THE EVOLUTION OF THE WEYL TENSOR UNDER THE RICCI FLOW

Transcription

1 THE EVOLUTION OF THE WEYL TENSO UNDE THE ICCI FLOW GIOVANNI CATINO AND CALO MANTEGAZZA ABSTACT. We compute the evolution equation of the Weyl tensor under the icci flow of a iemannian manifold and we discuss some consequences for the classification of locally conformally flat icci solitons. CONTENTS 1. The Evolution Equation of the Weyl Tensor 1 2. Locally Conformally Flat icci Solitons Compact LCF icci Solitons LCF icci Solitons with Constant Scalar Curvature Gradient LCF icci Solitons with Nonnegative icci Tensor The Classification of Steady and Shrinking Gradient LCF icci Solitons Singularities of icci Flow with Bounded Weyl Tensor 19 eferences THE EVOLUTION EQUATION OF THE WEYL TENSO The iemann curvature operator of a iemannian manifold (M n, g is defined as in [14] by iem(x, Y Z = Y X Z X Y Z [X,Y ] Z. In a local coordinate system the components of the (3, 1 iemann curvature tensor are given by l ijk = iem ( x l x, i x and we denote by j x k ijkl = g lm m ijk its (4, 0 version. In all the paper the Einstein convention of summing over the repeated indices will be adopted. With this choice, for the sphere S n we have iem(v, w, v, w = ijkl v i w j v k w l > 0. The icci tensor is obtained by the contraction ik = g jl ijkl and = g ik ik will denote the scalar curvature. The so called Weyl tensor is then defined by the following decomposition formula (see [14, Chapter 3, Section K] in dimension n 3, W ijkl = ijkl (n 1(n 2 (g ikg jl g il g jk 1 n 2 ( ikg jl il g jk jl g ik jk g il = ijkl A ijkl B ijkl, where we introduced the tensors A ijkl = (n 1(n 2 (g ikg jl g il g jk and B ijkl = 1 n 2 ( ikg jl il g jk jl g ik jk g il. The Weyl tensor satisfies all the symmetries of the curvature tensor and all its traces with the metric are zero, as it can be easily seen by the above formula. In dimension three W is identically zero for every iemannian manifold (M 3, g, it becomes Date: October 1,

2 2 GIOVANNI CATINO AND CALO MANTEGAZZA relevant instead when n 4 since its nullity is a condition equivalent for (M n, g to be locally conformally flat, that is, around every point p M n there is a conformal deformation g ij = e f g ij of the original metric g, such that the new metric is flat, namely, the iemann tensor associated to g is zero in U p (here f : U p is a smooth function defined in a open neighborhood U p of p. We suppose now that (M n, g(t is a icci flow in some time interval, that is, the time dependent metric g(t satisfies t g ij = 2 ij. We have then the following evolution equations for the curvature (see for instance [15], (1.1 where C ijkl = g pq g rs pijr slkq. = 2 ic 2 t t ij = ij 2 kl kilj 2g pq ip jq, t ijkl = ijkl 2(C ijkl C ijlk C ikjl C iljk g pq ( ip qjkl jp iqkl kp ijql lp ijkq, All the computations which follow will be done in a fixed local frame, not in a moving frame. The goal of this section is to work out the evolution equation under the icci flow of the Weyl tensor W ijkl. In the next sections we will see the geometric consequences of the assumption that a manifold evolving by the icci flow is locally conformally flat at every time. In particular, we will be able to classify the so called icci solitons under the hypothesis of locally conformally flatness. Since W ijkl = ijkl A ijkl B ijkl and we already have the evolution equation (1.1 for ijkl, we start differentiating in time the tensors A ijkl and B ijkl and t A 2 ic 2 ijkl = (n 1(n 2 (g ikg jl g il g jk t B ijkl = 1 (n 1(n 2 (2 ikg jl 2 jl g ik 2 il g jk 2 jk g il = A ijkl 2 ic 2 (n 1(n 2 (g ikg jl g il g jk 2 n 1 B ijkl ( ( ik 2 pq piqk 2g pq ip kq g jl n 2 ( il 2 pq piql 2g pq ip lq g jk ( jl 2 pq pjql 2g pq jp lq g ik ( jk 2 pq pjqk 2g pq jp kq g il 4 jk il 4 ik jl = B ijkl 2 ( ( pq piqk g pq ip kq g jl ( pq piql g pq ip lq g jk n 2 ( pq pjql g pq jp lq g ik ( pq pjqk g pq jp kq g il 4 n 2 ( ik jl jk il.

3 THE EVOLUTION OF THE WEYL TENSO UNDE THE ICCI FLOW 3 Now we deal with the terms like pq piqk. We have by definition pq piqk = pq W piqk pq A piqk pq B piqk and pq A piqk = (n 1(n 2 (pq g pq g ik pq g pk g iq = (n 1(n 2 (g ik ik, pq B piqk = 1 n 2 (pq pq g ik pq pk g iq pq ik g pq pq iq g pk = 1 n 2 ( ic 2 g ik ik 2g pq ip kq, hence, we get pq piqk = pq W piqk (n 1(n 2 (g ik ik 1 n 2 ( ic 2 g ik ik 2g pq ip kq = pq W piqk 1 n 2 ( ic 2 g ik 2g pq ip kq (n 1(n 2 (n ik g ik.

4 4 GIOVANNI CATINO AND CALO MANTEGAZZA Substituting these terms in the formula for t B ijkl we obtain t B ijkl = B ijkl 2 n 2 (pq W piqk g jl pq W piql g jk pq W pjql g ik pq W pjqk g il 2 ic 2 (n 2 2 (g ikg jl g il g jk g jl g ik g jk g il 4 (n 2 2 (gpq ip kq g jl g pq ip lq g jk g pq jp lq g ik g pq jp kq g il 2n (n 1(n 2 2 ( ikg jl il g jk jl g ik jk g il 2 2 (n 1(n 2 2 (g ikg jl g il g jk g jl g ik g jk g il 2 n 2 (gpq ip kq g jl g pq ip lq g jk g pq jp lq g ik g pq jp kq g il 4 n 2 ( ik jl jk il = B ijkl 2 n 2 (pq W piqk g jl pq W piql g jk pq W pjql g ik pq W pjqk g il 2n (n 2 2 (gpq ip kq g jl g pq ip lq g jk g pq jp lq g ik g pq jp kq g il 2n (n 1(n 2 2 ( ikg jl il g jk jl g ik jk g il 22 2(n 1 ic 2 (n 1(n 2 2 (g ik g jl g il g jk g jl g ik g jk g il 4 n 2 ( ik jl jk il = B ijkl 2 n 2 (pq W piqk g jl pq W piql g jk pq W pjql g ik pq W pjqk g il 2n (n 2 2 (gpq ip kq g jl g pq ip lq g jk g pq jp lq g ik g pq jp kq g il 2n (n 1(n 2 B ijkl 4 n 2 A ijkl 4 ic 2 (n 2 2 (g ikg jl g il g jk 4 n 2 ( ik jl jk il.

5 THE EVOLUTION OF THE WEYL TENSO UNDE THE ICCI FLOW 5 Hence, (1.2 ( ( t W ijkl = t ( ijkl A ijkl B ijkl = 2(C ijkl C ijlk C ikjl C iljk g pq ( ip qjkl jp iqkl kp ijql lp ijkq 2 ic 2 (n 1(n 2 (g ikg jl g il g jk 2 n 1 B ijkl 2 n 2 (pq W piqk g jl pq W piql g jk pq W pjql g ik pq W pjqk g il 2n (n 2 2 (gpq ip kq g jl g pq ip lq g jk g pq jp lq g ik g pq jp kq g il 2n (n 1(n 2 B ijkl 4 n 2 A ijkl 4 ic 2 (n 2 2 (g ikg jl g il g jk 4 n 2 ( ik jl jk il = 2(C ijkl C ijlk C ikjl C iljk g pq ( ip qjkl jp iqkl kp ijql lp ijkq 2 n 2 (pq W piqk g jl pq W piql g jk pq W pjql g ik pq W pjqk g il 2n (n 2 2 (gpq ip kq g jl g pq ip lq g jk g pq jp lq g ik g pq jp kq g il 4 (n 2 2 ( ikg jl il g jk jl g ik jk g il 42 2n ic 2 (n 1(n 2 2 (g ikg jl g il g jk 4 n 2 ( ik jl jk il. Now, in order to simplify the formulas, we assume to be in an orthonormal basis, then C ijkl = pijq qlkp and we have C ijkl = pijq qlkp = W pijq W qlkp A pijq A qlkp B pijq B qlkp A pijq B qlkp B pijq A qlkp W pijq A qlkp W pijq B qlkp A pijq W qlkp B pijq W qlkp. Substituting the expressions for the tensors A and B in the above terms and simplifying, we obtain the following identities. A pijq A qlkp = 2 (n 1 2 (n 2 2 (g ikg jl (n 2g ij g lk, 1 B pijq B qlkp = (n 2 2 ( pjg iq iq g pj pq g ij ij g pq ( qk g lp lp g qk pq g lk lk g pq 1 ( = 2ik (n 2 2 lj (n 4 ij lk pj pl g ik pk pi g lj 2 pj pi g lk 2 pl pk g ij ij g lk lk g ij ic 2 g ij g lk, ( A pijq B qlkp = ik (n 1(n 2 2 g lj lj g ik ij g lk (n 3 lk g ij g ij g lk, ( B pijq A qlkp = lj (n 1(n 2 2 g ik ik g lj lk g ij (n 3 ij g lk g ij g lk, W pijq A qlkp = (n 1(n 2 W lijk,

6 6 GIOVANNI CATINO AND CALO MANTEGAZZA A pijq W qlkp = (n 1(n 2 W ilkj, W pijq B qlkp = 1 n 2 (W lijp pk W pijk lp W pijq pq g lk, B pijq W qlkp = 1 n 2 (W ilkp pj W plkj pi W qlkp pq g ij where in these last four computations we used the fact that every trace of the Weyl tensor is null. Interchanging the indexes and summing we get A pijq A qlkp A pijq A qklp A pikq A qljp A pilq A qkjp 2 ( = (n 1 2 (n 2 2 g ik g jl (n 2g ij g lk g il g jk (n 2g ij g lk g ij g kl (n 2g ik g lj g ij g kl (n 2g il g jk = 2 (n 1(n 2 2 (g ikg jl g il g jk, B pijq B qlkp B pijq B qklp B pikq B qljp B pilq B qkjp 1 ( = (n ik lj (n 4 ij lk pj pl g ik pk pi g lj = 2 pj pi g lk 2 pl pk g ij ij g lk lk g ij ic 2 g ij g lk 2 il kj (n 4 ij lk pj pk g il pl pi g kj 2 pj pi g lk 2 pk pl g ij ij g lk lk g ij ic 2 g ij g lk 2 ij lk (n 4 ik lj pk pl g ij pj pi g lk 2 pk pi g lj 2 pl pj g ik ik g lj lj g ik ic 2 g ik g lj 2 ij kl (n 4 il jk pl pk g ij pj pi g kl 2 pl pi g jk 2 pk pj g il il g jk jk g il ic 2 g il g jk 1 ( (n 2 2 (n 2( ik lj il jk pj pl g ik pk pi g lj pl pi g jk pk pj g il ( ik g lj lj g ik il g jk jk g il ic 2 (g ik g lj g il g jk,

7 THE EVOLUTION OF THE WEYL TENSO UNDE THE ICCI FLOW 7 A pijq B qlkp B pijq A qlkp A pijq B qklp B pijq A qklp A pikq B qljp B pikq A qljp A pilq B qkjp B pilq A qkjp ( = (n 1(n 2 2 ik g lj lj g ik ij g lk (n 3 lk g ij g ij g lk lj g ik ik g lj lk g ij (n 3 ij g lk g lk g ij il g kj jk g il ij g kl (n 3 kl g ij g ij g kl kj g il il g kj kl g ij (n 3 ij g kl g kl g ij ij g lk lk g ij ik g lj (n 3 lj g ik g ik g lj lk g ij ij g lk lj g ik (n 3 ik g lj g lj g ik ij g kl lk g ij il g kj (n 3 kj g il g il g kj kl g ij ij g kl kj g il (n 3 il g kj g kj g il = ( ik g jl jl g ik jk g il il g jk (n 1(n (n 1(n 2 2 (g ikg jl g il g jk and W pijq A qlkp W pijq A qklp W pikq A qljp W pilq A qkjp = (n 1(n 2 (W lijk W kijl W likj W kilj = 0, since the Weyl tensor, sharing the same symmetries of the iemann tensor, is skew symmetric in the third fourth indexes. The same result holds for the other sum as hence, A pijq W qlkp = (n 1(n 2 W ilkj = (n 1(n 2 W lijk = W pijq A qlkp A pijq W qlkp A pijq W qklp A pikq W qljp A pilq W qkjp = 0. Finally, for the remaining two terms we have W pijq B qlkp B pijq W qlkp W pijq B qklp B pijq W qklp = 1 n 2 W pikq B qljp B pikq W qljp W pilq B qkjp B pilq W qkjp ( W lijp pk W pijk lp W pijq pq g lk W ilkp pj W plkj pi W qlkp pq g ij W kijp pl W pijl kp W pijq pq g kl W iklp pj W pklj pi W qklp pq g ij W likp pj W pikj lp W pikq pq g jl W iljp pk W pljk pi W qljp pq g ik W kilp pj W pilj kp W pilq pq g kj W ikjp pl W pkjl pi W qkjp pq g il = 1 (W pilq pq g kj W qkjp pq g il W pikq pq g jl W qljp pq g ik n 2

8 8 GIOVANNI CATINO AND CALO MANTEGAZZA where we used repeatedly the symmetries of the Weyl and the icci tensors. Hence, summing all these terms we conclude (1.3 2(C ijkl C ijlk C ikjl C iljk = 2(D ijkl D ijlk D ikjl D iljk 2 2 (n 1(n 2 2 (g ikg jl g il g jk 2 n 2 ( ik lj il jk 2 (n 2 2 ( pj pl g ik pk pi g lj pl pi g jk pk pj g il 2 (n 2 2 ( ikg lj lj g ik il g jk jk g il 2 ic 2 (n 2 2 (g ikg lj g il g jk 2 (n 1(n 2 ( ikg jl jl g ik jk g il il g jk 4 2 (n 1(n 2 2 (g ikg jl g il g jk 2 n 2 (W pilq pq g kj W qkjp pq g il W pikq pq g jl W qljp pq g ik = 2(D ijkl D ijlk D ikjl D iljk 2(n 1 ic (n 1(n 2 2 (g ik g jl g il g jk 2 n 2 ( ik lj il jk 2 (n 2 2 ( pj pl g ik pk pi g lj pl pi g jk pk pj g il 2 (n 1(n 2 2 ( ikg jl jl g ik jk g il il g jk 2 n 2 (W pilq pq g kj W qkjp pq g il W pikq pq g jl W qljp pq g ik, where D ijkl = W pijq W qlkp.

9 THE EVOLUTION OF THE WEYL TENSO UNDE THE ICCI FLOW 9 Then we deal with the following term appearing in equation (1.2, ip pjkl jp ipkl kp ijpl lp ijkp = ip W pjkl jp W ipkl kp W ijpl lp W ijkp ( ip (g pk g jl g pl g jk jp (g ik g pl g il g pk (n 1(n 2 ( kp (g ip g jl g il g jp lp (g ik g jp g ip g jk (n 1(n 2 1 n 2 ( ip( pk g jl pl g jk jl g pk jk g pl 1 n 2 ( jp( ik g pl il g pk pl g ik pk g il 1 n 2 ( kp( ip g jl il g jp jl g ip jp g il 1 n 2 ( lp( ik g jp ip g jk jp g ik jk g ip = ip W pjkl jp W ipkl kp W ijpl lp W ijkp 1 n 2 ( ip pk g jl ip pl g jk jl ik il jk 1 n 2 ( jl ik jk il jp pl g ik jp pk g il 1 n 2 ( kp ip g jl jk il ik jl kp jp g il 1 n 2 ( jl ik lp ip g jk lp jp g ik il jk 2 (n 1(n 2 ( ikg jl il g jk jl g ik jk g il = ip W pjkl jp W ipkl kp W ijpl lp W ijkp 2 n 2 ( ip kp g jl ip lp g jk jp lp g ik jp kp g il 4 n 2 ( ik jl jk il 2 (n 1(n 2 ( ikg jl il g jk jl g ik jk g il.

10 10 GIOVANNI CATINO AND CALO MANTEGAZZA Inserting expression (1.3 and this last quantity in equation (1.2 we obtain ( t W ijkl = 2(D ijkl D ijlk D ikjl D iljk 2(n 1 ic (n 1(n 2 2 (g ik g jl g il g jk 2 n 2 ( ik lj il jk 2 (n 2 2 ( pj pl g ik pk pi g lj pl pi g jk pk pj g il 2 (n 1(n 2 2 ( ikg jl jl g ik jk g il il g jk 2 n 2 (W pilq pq g kj W qkjp pq g il W pikq pq g jl W qljp pq g ik ip W pjkl jp W ipkl kp W ijpl lp W ijkp 2 n 2 ( ip kp g jl ip lp g jk jp lp g ik jp kp g il 4 n 2 ( ik jl jk il 2 (n 1(n 2 ( ikg jl il g jk jl g ik jk g il 2 n 2 ( pqw piqk g jl pq W piql g jk pq W pjql g ik pq W pjqk g il 2n (n 2 2 ( ip kp g jl ip lp g jk jp lp g ik jp kp g il 4 (n 2 2 ( ikg jl il g jk jl g ik jk g il 42 2n ic 2 (n 1(n 2 2 (g ikg jl g il g jk 4 n 2 ( ik jl jk il = 2(D ijkl D ijlk D ikjl D iljk ( ip W pjkl jp W ipkl kp W ijpl lp W ijkp 2(2 ic 2 (n 1(n 2 2 (g ikg jl g il g jk 2 n 2 ( ik lj il jk 2 (n 2 2 ( pj pl g ik pk pi g lj pl pi g jk pk pj g il 2 (n 2 2 ( ikg jl jl g ik jk g il il g jk. Hence, we resume this long computation in the following proposition, getting back to a standard coordinate basis.

11 THE EVOLUTION OF THE WEYL TENSO UNDE THE ICCI FLOW 11 Proposition 1.1. During the icci flow of an n dimensional iemannian manifold (M n, g, the Weyl tensor satisfies the following evolution equation ( t W ijkl = 2 (D ijkl D ijlk D ikjl D iljk where D ijkl = g pq g rs W pijr W slkq. g pq ( ip W qjkl jp W iqkl kp W ijql lp W ijkq 2 (n 2 2 gpq ( ip qk g jl ip ql g jk jp ql g ik jp qk g il 2 (n 2 2 ( ikg jl il g jk jl g ik jk g il 2 n 2 ( ik jl jk il 2(2 ic 2 (n 1(n 2 2 (g ikg jl g il g jk, From this formula we immediately get the following rigidity result on the eigenvalues of the icci tensor. Corollary 1.2. Suppose that under the icci flow of (M n, g of dimension n 4, the Weyl tensor remains identically zero. Then, at every point, either the icci tensor is proportional to the metric or it has an eigenvalue of multiplicity (n 1 and another of multiplicity 1. Proof. By the above proposition, as every term containing the Weyl tensor is zero, the following relation holds at every point in space and time 0 = 2 (n 2 2 gpq ( ip qk g jl ip ql g jk jp ql g ik jp qk g il 2 2 (n 1(n 2 2 (g 2 ic 2 ikg jl g il g jk (n 1(n 2 2 (g ikg jl g il g jk 2 (n 2 2 ( ikg jl il g jk jl g ik jk g il 2 n 2 ( ik jl jk il. In normal coordinates such that the icci tensor is diagonal we get, for every couple of different eigenvectors v i with relative eigenvalues λ i, (1.4 (n 1[λ 2 i λ 2 j] (n 1(λ i λ j (n 1(n 2λ i λ j 2 ic 2 = 0. As n 4, fixing i, then the equation above is a second order polynomial in λ j, hence it can only have at most 2 solutions, hence, we can conclude that there are at most three possible values for the eigenvalues of the icci tensor. Since the dimension is at least four, at least one eigenvalues must have multiplicity two, let us say λ i, hence the equation (1.4 holds also for i = j, and it remains at most only one possible value for the other eigenvalues λ l with l i. In conclusion, either the eigenvalues are all equal or they divide in only two possible values, λ with multiplicity larger than one, say k and µ λ. Suppose that µ also has multiplicity larger than one, that is, k < n 1, then we have nλ 2 2λ = ic 2 2 (1.5 n 1 nµ 2 2µ = ic 2 2 n 1 taking the difference and dividing by (λ µ we get n(λ µ = 2 = 2[kλ (n kµ] then, (n 2kλ = (n 2kµ hence, n = 2k, but then getting back to equation (1.5, = n(µ λ/2 and nλ 2 n(µ λλ = n(λ2 µ 2 /2 n 2 (µ 2 λ 2 2λµ/4 n 1

12 12 GIOVANNI CATINO AND CALO MANTEGAZZA which implies that is, after some computation, 4nλµ = n(n 2 n 1 (λ2 µ 2 2n2 n 1 µλ 2n(n 2 n(n 2 µλ = n 1 n 1 (λ2 µ 2, which implies λ = µ. At the end we conclude that at every point of M n, either ic = λg or there is an eigenvalue λ of multiplicity (n 1 and another µ of multiplicity 1. emark 1.3. Notice that in dimension three equation (1.4 becomes 2[λ 2 i λ 2 j] 2(λ i λ j 2λ i λ j 2 ic 2 = 2(λ i λ j 2 2(λ i λ j 2λ i λ j 2 ic 2 = 2λ l (λ i λ j 2λ i λ j 2 ic 2 = 0, where λ i, λ j and λ l are the three eigenvalues of the icci tensor. Hence, the condition is void and our argument does not work. This is clearly not unexpected as the Weyl tensor is identically zero for every three dimensional iemannian manifold. 2. LOCALLY CONFOMALLY FLAT ICCI SOLITONS Let (M n, g, for n 4, be a connected, complete, icci soliton, that is, there exists a smooth 1 form ω and a constant α such that ij 1 2 ( iω j j ω i = α n g ij. If α > 0 we say that the soliton is shrinking, if α = 0 steady, if α < 0 expanding. If there exists a smooth function f : M n such that df = ω we say that the soliton is a gradient icci soliton and f its potential function, then we have ij 2 ijf = α n g ij. If the metric dual field to the form ω is complete, then a icci soliton generates a self similar solution to the icci flow (if the soliton is a gradient soliton this condition is automatically satisfied [33]. In all this section we will assume to be in this case. In this section we discuss the classification of icci solitons (M n, g, for n 4, which are locally conformally flat (LCF. As a consequence of Corollary 1.2 we have the following fact. Proposition 2.1. Let (M n, g be a complete, LCF icci soliton of dimension n 4. Then, at every point, either the icci tensor is proportional to the metric or it has an eigenvalue of multiplicity (n 1 and another of multiplicity 1.

13 THE EVOLUTION OF THE WEYL TENSO UNDE THE ICCI FLOW 13 If a manifold (M n, g is LCF, it follows that 0 = l W ijkl = l( ijkl (n 1(n 2 (g ikg jl g il g jk 1 n 2 ( ikg jl il g jk jl g ik jk g il j = i jk j ik (n 1(n 2 g i ik (n 1(n 2 g jk 1 n 2 ( j ik l il g jk l jl g ik i jk g il = n 3 n 2 ( i jk j ik j (n 1(n 2 g ik i (n 1(n 2 g jk 1 2(n 2 ( ig jk /2 j g ik /2 = n 3 [ i jk j ik ( ig jk j g ik ] n 2 2(n 1 = n 3 ( 1 ( [ j ik n 2 2(n 1 g 1 ] ik i jk 2(n 1 g jk, where we used the second Bianchi identity and Schur s Lemma = 2 div ic. Hence, since we assumed that the dimension n is at least four, the Schouten tensor defined by S = ic 1 2(n1 g satisfies the equation ( X S Y = ( Y S X, X, Y T M. Any symmetric two tensor satisfying this condition is called a Codazzi tensor (see [2, Chapter 16] for a general overview of Codazzi tensors. Suppose that we have a local orthonormal frame {E 1,..., E n } in an open subset Ω of M n such that ic(e 1 = λe 1 and ic(e i = µe i for i = 2,..., n and λ µ. For every point in Ω also the Schouten tensor S has two distinct eigenvalues σ 1 of multiplicity one and σ 2 of multiplicity (n 1, with the same eigenspaces of λ and µ respectively, and σ 1 = 2n 3 2(n 1 λ 1 2 µ and σ 2 = 1 2 µ 1 2(n 1 λ. Splitting results for iemannian manifolds admitting a Codazzi tensor with only two distinct eigenvalues were obtained by Derdzinski [11] and Hiepko eckziegel [20, 21] (see again [2, Chapter 16] for further discussion. In particular, it can be proved that, if the two distinct eigenvalues σ 1 and σ 2 are both constant along the eigenspace span{e 2,..., E n } then the manifold is locally a warped product on an interval of of a (n 1 dimensional iemannian manifold (see [2, Chapter 16] and [31]. Since σ 2 has multiplicity (n 1, larger than 2, we have for any two distinct indexes i, j 2, i σ 2 = i S(E j, E j = i S jj 2S( Ei E j, E j = j S ij 2σ 2 g( Ei E j, E j = j S(E i, E j S( Ej E i, E j S(E i, Ej E j = σ 2 g( Ej E i, E j σ 2 g(e i, Ej E j = 0, hence, σ 2 is always constant along the eigenspace span{e 2,..., E n }. The eigenvalue σ 1 instead, for a general LCF manifold, can vary, for example n endowed with the metric dx 2 g = [1 (x 2 1 x2 2 x2 n1 ]2

14 14 GIOVANNI CATINO AND CALO MANTEGAZZA is LCF and g ij = (n 2( 2 ij log A i log A j log A ( log A (n 2 log A 2 δ ij where the derivatives are the standard ones of n and A(x = 1 (x 2 1 x 2 2 x 2 n1 (see [2, Theorem 1.159]. Hence, this icci tensor factorizes on the eigenspaces e 1,..., e n1 and e n but the eigenvalue σ 1 of the Schouten tensor, which is given by σ 1 = g nn g nn = ( log A (n 2 log A 2 A 2 = A A (n 1 A 2 = 2(n 1A 4(n 1(A 1 = 2(n 1(A 2, is clearly not constant along the directions e 1,..., e n1. The best we can say in general is that the metric of (M n, g locally around every point can be written as I N and g(t, p = dt2 σ K (p [α(t β(p] 2 where σ K is a metric on N of constant curvature K, α : I and β : N are smooth functions such that Hess K β = fσ K, for some function f : N and where Hess K is the Hessian of (N, σ K Compact LCF icci Solitons. A compact icci soliton is actually a gradient soliton (by the work of Perelman [27]. In general (even if they are not LCF, steady and expanding compact icci solitons are Einstein, hence, when also LCF, they are of constant curvature (respectively zero and negative. In [7, 12] it is proved that also shrinking, compact, LCF icci solitons are of constant positive curvature, hence quotients of spheres. Any compact, n dimensional, LCF icci soliton is a quotient of n, S n and H n with their canonical metrics, for every n N LCF icci Solitons with Constant Scalar Curvature. Getting back to the Schouten tensor, if the scalar curvature of an LCF icci soliton (M n, g is constant, we have that also the other eigenvalue σ 1 of the Schouten tensor is constant along the eigenspace span{e 2,..., E n }, that is, i σ 1 = 0, by simply differentiating the equality = 2(n1 n2 (σ 1 (n 1σ 2. Hence, by the above discussion, we can conclude that around every point of M n in the open set Ω M n where the two eigenvalues of the icci tensor are distinct the manifold (M n, g is locally a warped product I N with g(t, p = dt 2 h 2 (tσ(p (this argument is due to Derdzinski [11]. Then the LCF hypothesis implies that the warp factor (N, σ is actually a space of constant curvature K (see for instance [4]. As the scalar curvature is constant, by the evolution equation t = 2 ic 2 we see that also ic 2 is constant, that is, locally = λ (n 1µ = C 1 and ic 2 = λ 2 (n 1µ 2 = C 2. Putting together these two equations it is easy to see that then both the eigenvalues µ and λ are locally constant in Ω. Hence, by connectedness, either (M n, g is Einstein, so a constant curvature space, or the icci tensor has two distinct constant eigenvalues everywhere. Using now the local warped product representation, the icci tensor is expressed by (see [2, Proposition 9.106] or [10, p. 65] or [5, p. 168] (2.1 ic = (n 1 h h dt2 ( (n 2K h h (n 2(h 2 σ K. hence, h /h and ((n2kh h (n2(h 2 /h 2 are constant in t. This implies that (K(h 2 /h 2 is also constant and h = Ch, then locally either the manifold (M n, g is of constant curvature or it is the iemannian product of a constant curvature space with an interval of.

15 THE EVOLUTION OF THE WEYL TENSO UNDE THE ICCI FLOW 15 By a maximality argument, passing to the universal covering of the manifold, we get the following conclusion. If n 4, any n dimensional, LCF icci soliton with constant scalar curvature is either a quotient of n, S n and H n with their canonical metrics or a quotient of the iemannian products S n1 and H n1 (see also [29] Gradient LCF icci Solitons with Nonnegative icci Tensor. Getting back again to the Codazzi property of the Schouten tensor S, for every index i > 1, we have locally 0 = 1 i1 i (n 1 g i1 i 2(n 1 g 11 = 1 i1 i 11 i 2(n 1. If the soliton is a gradient LCF icci soliton, that is, ic = 2 f α n g, we have = f α and taking the divergence of both sides i /2 = div ic i = g jk k ij = g jk k i j f = g jk i k j f g jk kijl l f = i f il l f = i il l f, where we used Schur s Lemma i = 2 div ic i and the formula for the interchange of covariant derivatives. Hence, the relation i = 2 il l f holds and 1 2 i1f i 2 11f = ij j f n 1. By means of the fact that W = 0, we compute now for i > 1 (this is a special case of the computation in Lemma 3.1 of [6], µ n 1 if = ij j f n 1 = 1 2 i1f i 2 11f = 1i1j j f [ ] 1 = n 2 ( 11g ij 1j g i1 ij g 11 i1 g 1j (n 1(n 2 (g 11g ij g 1j g i1 j f [ ] 1 = n 2 (λg ij µg ij (n 1(n 2 g ij j f [ ] λ µ = n 2 i f (n 1(n 2 (n 1λ (n 1µ λ (n 1µ = i f (n 1(n 2 = λ n 1 if. Then, in the open set Ω M n where the two eigenvalues of the icci tensor are distinct, the vector field f is parallel to E 1, hence it is an eigenvector of the icci tensor and i = 2 il l f = 0, for every index i > 1. As σ 1 = n2 2(n1 (n 1σ 2 we get that also i σ 1 = 0 for every index i > 1. The set Ω is dense, otherwise its complement where ic g/n = 0 has interior points and, by Schur s Lemma, the scalar curvature would be constant in some open set of M n. Then, strong

16 16 GIOVANNI CATINO AND CALO MANTEGAZZA maximum principle applied to the equation t = 2 ic 2 implies that is constant everywhere on M n, and we are in the previous case. So we can conclude also in this case by the previous argument that the manifold, locally around every point in Ω, is a warped product on an interval of of a constant curvature space L K. Moreover, Ω is obviously invariant by translation in the L K direction. We consider a point p Ω and the maximal geodesic curve γ(t passing from p orthogonal to L K, contained in Ω. It is easy to see that for every compact, connected segment of such geodesic we have a neighborhood U and a representation of the metric in g as g = dt 2 h 2 (tσ K, covering the segment with the local charts and possibly shrinking them in the orthogonal directions. Assuming from now on that the the icci tensor is nonnegative, by the local warped representation formula (2.1 we see that h 0 along such geodesic, as tt 0. If such geodesic has no endpoints, being concave the function h must be constant and we have either a flat quotient of n or the iemannian product of with a quotient of S n1. The same holds if the function h is constant in some interval, indeed, the manifold would be locally a iemannian product and the scalar curvature would be locally constant (hence we are in the case above. If there is at least one endpoint, one of the following two situations happens: the function h goes to zero at such endpoint, the geodesic hits the boundary of Ω. If h goes to zero at an endpoint, by concavity (h 2 must converge to some positive limit and by the smoothness of the manifold, considering again formula (2.1, the quantity K (h 2 must go to zero as h goes to zero, hence K > 0 and the constant curvature space L K must be a quotient of the sphere S n1 (if the same happens also at the other endpoint, the manifold is compact. Then, by topological reasons we conclude that actually the only possibility for L K is the sphere S n1 itself. Assuming instead that h does not go to zero at any endpoint, where the geodesic hits the boundary of Ω the icci tensor is proportional to the metric, hence, again by the representation formula (2.1, the quantity K (h 2 is going to zero and either K = 0 or K > 0. The case K = 0 is impossible, indeed h would tend to zero at such endpoint, then by the concavity of h the function h has a sign, otherwise h is constant in an interval, implying that in some open set (M n, g is flat, which cannot happens since we are in Ω. Thus, being h 0, h concave and we assumed that h does not go to zero, there must be another endpoint where the geodesic hits the boundary of Ω, which is in contradiction with K = 0 since also in this point K (h 2 must go to zero but instead h tends to some nonzero value. Hence, K must be positive and also in this case we are dealing with a warped product of a quotient of S n1 on an interval of. esuming, in the non product situation, every connected piece of Ω is a warped product of a quotient of the sphere S n1 on some intervals of. Then, we can conclude that the universal cover ( M, g can be recovered gluing together, along constant curvature spheres, warped product pieces that can be topological caps (when h goes to zero at an endpoint and cylinders. Nontrivial quotients (M, g of ( M, g are actually possible only when there are no caps in this gluing procedure. In such case, by its concavity, the function h must be constant along every piece of geodesic and the manifold ( M, g is a iemannian product. If there is at least one cap, the whole manifold is a warped product of S n1 on an interval of. emark 2.2. We do not know if the condition on (M n, g to be a gradient LCF icci soliton is actually necessary to have locally a warped product. We conjecture that such conclusion should hold also for nongradient LCF icci solitons. If n 4, any n dimensional, LCF gradient icci soliton with nonnegative icci tensor is either a quotient of n and S n with their canonical metrics, or a quotient of S n1 or it is a warped product of S n1 on a proper interval of.

17 THE EVOLUTION OF THE WEYL TENSO UNDE THE ICCI FLOW The Classification of Steady and Shrinking Gradient LCF icci Solitons. The class of solitons with nonnegative icci tensor is particularly interesting as it includes all the shrinking and steady icci solitons. Indeed, by the same arguments of [32] (keeping in mind, in following the proof of the main Proposition 3.2 in such paper, that the nonnegativity of the scalar curvature for every complete, ancient icci was proved in [8, Corollary 2.5], where the author generalizes the well known Hamilton Ivey curvature estimate to locally conformally flat, gradient, shrinking icci solitons (Corollary 3.3 in the same paper [32], it follows that actually every complete ancient solution g(t to the icci flow whose Weyl tensor is identically zero for all times, is forced to have nonnegative curvature operator for every time t. In particular, this holds for any complete, steady or shrinking icci soliton (even if not gradient as they generate self similar ancient solutions of icci flow. By the previous discussion and the analysis of Bryant in the steady case [5] (see also [9, Chapter 1, Section 4] showing that there exists a unique (up to dilation of the metric nonflat, steady, gradient icci soliton which is a warped product of S n1 on a halfline of, called Bryant soliton, we get the following classification. Proposition 2.3. The steady, gradient, LCF icci solitons of dimension n 4 are given by the quotients of n and the Bryant soliton. This classification result, including also the three dimensional LCF case, was first obtained recently by H.-D. Cao and Q. Chen [6]. In the shrinking case, the analysis of Kotschwar [22] of rotationally invariant shrinking, gradient icci solitons gives the following classification where the Gaussian soliton is defined as the flat n with a potential function f = α x 2 /2n, for a constant α. Proposition 2.4. The shrinking, gradient, LCF icci solitons of dimension n 4 are given by the quotients of S n, the Gaussian solitons with α > 0 and quotients of S n1. This classification of shrinking, gradient, LCF icci solitons follows by the works of L. Ni and N. Wallach [26], P. Petersen and W. Wylie [29] and Z.-H. Zhang [32]. Several other authors contributed to the subject, including X. Cao, B. Wang and Z. Zhang [7], B.- L. Chen [8], M. Fernández López and E. García ío [13], M. Eminenti, G. La Nave and C. Mantegazza [12], O. Munteanu and N. Sesum [24] and again P. Petersen and W. Wilye [28]. We show now that every complete, warped, LCF icci soliton with nonnegative icci tensor is actually a gradient soliton. Proving our conjecture in emark 2.2 that every icci soliton is locally a warped product would then lead to have a general classification of also nongradient icci solitons, in the steady and shrinking cases. emark 2.5. In the compact case, the fact that every icci soliton is actually a gradient is a consequence of the work of Perelman [27]. Naber [25] showed that it is true also for shrinking icci solitons with bounded curvature. For examples of nongradient icci solitons see Baird and Danielo [1]. Proposition 2.6. Let (M n, g be a complete, warped, LCF icci soliton with nonnegative icci tensor, then it is a gradient icci soliton with a potential function f : M n depending only on the t variable of the warping interval. Proof. We assume that (M n, g is globally described by M n = I L K and g = dt 2 h 2 (tσ K, where I is an interval of or S 1 and (L K, σ K is a complete space of constant curvature K. In the case h is constant, which clearly follows if I = S 1, as h 0 the conclusion is trivial. We deal then with the case where h : I is zero at some point, let us say h(0 = 0 and I = [0,, (if the interval I is bounded the manifold M n is compact and we are done. Then,

18 18 GIOVANNI CATINO AND CALO MANTEGAZZA L K = S n1 with its constant curvature metric σ K. As a consequence, we have M n = n, simply connected. We consider the form ω satisfying the structural equation γβ 1 2 ( γω β β ω γ = α n g γβ, If ϕ : S n1 S n1 is an isometry of the standard sphere, the associated map φ : M n M n given by φ(t, p = (t, ϕ(p is also an isometry, moreover, by the warped structure of M n we have that the 1 form φ ω also satisfies γβ 1 [ ( φ ω γβ ( φ ] α ω βγ = 2 n g γβ, Calling I the Lie group of isometries of S n1 and ξ the Haar unit measure associated to it, we define the following 1 form θ = φ ω dξ(ϕ. By the linearity of the structural equation, we have I γβ 1 2 ( γθ β β θ γ = α n g γβ, moreover, by construction, we have L X θ = 0 for every vector field X on M n which is a generator of an isometry φ of M n as above (in other words, θ depends only on t. Computing in normal coordinates on S n1, we get Hence, i θ j = θ( j i = Γ t ijθ t = hh σ K ij θ t, i θ t = θ( t i = Γ j ti θ j = h h θ i. α n = tt t θ t = (n 1 h h tθ t, 0 = i θ t t θ i = t θ i 2 h h θ i, α n gk ij = ij 1 2 ( iθ j j θ i = ( (n 2(K (h 2 hh hh θ t g K ij. It is possible to see that, by construction, actually θ i = 0 for every i at every point, but it is easier to consider directly the 1 form σ = θ t dt on M n and checking that it also satisfies these three equations as θ, hence the structural equation γβ 1 2 ( γσ β β σ γ = α n g γβ. It is now immediate to see that, dσ it = i σ t t σ i = 0 and dσ ij = i σ j j σ i = 0, so the form σ is closed and being M n simply connected, there exists a smooth function f : M such that df = σ, thus γβ 2 γβf = α n g γβ, that is, the soliton is a gradient soliton. It is also immediate to see that the function f depends only on t I. In the expanding, noncompact case (in the compact case the soliton can be only a quotient of the hyperbolic space H n, if the icci tensor is nonnegative and (M n, g is a gradient soliton, then either it is a warped product of S n1 (and M n = n or it is the product of with a constant curvature space, but this last case is possible only if the soliton is the Gaussian expanding icci soliton, α < 0, on the flat n. For a discussion of the expanding icci solitons which are warped products of S n1 see [9, Chapter 1, Section 5], where the authors compute, for instance, an example with positive icci tensor (analogous to the Bryant soliton.

19 THE EVOLUTION OF THE WEYL TENSO UNDE THE ICCI FLOW 19 To our knowledge, the complete classification of complete, expanding, gradient, LCF icci solitons is an open problem, even if they are rotationally symmetric. 3. SINGULAITIES OF ICCI FLOW WITH BOUNDED WEYL TENSO Let (M n, g(t be a icci flow with M n compact on the maximal interval [0, T, with T <. Hamilton proved that max m (, t M as t T. We say that the solution has a Type I singularity if max (T t m (p, t <, M [0,T otherwise we say that the solution develops a Type IIa singularity. By Hamilton s procedure in [19], one can choose a sequence of points p i M n and times t i T such that, dilating the flow around these points in space and time, such sequence of rescaled icci flows (using Hamilton Cheeger Gromov compactness theorem in [18] and Perelman s injectivity radius estimate in [27] converges to a complete maximal icci flow (M, g (t in an interval t (, b where 0 < b. Moreover, in the case of a Type I singularity, we have 0 < b <, m (p, 0 = 1 for some point p M and m (p, t 1 for every t 0 and p M. In the case of a Type IIa singularity, b =, m (p, 0 = 1 for some point p M and m (p, t 1 for every t and p M. These ancient limit flows were called by Hamilton singularity models. We want now to discuss them in the special case of a icci flow with uniformly bounded Weyl tensor (or with a blow up rate of the Weyl tensor which is of lower order than the one of the icci tensor. The icci flow under this condition is investigated also in [23]. Clearly, any limit flow consists of LCF manifolds, hence, by Corollary 1.2 and the cited results of Chen [8] and Zhang [32] at every time and every point the manifold has nonnegative curvature operator and either the icci tensor is proportional to the metric or it has an eigenvalue of multiplicity (n 1 and another of multiplicity 1. We follow now the argument in the proof of Theorem 1.1 in [6]. We recall the following splitting result (see [10, Chapter 7, Section 3] which is a consequence of Hamilton s strong maximum principle for systems in [16]. Theorem 3.1. Let (M n, g(t, t (0, T be a simply connected complete icci flow with nonnegative curvature operator. Then, for every t (0, T we have that (M n, g(t is isometric to the product of the following factors, (1 the Euclidean space, (2 an irreducible nonflat compact Einstein symmetric space with nonnegative curvature operator and positive scalar curvature, (3 a complete iemannian manifold with positive curvature operator, (4 a complete Kähler manifold with positive curvature operator on real (1, 1 forms. Since we are in the LCF case, every Einstein factor above must be a sphere (the scalar curvature is positive. The Kähler factors can be excluded as the following relation holds for Kähler manifolds of complex dimension m > 1 at every point (see [2, Proposition 2.68] W 2 3(m 1 m(m 1(2m 1 2. Thus, any Kähler factor would have zero scalar curvature, hence would be flat. Finally, by the structure of the icci tensor and the fact that these limit flows are nonflat, it is easy to see that only a single Euclidean factor of dimension one is admissible, moreover, in this case there is only another factor S n1.

20 20 GIOVANNI CATINO AND CALO MANTEGAZZA In conclusion, passing to the universal cover, the possible limit flows are quotients of S n1 or have a positive curvature operator. Proposition 3.2 (LCF Type I singularity models. Let (M n, g(t, for t [0, T, be a compact smooth solution to the icci flow with uniformly bounded Weyl tensor. If g(t develops a Type I singularity, then there are two possibilities: (1 M n is diffeomorphic to a quotient of S n and the solution to the normalized icci flow converges to a constant positive curvature metric. In this case the singularity model must be a shrinking compact icci soliton by a result of Sesum [30], hence by the analysis in the previous section, a quotient of S n (this also follows by the work of Böhm and Wilking [3]. (2 There exists a sequence of rescalings which converges to the flow of a quotient of S n1. Proof. By the previous discussion, either the curvature operator is positive at every time or the limit flow is a quotient of S n1. Hence, we assume that every manifold in the limit flow has positive curvature operator. The family of metrics g (t is a complete, nonflat, LCF, ancient solution with uniformly bounded positive curvature operator which is k non collapsed at all scales (hence a k solution in the sense of [27]. By a result of Perelman in [27], we can find a sequence of times t i such that a sequence of suitable dilations of g (t i converges to a nonflat, gradient, shrinking, LCF icci soliton. Hence, we can find an analogous sequence for the original flow. By the classification in the previous section, the thesis of the proposition follows. emark 3.3. Notice that in case (2 we are not claiming that every Type I singularity model is a gradient shrinking icci soliton. This problem is open also in the LCF situation. Proposition 3.4 (LCF Type IIa singularity models. Let (M n, g(t, for t [0, T, be a compact smooth solution to the icci flow with uniformly bounded Weyl tensor. If the flow develops a Type IIa singularity, then there exists a sequence of dilations which converges to the Bryant soliton. Proof. As we said, if the curvature operator gets some zero eigenvalue, the limit flow is a quotient of S n1 which cannot be a steady soliton as it is not eternal. Hence, the curvature operator is positive. By Hamilton s work [17], any Type IIa singularity model with nonnegative curvature operator and positive icci tensor is a steady, nonflat, gradient icci soliton. Since in our case such soliton is also LCF, by the analysis of the previous section, it must be the Bryant soliton. Acknowledgments. We thank Peter Petersen and Gérard Besson for several valuable suggestions. We also wish to thank Fabrizio Bracci, Alessandro Cameli and Paolo Dell Anna for several interesting comments on earlier versions of the paper. The authors are partially supported by the Italian project FIB IDEAS Analysis and Beyond. The second author is partially supported by the Italian GNAMPA (INdAM Group. EFEENCES 1. P. Baird and L. Danielo, Three dimensional icci solitons which project to surfaces, J. eine Angew. Math. 608 (2007, A. L. Besse, Einstein manifolds, Springer Verlag, Berlin, C. Böhm and B. Wilking, Manifolds with positive curvature operators are space forms, Ann. of Math. (2 167 (2008, no. 3, M. Brozos-Vázquez, E. García-ío, and. Vázquez-Lorenzo, Some remarks on locally conformally flat static space-times, J. Math. Phys. 46 (2005, no. 2, , L. Bryant, Local existence of gradient icci solitons, Unpublished work, H.-D. Cao and Q. Chen, On locally conformally flat gradient steady icci solitons, Trans. Amer. Math. Soc. 364 (2012, X. Cao, B. Wang, and Z. Zhang, On locally conformally flat gradient shrinking icci solitons, Commun. Contemp. Math. 13 (2011, no. 2,

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