Rigidity of positively curved shrinking Ricci solitons in dimension four

We classify four-dimensional shrinking Ricci solitons satisfying $Sec \geq \frac{1}{24} R$, where $Sec$ and $R$ denote the sectional and the scalar curvature, respectively. They are isometric to either $\mathbb{R}^{4}$ (and quotients), $\mathbb{S}^{4}$, $\mathbb{RP}^{4}$ or $\mathbb{CP}^{2}$ with their standard metrics.


Introduction
In this paper we investigate gradient shrinking Ricci solitons with positive sectional curvature. We recall that a Riemannian manifold (M n , g) of dimension n ≥ is a gradient Ricci soliton if there exists a smooth function f on M n such that Ric + ∇ f = λ g for some constant λ. If ∇f is parallel, then (M n , g) is Einstein. The Ricci soliton is called shrinking if λ > , steady if λ = and expanding if λ < . Ricci solitons generate self-similar solutions of the Ricci ow, play a fundamental role in the formation of singularities and have been studied by many authors (see H.-D. Cao [5] for an overview). It is well known that (compact) Einstein manifolds can be classi ed, if they are enough positively curved. Su cient conditions are non-negative curvature operator (S. Tachibana [18]), non-negative isotropic curvature (M. J. Micallef and Y. Wang [13] in dimension four and S. Brendle [3] in every dimension) and weakly -pinched sectional curvature [1] (if Sec and R denote the sectional and the scalar curvature, respectively, this condition in dimension four is implied by Sec ≥ R). Moreover, in dimension four, it is proved by D. Yang [19]) that four-dimensional Einstein manifolds satisfying Sec ≥ εR are isometric to either S , RP or CP with their standard metrics, if ε = √ − . The lower bound has been improved to ε = − √ by E. Costa [8] and, more recently, to ε = by E. Ribeiro [16] (see also X. Cao and P. Wu [6]). It is conjectured in [19] that the result should be true assuming positive sectional curvature.
In dimension n ≤ , complete shrinking Ricci solitons are classi ed. In the last years there have been a lot of interesting results concerning the classi cation of shrinking Ricci solitons which are positively curved. For instance, it follows by the work of C. Böhm and B. Wilking [2] that the only compact shrinking Ricci solitons with positive (two-positive) curvature operator are quotients of S n . In dimension four, A. Naber [14] classi ed complete shrinkers with non-negative curvature operator. Four dimensional shrinkers with nonnegative isotropic curvature were classi ed by X. Li, L. Ni and K. Wang [12].
Recently, O. Munteanu and J.P. Wang [17] showed that every complete shrinking Ricci solitons with positive sectional curvature are compact. It is natural to ask the following question: given ε > , are there four dimensional non-Einstein shrinking Ricci solitons satisfying Sec ≥ εR?.
In this paper we give an answer to this question proving the following Theorem 1.1. Let (M , g) be a four-dimensional complete gradient shrinking Ricci soliton with Sec ≥ R. Then (M , g) is necessarily Einstein, thus isometric to either R (and quotients), S , RP or CP with their standard metrics.
Note that, by the work of S. Brendle and R. Schoen [4], using the Ricci ow, one can show that compact Ricci shrinkers with weakly -pinched sectional curvature are isometric to S , RP or CP with their standard metrics. The condition Sec ≥ R is a little stronger, but the proof of Theorem 1.1 that we present is completely "elliptic".
Estimates on manifolds with positive sectional curvature To x the notation we recall that the Riemann curvature operator of a Riemannian manifold (M n , g) is de ned as in [10] by In a local coordinate system the components of the ( , )-Riemann curvature tensor are given by R l ∂ ∂x k and we denote by R ijkl = g lp R p ijk its ( , )-version. Throughout the paper the Einstein convention of summing over the repeated indices will be adopted. The Ricci tensor Ric is obtained by the contraction (Ric) ik = R ik = g jl R ijkl , R = g ik R ik will denote the scalar curvature and ( • Ric) ik = R ik − n R g ik the traceless Ricci tensor. The Riemannian metric induces norms on all the tensor bundles, in coordinates this norm is given, for a tensor T = T j ...j l i ...i k , by |T| g = g i m · · · g i k m k g j n . . . g j l n l T j ...j l i ...i k T n ...n l m ...m k .
The rst key observation are the following pointwise estimates which are satis ed by every metric with Sec ≥ εR for some ε ∈ R. Proposition 2.1. Let (M n , g) be a Riemannian manifold of dimension n ≥ . If the sectional curvature satis es Sec ≥ εR for some ε ∈ R, then the following two estimates hold In particular, in dimension four Proof. Let {e i }, i = , . . . , n, be the eigenvectors of • Ric and let λ i be the corresponding eigenvalues. Moreover, let σ ij be the sectional curvature de ned by the two-plane spanned by e i and e j . Since the sectional curvature satisfy Sec ≥ εR, it is natural to de ne the tensor In particular Moreover, if µ k and µ k are the eigenvalues with eigenvector e k of Ric and Ric, respectively, one has Denoting by R ijkl the components of Rm, we get Using the de nition of Rm and Ric, we obtain

Ric|
and this proves the rst inequality of this proposition.
In order to show the second one, we will follow the proof of [7, Proposition 3.1]. We observe that Since the modi ed scalar curvature R can be written as one has the following n i,j= On the other hand, one has Moreover, using the Cauchy-Schwarz inequality and the fact that n k= λ k = , we obtain with equality if and only if λ k = λ k for every k, k ≠ i, j. Hence, the following estimate holds Using this, since σ ij ≥ , it follows that n i,j= where in the last equality we have used equation (2.1). Hence, we proved Finally, substituting Rm, Ric and R we obtain the the second inequality of this proposition.
Taking the convex combination of the two previous estimates we obtain the following.

Corollary 2.2. Let (M n , g) be a Riemannian manifold of dimension n ≥ . If the sectional curvature satis es
Sec ≥ εR for some ε ∈ R, then, for every s ∈ [ , ], one has In particular, in dimension four, for every s ∈ [ , ], one has 3. Taking ε = and s = n− n , we recover the estimate on manifolds with non-negative sectional curvature which was proved in [7].

Some formulas for Ricci solitons
Let (M n , g) be a n-dimensional complete gradient shrinking Ricci solitons for some smooth function f and some positive constant λ > . First of all we recall the following well known formulas (for the proof see [9]) Lemma 3.1. Let (M n , g) be a gradient Ricci soliton. Then the following formulas hold where the ∆ f denotes the f -Laplacian, ∆ f = ∆ − ∇ ∇f .
In particular, de ning • R ij = R ij − n R g ij , a simple computation shows the following equation for the f -Laplacian of the squared norm of the trece-less Ricci tensor • Ric Lemma 3.2. Let (M n , g) be a gradient Ricci soliton. Then the following formula holds Moreover we have the following scalar curvature estimate [15]. Finally, we show this simple identity.

In particular, in dimension four
Proof. Integrating by parts and using Lemma 3.1 we obtain

Proof of Theorem 1.1
Let (M , g) be a complete gradient shrinking Ricci soliton of dimension four and assume that Sec ≥ εR on M for some ε > . By Lemma 3.3 either g is at or R > . In this second case, by the result in [17] we know that M must be compact. From now on we can assume that (M , g) is compact with Sec ≥ εR > . Lemma 3.2 gives Integrate over M and using equation (3.1) we obtain On the other hand, given a , a , b , b , b ∈ R we de ne the three tensor Using the Bianchi identity ∇ i • R ij = ∇ j R, a computation gives In particular, where, in the last equality we have used Lemma 3.4. On the other hand, integrating by parts and commuting the covariant derivatives, one has From equation (4.2), we obtain Using this inequality in (4. with In particular, the maximum is attained at b = − / and is given by Actually a (long) computation gives that the maximum of the function Q de ned for general variables (a , a , b , b , b ) is attained at the point a , b , b , b ) where we used (4.