FOUR-DIENIONAL GRADIENT HRINKING OLITON WITH POITIVE IOTROPIC CURVATURE XIAOLONG LI, LEI NI, AND KUI WANG Abstract. We show that a four-dimensional complete gradient shrinking Ricci soliton with positive isotropic curvature is either a quotient of 4 or a quotient of 3 R. This gives a clean classification result removing the earlier additional assumptions in [4] by Wallach and the second author. The proof also gives a classification result on gradient shrinking Ricci solitons with nonnegative isotropic curvature.. Introduction A gradient shrinking Ricci soliton is a triple, g, f, a complete Riemannian manifold with a smooth potential function f whose Hessian satisfying. Ric + f = g. Gradient shrinking solitons arise naturally in the study of the singularity analysis of the Ricci flow [4, 5]. It also attracts the study since. generalizes the notion of Einstein metrics. The main purpose of this article concerns a classification of such four-manifolds with positive isotropic curvature. The positive isotropic curvature condition was first introduced by icalleff and oore [7] in applying the index computation of harmonic spheres to the study of the topology of manifolds. The Ricci flow on four-manifolds with positive isotropic curvature was studied by Hamilton [5]. This condition was proven to be invariant under Ricci flow in dimension four by Hamilton and in high dimensions by Brendle-choen [] and Nguyen []. It is hence then interesting to understand the solitons under the positive isotropic curvature condition. In [4], Wallach and the second author classified the fourdimensional gradient shrinking solitons with positive isotropic curvature under some extra assumptions, including the nonnegative curvature operator, a pinching condition and a curvature growth condition. ince then, there have been much progresses in understanding the general shrinking solitons [3,6,8,0,] and particularly the four-dimensional ones [9]. In particular, a classification result was obtained in [0] for solitons with nonnegative curvature operator for all dimensions. The purpose of this article is to prove the following classification result on shrinking solitons with positive isotropic curvature by removing all the additional assumptions in [4]. Theorem.. Any four-dimensional complete gradient shrinking Ricci soliton with positive isotropic curvature is either a quotient of 4 or a quotient of 3 R. The research of the first author is partially supported by an Inamori fellowship and NF grant D- 40500. The research of the second author is partially supported by NF grant D-40500. The third author s research is partially supported by NFC grant No. 60359 and Natural cience Foundation of Jiangsu Province grant No. BK06030.
XIAOLONG LI, LEI NI, AND KUI WANG It remains an interesting question whether or not the same result holds in high dimensions. We plan to return to this in the future.. Preliminaries In this section, we collect some results on gradient shrinking Ricci solitons that will be used in this paper. After normalizing the potential function f via translating, we have the following identities [4]: Lemma.. where denotes the scalar curvature of. + f = n, + f = f, Regarding the growth of the potential function f and the volume of geodesic balls, Cao and Zhou [3] showed that Lemma.. Let n, g be a complete gradient shrinking Ricci soliton of dimension n and p. Then there are positive constants c, c and C such that 4 dx, p c + fx 4 dx, p + c, VolB p r Cr n. unteanu and esum [8] proved the following integral bound for the Ricci curvature. Lemma.3. Let, g be a complete gradient shrinking Ricci soliton. Then for any λ > 0, we have Ric e λf <. The above mentioned results are valid in all dimensions. In the following, we recall some special properties in four dimensions. It is well known that, in dimension four, the curvature operator R can be written as A B R = B t C according to the natural splitting R 4 = +, where + and are the self-dual and anti-self-dual parts respectively. It is easy to see that A and C are symmetric. Denote A A A 3 and C C C 3 the eigenvalues of A and C, respectively. Also let 0 B B B 3 be the singular eigenvalues of B. Note that a direct consequence of the first Bianchi identity is that tra = trc = 4, where is the scalar curvature. oreover, has positive isotropic curvature amounts to A + A > 0 and C + C > 0. In [4], Wallach and the second author computed that Ric can be expressed in terms of components of B, where Ric is the traceless part of the Ricci tensor. In particular, 4 B = Ric and 4 Ric. It was also observed λ3 i = 4 det B, where λ i s are the eigenvalues of in [4, Proposition 3.] that these components of the curvature operator satisfy certain differential inequalities, which play a significant role in the classification results.
3 Proposition.. If, gt is a solution to the Ricci flow, then we have the following differential inequalities t A + A A + A + A + A A 3 + B + B, t C + C C + C + C + C C 3 + B + B, t B 3 A 3 B 3 + C 3 B 3 + B B in the distributional sense. We now summarize the ideas used to prove the classification result in [4] and explain our strategy of removing all the assumptions except positive isotropic curvature. First of all, it was shown in [3, Proposition 4.] that if 0, then t Rijkl where P is defined by = 4P 3 4 pr ijkl p R ijkl + P = 4 R + R, R Ric R ijkl. Rijkl, log, Here # is the operator defined using the Lie algebra structure of T p see for example [4]. Under a certain curvature growth assumption, the classification result follows from the proof of the main theorem in [3] if one can show that P 0. In dimension four, P can be expressed in terms of A, B and C. Let A and C be the traceless parts of A and C, respectively. By choosing suitable basis of + and, we may diagonalize A and C such that + a 0 0 A = 0 + a + c 0 0 0, C = 0 0 0 + a + c 0. 3 0 0 + c 3 Then P can be written as ee [4] P = 4 λ i + a i + c i 6 +4 a 3 i + c 3 i + 6a a a 3 + 6c c c 3 4 + a b + a b + a 3 b 3 + c b + c b + c 3 b 3 4 4 λ i 4 a i + c i, where 3 a i = 3 c i = 4 λ i = 0, b i = 3 j= B ij and b i = 3 j= B ji. A key observation in [4] is that for several basic examples, one always has BB t = b id for some constant b and P = 0. By considering the evolution equation of the quantity B 4 3 A +A C +C and applying integration by parts, Wallach and the second author proved λ i λ 3 i
4 XIAOLONG LI, LEI NI, AND KUI WANG that BB t = b id is indeed valid for all four-dimensional shrinking Ricci solitons satisfying two very weak curvature assumptions:. B3 x exparx +, A + A C + C. R ijkl x exparx + for some a > 0, where rx is the distance function to a fixed point on the manifold. These two assumptions are only needed to ensure that all the integrals involved in the integration by parts argument are finite. To remove these two curvature assumptions, we show that BB t = b id in fact holds for all four-dimensional shrinking Ricci solitons with positive B isotropic curvature. Our strategy is to consider the evolution equation of 3 A +A C +C and sharpen the integration by parts argument in [4]. In the proof, in order to guarantee that all the integrals involved are finite, we use the integral bound of the Ricci curvature in Lemma.3. Now when BB t = b id, P has a much simpler expression: P = a i + c i + a 3 i + c 3 i b + b 48b a i + c i a i c i where we have used the following 3b = a 3 i b i = b i = 4 a 3 i + 6a a a 3 = 3 4 λ i, a 3 i, 4 λ 3 i = 4 det B = 4b 3. Then by further assuming that has nonnegative curvature operator, namely, A and C are positive semidefinite, P 0 was proved in [4] by solving two optimization problems with constraints. ore precisely, they showed that under the constraints 3 a i = 0 and + a i 0, the maximum of 3 a i is 4, while under further normalization 3 a i = it holds that 3 a3 i 3 a i 6 Combining these results, one easily obtains a i a i. a 3 i 0. The terms with c i s can be handled similarly and thus they arrived at that P 0. c 3 i,
5 By solving another optimization problem under weaker constraints, we achieve in this paper that P 0 without assuming has nonnegative curvature operator. In the last step we also sharpen the integration by parts argument in [3] instead of appealing the result in [3] as in [4] since we no longer assume a curvature growth condition.. These three improvements allow us to prove the main result without any additional assumptions. 3. Proof of Theorem. For brevity, we introduce the same notations as in [4]: ψ = A + A, ψ = C + C, = B 3 and E = 4B B 3 B B 3 A B + A B + A B B A + A C B + C B + C B B C + C. It is clear that E 0 with equality holds only if A = C = B = B = A = C = B 3. In particular we have BB t = b id for some b. Also notice that has positive isotropic curvature amounts to ψ > 0 and ψ > 0. Proposition 3.. The following differential inequality holds in the sense of distribution: t E ψ ψ ψ ψ 3. ψ ψ ψ ψ 4 ψ ψ 5 +, logψ ψ. ψ ψ Proof. traightforward calculations yield t = ψ ψ t ψ t ψ + ψ t ψ ψ ψ ψ ψ 3 ψ ψ ψ ψ + 4 ψ ψ 5, logψ ψ ψ ψ ubstituting the differential inequalities in Proposition. into the above equation gives, after some cancelations, that t E ψ ψ ψ ψ ψ ψ ψ ψ 4 ψ ψ 5 +, logψ ψ. ψ ψ This proves the proposition. Proposition 3.. Let, g, f be a four-dimensional gradient shrinking Ricci soliton with positive isotropic curvature. Then BB t = b id. Proof. Note that on a gradient Ricci soliton, it holds that = f, ψ ψ t ψ ψ.
6 XIAOLONG LI, LEI NI, AND KUI WANG Fix a point x 0 in. For any r > 0, one can choose a smooth cut-off function η with support in {x : dx, x 0 r} and η C/r. Then multiplying both sides of 3. by ψψ e f+logψ ψ η and integrating over, f, e f+logψ ψ η ψ ψ ψ ψ ψ ψ Ee f η +, logψ ψ e f+logψψ η ψ ψ ψ ψ After integration by parts and some cancelations, we arrive at 0 e f+logψ ψ η Ee f η ψ ψ, η e f+logψψ η ψ ψ ψ ψ η e f Ee f η C r e f Ee f η. ince 4 = 4B3 4 B = Ric = Ric 4, we obtain, in view of Lemma., Lemma. and Lemma.3, e f Ric e f <. 4 4 Letting r implies that Ee f = 0. Therefore, we must have either B 3 = 0 or E = 0. It then follows that BB t = b id. Proposition 3.3. Let, g, f be a four-dimensional gradient shrinking Ricci soliton with positive isotropic curvature. Then P 0. Proof. Recall from ection, since BB t = b id, we have P a i c i a 3 i c 3 i. In order to prove P 0, it suffices to show that a i a 3 i 0. With suitable choices of the orthonormal basis for +, we can assume that A i = +a i.note that we have the constraints 3 a i = 0 and A i + A j = 6 + a i + a j > 0 for i j because has positive isotropic curvature. By the change of variables x i = 6 a i, the constraints
become 3 x i = 3 and x i > 0 for i 3, and the objective function becomes F x, x, x 3 := a i a 3 i = 3 x i x i 3 36 = 3 x 3 i 5 x i + 9. 36 Using Lagrange multipliers, we find two critical points Z =,, and W = 3, 4 3, 4 3 with F Z = 0 and F W = 3 6. On the boundary, we have x i = 0 for some i. ince F is symmetric, we can assume without loss of generality that x = 0. Then we have, using x 3 = 3 x, that F x, x, x 3 = 3 x 3 36 + 3 x 3 5x + 3 x + 9 = 3 8 x 3 0. Therefore, under the constraints 3 a i = 0 and 6 + a i + a j 0 for i j, 7 F x, x, x 3 = a i a 3 i 0. The terms involving c i s can be handled similarly. Hence P 0. Proof of Theorem.. Recall it was shown in [3, Proposition 4.] that if 0, then 3. t Rijkl = 4P 3 4 pr ijkl p R ijkl Rijkl +, log, Trying to use integration by parts here would require a stronger integral bound of the Ricci curvature than we actually have in Lemma.3. To overcome this difficulty, we adopt a similar idea that was used to prove BB t = b id. We consider, instead, u = R ijkl and T = R ijkl. A direct calculation shows that t u = P u 3 + u, log + u T u P u 3 + u, log where we have used Kato s inequality in the last line. Now let η be a smooth cut-off function with support in {x : dx, x 0 r} and η C/r. ultiplying the above inequality by ue f+log η and integrating over, we obtain, f, u ue f+log η u ue f+log η P 3 e f+log η + u, log ue f+log η.
8 XIAOLONG LI, LEI NI, AND KUI WANG Integration by parts then yields u e f+log η u e f+log η 8 η u f+log e r R ijkl e f. P e f f+log u, η uηe The integral R ijkl e f is finite in view of Lemma.3, since if has positive isotropic curvature, then the components of curvature operator A, B and C can be estimated by 4 A A A 3 4, 4 C C C 3 4, 4 B Ric. Therefore we know that u is a positive constant and P = 0 by letting r. Then it follows from 3. that p R ijkl p R ijkl = 0. Theorem. then follows from the proof of the main theorem in [3]. The strong maximum principle of [] together with the classification of positive case implies the following corollary for the solitons with nonnegative isotropic curvature. Corollary 3.. If, g, f is a complete gradient shrinking soliton with nonnegative isotropic curvature then its universal cover must be one of the following spaces R 4, 4, CP,, R and 3 R. This provides a more general classification result than that of [], where the shrinking solitons are assumed to have bounded nonnegative curvature operator. References [] imon Brendle and Richard choen, anifolds with /4-pinched curvature are space forms, J. Amer. ath. oc. 009, no., 87 307. R449060 [], Classification of manifolds with weakly /4-pinched curvatures, Acta ath. 00 008, no., 3. 8 [3] Huai-Dong Cao and Detang Zhou, On complete gradient shrinking Ricci solitons, J. Differential Geom. 85 00, no., 75 85. R73975, [4] Richard. Hamilton, The formation of singularities in the Ricci flow, urveys in differential geometry II 993, 7 36. R37555,, 3 [5], Four-manifolds with positive isotropic curvature, Comm. Anal. Geom. 5 997, no., 9. R456308 [6] Brett Kotschwar and Lu Wang, Rigidity of asymptotically conical shrinking gradient Ricci solitons, J. Differential Geom. 00 05, no., 55 08. R336574 [7] ario J. icallef and John Douglas oore, inimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of ath. 7 988, no., 99 7. R094677 [8] Ovidiu unteanu and Natasa esum, On gradient Ricci solitons, J. Geom. Anal. 3 03, no., 539 56, DOI 0.007/s0-0-95-6. R303848,
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