly, it follows from (3.11) We move on to estimate II in (3.37).Note that On the other hand, we have Step 6. Fix ζ ∈ (0, 0.1).Suppose B(z

Direct calculation and the volume ratio estimate imply that
As z * is outside of Ω , integration by parts implies that ˆΩ In light of (3.11) and (3.27), it is clear that (3.45) follows directly from the above inequality.
Step 7. Fix ζ ∈ (0, 0.1).Suppose B(z Step 8.As δ → 0, the limit space (R m , x ∞ , g E ) admits a limit function ϕ ∞ , which is supported on B(x ∞ , A) and satisfies the normalization condition.Furthermore, ϕ ∞ satisfies the integration equation whenever z = y and z * is the symmetric point of z with respect to ∂B(x ∞ , A).

By continuity, it suffices to show (3.48) for all
where Plugging the proper forms of (3.8) and (3.9) into (3.56)implies that In light of (3.8) and (3.11), we know that λk is uniformly bounded.Due to (3.11), the term ϕ k log ϕ k is uniformly bounded.Therefore, it follows from (3.57) that where C = C (ζ, d(z, x ∞ ), m, p).Note that we have used the volume comparison in the last step.Since ζ ∈ (0, 0.1), we can combine the last terms and arrive at By volume convergence, the gradient estimate and the boundary C α -estimate of ϕ k , we can take limit of the above inequality and obtain Putting the corresponding formula of G ∞ (cf.(3.50) and (3.51)) into the above inequality, and letting ζ → 0, we obtain (3.48).
Step 9.The limit function ϕ ∞ is a strictly positive smooth function on B(x ∞ , A) and satisfies It is clear that (3.58) and (3.59) follow from (3.48) directly.By standard regularity theory of elliptic PDE, we know that ϕ ∞ is a smooth function on B(x ∞ , A).Consequently, the estimates (3.11), (3.12), (3.13) and (3.14) also hold true by ϕ ∞ .In short, we have the estimates As a limit of non-negative functions ϕ i , it is clear that ϕ ∞ ≥ 0. Since ϕ ∞ solves (3.58), its zero set is open (cf. the lemma on page 114 of [30]).Therefore, either ϕ ∞ ≡ 0 or ϕ ∞ > 0 everywhere on B(x ∞ , A).However, together with volume comparison and volume convergence, the estimates (3.62) and (3.63) guarantee that we can take limit of the normalization condition (3.6) to obtain (3.60).In particular, ϕ ∞ is not the zero function.Therefore, ϕ ∞ > 0 everywhere on B(x ∞ , A).
We move on to show (3.61).Fix an arbitrary > 0. By Sard's theorem, we can find an s ∈ (0, ) such that the level set {x|ϕ ∞ (x) = s} is a smooth hypersurface.

Then standard integration by parts implies
Plugging (3.58) into the above equations, we obtain Note that B(x ∞ , A) = {x|ϕ ∞ (x) > 0} by the strict positivity of ϕ ∞ on B(x ∞ , A).Since 0 < s < , we obtain (3.61) by letting → 0 in the above inequality.
Step 10.Derive the desired contradiction.
Applying the normalization condition and the Sobolev inequality, we arrive at  μ(Ω, g, τ ) − μ(Ω, g, σ) Plugging the particular value σ = e − 0 C S τ into the above inequality yields that Proof.Note that (3.72) follows from (3.71), in light of scaling invariant property of μ and the almost monotonicity of κ(p, r) and the volume ratio (cf.Lemma 2.6 and Lemma 2.7).Therefore, it suffices to prove (3.71).
In light of (3.72), we can apply Theorem 1.2 of [36] and derive Theorem 3.4.
Theorem 3.4.For each p > m 2 and small, there exists a number δ = δ(m, p, ) satisfying the following properties.
Suppose {(M m , g(t)), t ∈ [0, 1]} is a solution of Ricci flow.Suppose under the metric g(0) it holds that We close this section by the proof of Theorem 1.3.

Estimate of volume and scalar curvature integral
Suppose {(M m , g(t)), t ∈ [0, 1]} is a solution of Ricci flow with an initial data (M, g(0)).In Section 3, we proved that if (M, g(0)) satisfies the integral Ricci curvature condition (3.75a) and the volume lower bound condition (3.75b), then we have the pseudo-locality estimate (3.76).In this section, we shall provide local estimates of volume and L p -norm of scalar curvature along the Ricci flow.In [36], similar estimates were obtained under the conditions |Rc| ≤ t , and R g(0) ≥ − , with the help of localized maximum principle (cf.Theorem 5.1 of [36]).In this section, we use the conditions which can be deduced from (3.75a) and (3.75b) naturally.
The local estimates of volume and L p -norm of scalar curvature are key new ingredients of this paper.Before we delve into the details of proving these estimates, let us detour for elementary technical preparations.We shall prove them step by step.
Since d dt log dv t = 1 2 tr g ġ and {(M m , g(t)), 0 ≤ t ≤ 1} is a smooth flow, it is clear that Consequently, by continuity of ηf p , there holds that which is nothing but (4.4).

Lemma 4.2. Suppose
Proof.We first assume Ω has smooth boundary.Recall that Ω 0 = {x ∈ Ω : f (x) ≥ 0}.If ∂Ω 0 is smooth, then equation (4.9) holds obviously.Otherwise, we shall show (4.9) by approximation.Suppose h is a regular value of f , then ∂Ω h is smooth.Integration by parts yields that (4.10) Since f = h on ∂Ω h , there holds (4.11) Consequently we have (4.12) In Ω 0 \Ω h , since 0 ≤ f ≤ h , there holds (4.13) By Sard's theorem, we can choose regular values h → 0. It follows from (4.12) and (4.13) that ´Ω0 ηΔf p dv + ´Ω0 ∇η, ∇f p dv exists and which is equivalent to (4.9).Now we consider the general situation.Note that Ω can be exhausted by a sequence of domains Ωk , which have smooth boundaries (cf. the proof of Lemma 2.6 of [35]).Namely, we have is an exhaustion of Ω 0 , we can take limit of both sides of the above equation to obtain (4.9).The proof of Lemma 4.2 is complete.Now we are ready to prove the main theorem of this section.
Proof.The proof consists of five steps.We shall first construct a cutoff function η with proper properties in Step 1. Then we calculate the evolution of ´M ηdv and ´M η Furthermore, we can choose an η ∈ C 3 such that Abusing notation, we define Step 2. It holds that Direct calculation shows that Since η ≤ 0 and α ∈ (0, 1), it follows from (4.15) and (4.20) that Plugging (4.24) into (4.23) and noting that −R ≤ R − , we obtain Now we divide the support of η into two parts: It follows from (4.18) that η = −η 4p+3 4p+4 on M 2 .Then (4.25) can be written as Hölder's inequality and Young's inequality imply that which is nothing but (4.22).
Step 3. It holds that Along the Ricci flow, we have Putting them into (4.31) and noting that p > m 2 , we obtain Plugging the above equation into (4.32)yields that Since (−R) p = 0 and ∇(−R) p = 0 on the boundary of M − (t), it follows from Lemma 4.2 and elementary inequality that Hölder's inequality yields that ˆM− η . Consequently, we have Note that 2p m − 1 > 0, we can apply Young's inequality to obtain (4.37) It follows from the combination of (4.36) and (4.37) that which directly implies (4.30).
Step 4. For sufficiently large L = L(p, m), the values of ´M ηdv and ´M η In light of (4.14a) and (4.14b), it is clear that h 1 (0) ≥ 0 and h 2 (0) ≥ 0. In particular, we have In light of (4.40), the above inequalities mean that ˆM η Step 5. Solve the ODE and finish the proof of (4.16a) and (4.16b).

.56)
Then the Ricci flow initiated from (M, g) exists on [0, T ].Furthermore, there exists a family of diffeomorphisms {Φ Sketch of proof.Since the proof is very similar to that in section 6 of [36], we shall only sketch the proof and highlight the key points.
In [36], the second named author proved estimate (5.7) when initial Ricci curvature has a uniform point-wise lower bound.The key idea there is to construct diffeomorphisms between locally almost flat manifolds with rough Gromov-Hausdorff approximations (cf.Lemma 6.2 and 6.3 of [36]).In the current situation, the point-wise Ricci lower bound condition Rc M > −(m − 1)A is replaced by the weaker integral Ricci curvature condition κ(p, ) < 2− m p .The almost flatness and rough Gromov-Hausdorff approximation conditions are realized by the curvatureinjectivity-radius estimate in Theorem 3.4.
By the smoothness and compactness of (N, h), for each small δ, there exists a small r 1 (δ, N, h) satisfying Choosing η = η(h, ) sufficiently small.Since (5.6) holds, the volume continuity (cf.Lemma 2.15) and volume comparison (cf.Lemma 2.3 of [28]) guarantees that Define r 2 := min{ , r 1 8 }.Then we have Thus we can choose an r 3 (δ, ) which satisfies κ(p, r 3 ) ≤ δ 2 .By scaling, on the manifold (M, g δ 2 r 2 3 ), (3.75a) and (3.75b) holds.Thus we can apply the pseudolocality property in Theorem 3.4 and obtain the distance distortion estimate in Theorem 5.3.Note that this estimate is the key point (cf.Lemma 6.2 in [36]) for the construction of diffeomorphism Φ : N → M .Furthermore, Φ * (g(ξ)) is very close to h in C 0 -topology, for a very small time ξ = ξ(N , ).Then we apply the Ricci-Deturck flow technique to obtain (5.7), following exactly the same steps as that in Theorem 6.1 of [36].|N | dv h(0) ≤ ψ(η| , h). (5.9) From the definition of κ(p, ), we know that So for any points x ∈ M , ´B(x, ) Rc p − dv ≤ 2ω m .By Vitali covering method, we can choose some disjoint balls {B(x i , ), i ∈ I} such that {B(x i , 3 ), i ∈ I} covers the manifold M .Then we have By (5.6) and Theorem 4.3, we know for ξ small enough, there holds Then we calculate As for the Ricci flow (N, h(t)), it is clear that Thus direct estimate of volume element yields that For each t ∈ (0, ξ), it follows from the combination of (5.12) and (5.14) that Plugging (5.13) into the above inequality and applying (5.9), we obtain , which implies (5.8) immediately.Thus the normalized Ricci flow initiated from (M, g) exists on [0, T ].Since g( t) = λ M (t)g(t) and λ M (t) is uniformly bounded by (5.17), up to adjusting constant slightly, it is clear that (5.15) follows immediately from (5.7).
Note that Theorem 1.6 is nothing but the combination of Theorem 5.4 and Theorem 5.6.Therefore, we have already finished the proof of Theorem 1.6.

Proof of the main theorem
In this section, we study the global behavior of the normalized Ricci flow near a given immortal solution initiated from (N, h).In particular, we shall show a stability theorem nearby a weakly stable Einstein manifold under appropriate L p -Ricci curvature conditions.Then we apply this stability theorem to prove Theorem 1.1.

3 2 3 2
R p − dv in Step 2 and Step 3, respectively.In Step 4, we dominate ´M ηdv and ´M η R p − dv by ODE solutions which can be calculated explicitly.Finally, in Step 5, we focus on the domain where η ≡ 1 and finish the proof of the estimate (4.16a) and (4.16b).All constants C in this proof depend only on m and p and may vary from line to line.Step 1. Construction of a proper cutoff function η.Define a cutoff function η : R → R satisfying η ≤ 0 and

3 2
R p − dv are dominated by the solutions of the ODE d
Now we are ready to finish the proof of main theorem.Proof of Theorem 1.1.By the diameter estimate of Aubry[1], we know from (1.3) that the diameter of (M, g) is bounded by 2π.Then we can apply the volume comparison of Petersen-Wei (cf.Theorem 1.1 of[28]) to obtaininf x∈M |B(x, 1)| ≥ c 0 (m, p),which in turn means that the average L p -norm of {Rc−(m−1)g} − is small.Namely, (1.6) holds.Consequently, the work of Petersen and Sprouse (introduction of[27]) applies and we have d GH {(M, g), (S m , g round )} ≤ ψ( |m, p).
Theorem 5.4 can be generalized to a version under normalized Ricci flow.Lemma 5.5 of volume estimate is needed to achieve such generalization.
.13)On the other hand, ξ ≤ implies that κ(p, ξ) ≤ ξ2− m Definition 6.1.A closed Einstein metric (N, h E ) is called weakly stable if there exists an = (N, h E ) with the following properties.For any smooth Riemannian metric h satisfying (6.1)|N | dv h = |N | dv h E , h − h E C [ −1 ] (h E ) < ,the normalized Ricci flow initiated from (N, h) exists immortally and converges (in smooth topology) to an Einstein metric (N, h E ).Furthermore, (N, h E ) is called strictly stable if each h E is isometric to h E for some sufficiently small .(Stabilitynearby a stable Ricci flow).Suppose {(N m , h(t)), t ∈ [0, ∞)} is an immortal solution of normalized Ricci flow with the initial metric h(0) = h, and p > m 2 .The immortal solution (N, h(t)) converges to a weakly stable Einstein manifold (N, h E ).Then for any small, there exists an η = η (N, h, p, ) with the following properties.Then the normalized Ricci flow solution initiated from (M, g) exists immortally and converges to an Einstein manifold(N, h E ).Moreover, if h E is strictly stable, then h E is homothetic to h E .Proof.Because the immortal solution (N, h(t)) converges to an Einstein manifold (N, h E ), we can choose T = T ( ) such that(6.3)h(T)−hEC[ −1 ] (h E ) < .On the other hand, by Theorem 5.6, we can find η = η (N, h, p, ) such that if (M, g) satisfies κ(p, ) < 2− m (6.3) and (6.4), we have(6.5)hE−Φ* g(T ) C [ −1 ] (h E ) < 2 .Scaling the metric (N, Φ * g(T )) slightly, we have|N | dv (θΦ * g(T )) = |N | dv h E , h E − θΦ * g(T ) C [ −1 ] (h E ) < 3for an = (N, h E ) which is sufficiently small.Because (N, h E ) is weakly stable (cf.Definition 6.1), the normalized Ricci flow initiated from (N, Φ * g(T )) exists immortally and converges to an Einstein manifold (N, h E ).If (N, h E ) is strictly stable, then we have θh E = h E where θ is the scaling number nearby 1 and controlled by .Concatenating the flows, we obtain a normalized Ricci flow which initiates from (M, g) and converges to (N, h E ).In Theorem 6.2, if (N, h) itself is a weakly stable Einstein manifold, then we have Corollary 6.3.Corollary 6.3 (Stability nearby a stable Einstein manifold).Suppose (N m , h) is a weakly stable Einstein manifold, p > m 2 and ξ < 0 (m, p) is sufficiently small.Then there exists δ = δ(h, p, ξ) with the following properties.If (M Suppose a Riemannian manifold (M, g) satisfying(6.2) κ(p, ) < 2− m p , d GH {(M, g), (N, h)} < η. p , d GH {(M, g), (N, h)} < η, then the normalized Ricci flow initiated from (M, g) exists on [0, T ] and (6.4) h(T ) − Φ * g(T ) C [ −1 ] (h E ) < for a diffeomorphism Φ : N → M .Combining m , g) is a Riemannian manifold satisfying κ(p, ξ) < ξ 2− m p , and d GH {(M, g), (N, h)} < δ,then (M, g) can be smoothly deformed to an Einstein metric (N, h E ) by normalized Ricci flow.Moreover, if (N, h) is strictly stable, then the limit metric is homothetic to (N, h).