We show that any locally conformally flat ancient solution to the Ricci flow
must be rotationally symmetric. As a by-product, we prove that any locally
conformally flat Ricci soliton is a gradient soliton in the shrinking and
steady cases as well as in the expanding case, provided the soliton has
nonnegative curvature.Comment: Final version, to appear on Anal. PD

Catino, Giovanni

Mantegazza, Carlo

Mazzieri, Lorenzo

English

arXiv

ABSTRACT. We show that any locally conformally flat ancient solution to the Ricci flow must be rotationally symmetric. As a by–product, we prove that any locally conformally flat Ricci soliton is a gradient soliton in the shrinking and steady cases as well as in the expanding case, provided the soliton has nonnegative curvature. Key Words: Ricci solitons, Ricci flo

Giovanni Catino

Carlo Mantegazza

Lorenzo Mazzieri

CiteSeerX

LOCALLY CONFORMALLY FLAT ANCIENT RICCI FLOWS

We discuss a gap in Besse’s book (Einstein manifolds, 2008), recently pointed out by Merton in (Proc Am Math Soc 141:3265–3273, 2013), which concerns the classification of Riemannian manifolds admitting a Codazzi tensors with exactly two distinct eigenvalues. For such manifolds, we prove a structure theorem, without adding extra hypotheses and then we conclude with some application of this theory to the classification of three-dimensional gradient Ricci solitons

CATINO, GIOVANNI

MANTEGAZZA, Carlo Maria

MAZZIERI, LORENZO

Archivio istituzionale della Ricerca - Scuola Normale Superiore

arXiv:1308.2374v5  [math.DG]  11 Feb 2015 DraftLOCALLY CONFORMALLY FLAT ANCIENT RICCI FLOWSGIOVANNI CATINO, CARLO MANTEGAZZA, AND LORENZO MAZZIERIABSTRACT. We show that any locally conformally flat ancient solution to the Ricci flow mustbe rotationally symmetric. As a by–product, we prove that any locally conformally flat Riccisoliton is a gradient soliton in the shrinking and steady cases as well as in the expanding case,provided the soliton has nonnegative curvature.Key Words: Ricci solitons, Ricci flowAMS subject classification: 53C24, 53C25, 53C441. INTRODUCTIONIn this paper, we study ancient solutions to the Ricci flow. We recall that a time–dependentmetric g(t) on a Riemannian manifold M is a solution to the Ricci flow if it evolves by theequation∂∂tg(t) = −2Ricg(t) .A solution is called ancient if it is defined for every negative time. Ancient solutions typi-cally arise as the limit of a sequence of suitable blow–ups as the time approaches a singulartime for the Ricci flow. In dimension two there exists a compact, rotationally symmetric,ancient solution due to King [22], Rosenau [32] and Fateev, Onofri and Zamolodchikov [18].In dimension three Perelman [30] constructed a compact, rotationally symmetric, ancientsolution on the three sphere. In the non–rotationally symmetric case, the first constructionis due to Fateev [17] in dimension three. Motivated by this construction, Bakas, Kong andNi [4] produced high dimensional compact ancient solutions to the Ricci flow which are notrotationally symmetric.In dimension two, Daskalopoulos, Hamilton and Sesum [15] have obtained a completeclassification of all compact ancient solutions to the Ricci flow. Ni [27] showed that any com-pact ancient solution to the Ricci flow which is of type I, is k–noncollapsed, and has positivecurvature operator has constant sectional curvature. In [7] Brendle, Huisken and Sinestrariproved that any compact ancient solution which satisfies a suitable pinching condition musthave constant sectional curvature.In this article, we show that any complete ancient solution to the Ricci flow in dimensionn ≥ 4 which is locally conformally flat along the flow must be rotationally symmetric.Theorem 1.1. Let (Mn, g(t)), n ≥ 4, be a complete ancient solution to the Ricci flow which is locallyconformally flat at every time. Then (Mn, g(t)) is rotationally symmetric.The non–rotationally symmetric examples of Bakas, Kong and Ni show that the locallyconformally flatness assumption cannot be removed. The proof of Theorem 1.1 relies on aprevious work of the first two authors [12] about the behavior of the Weyl tensor under theDate: February 12, 2015.1 Draft2 GIOVANNI CATINO, CARLO MANTEGAZZA, AND LORENZO MAZZIERIRicci flow, combined with a more recent result [13] concerning the classification of Riemann-ian manifolds admitting a Codazzi tensor with exactly two distinct eigenvalues.As a consequence of Theorem 1.1, we classify locally conformally flat Ricci solitons. Werecall that a complete Riemannian manifold (Mn, g) is a Ricci soliton if there exists a vectorfield X on Mn such thatRic +12LXg = λ gfor some constant λ. The Ricci soliton is called shrinking if λ > 0, steady if λ = 0 and ex-panding if λ < 0. If X = ∇f for some smooth function f , then the soliton is called a gradientRicci soliton. It follows from the work of Perelman [29] (see [16], for instance, for a directproof) that any compact Ricci soliton is actually a gradient soliton. Moreover, Naber [26]has showed that any shrinking Ricci soliton with bounded curvature has a gradient solitonstructure. On the other hand, steady and expanding Ricci solitons which do not support agradient structure were found by Lauret [24], Baird and Danielo [3], Lott [25] and Baird [2].In this article we prove the following result.Theorem 1.2. Let (Mn, g,X), n ≥ 4, be a complete, locally conformally flat, shrinking or steadyRicci soliton. Then, it is a gradient Ricci soliton. The conclusion still holds in the expanding case,provided the soliton has nonnegative curvature operator.In particular, from the classification results of locally conformally flat gradient Ricci soli-tons in the shrinking case [11, 16, 28, 31, 33], in the steady case [10, 12], as well as in theexpanding case [12], we obtain the following corollaries.Corollary 1.3. Let (Mn, g,X), n ≥ 4, be a complete, locally conformally flat, shrinking Ricci soliton.Then, it is isometric to a quotient of Sn, R× Sn−1 or Rn.Corollary 1.4. Let (Mn, g,X), n ≥ 4, be a complete, locally conformally flat, steady Ricci soliton.Then, it is isometric to a quotient of Rn or the Bryant soliton.Corollary 1.5. Let (Mn, g,X), n ≥ 4, be a complete, locally conformally flat, expanding Riccisoliton with nonnegative curvature operator. Then, it is a rotationally symmetric gradient expandingRicci soliton.We notice that, rotationally symmetric gradient expanding Ricci solitons were constructedin [8, 9, 19].2. NOTATIONS AND PRELIMINARIESThe Riemann curvature operator of a Riemannian manifold (Mn, g) is defined as in [20]byRiem(X,Y )Z = ∇Y ∇XZ −∇X∇Y Z +∇[X,Y ]Z .In a local coordinate system the components of the (3, 1)–Riemann curvature tensor are givenby Rlijk∂∂xl= Riem(∂∂xi, ∂∂xj)∂∂xkand we denote by Rijkl = glmRmijk its (4, 0)–version.With this choice, we have that the round sphere Sn has positive curvature, meaning thatRiem(v,w, v, w) = Rijklviwjvkwl > 0, for every couple u and v of non parallel vector fields.The Ricci tensor is obtained by the contraction Rik = gjlRijkl and R = gikRik will denote thescalar curvature. The so called Weyl tensor is then defined by the following decomposition DraftLOCALLY CONFORMALLY FLAT ANCIENT RICCI FLOWS 3formula (see [20, Chapter 3, Section K]) in dimension n ≥ 3,Wijkl = Rijkl +R(n− 1)(n − 2)(gikgjl − gilgjk)−1n− 2(Rikgjl − Rilgjk +Rjlgik − Rjkgil) . (2.1)The Weyl tensor shares the symmetries of the curvature tensor. Moreover, as it can be easilyseen by the formula above, all of its contractions with the metric are zero.In dimension three, W is identically zero on every Riemannian manifold, whereas, whenn ≥ 4, the vanishing of the Weyl tensor is a relevant condition, since it is equivalent to thelocally conformally flatness of (Mn, g). We recall that this latter condition means that aroundevery point p ∈ Mn there exists a smooth function f defined in a open neighborhood Up of p,such that the conformal deformation g̃ of the original metric g defined by g̃ij = efgij is flat.In particular, the Riemann tensor associated to g̃ is zero in Up.We also recall that in dimension n = 3, locally conformally flatness is equivalent to thevanishing of the Cotton tensorCijk = ∇kRij −∇jRik −12(n− 1)(∇kR gij −∇jR gik).By direct computation, we can see that the tensor Cijk satisfies the following symmetriesCijk = −Cikj, Cijk +Cjki +Ckij = 0 , (2.2)moreover it is trace–free in any two indices,gijCijk = gikCijk = gjkCijk = 0 , (2.3)by its skew–symmetry and Schur lemma. We notice that, for n ≥ 4, there holds∇lWijkl = −n− 3n− 2Cijk , (2.4)and we refer the reader to [6] for the detailed computation. It follows from this formula that,in every dimension n ≥ 3, the vanishing of the Cotton tensor is a necessary condition for aRiemannian manifold (Mn, g) to be locally conformally flat. We also notice that the vanishingof the Cotton tensor can be rephrased in terms of the so called Schouten tensorSij = Rij −12(n − 1)R gij ,by saying that S must satisfy the Codazzi equation(∇XS)Y = (∇Y S)X, X, Y ∈ TM .Any symmetric two tensor satisfying this condition is called a Codazzi tensor (see [6, Chap-ter 16] for a general overview on Codazzi tensors). Hence, if (Mn, g), n ≥ 3, is a locallyconformally flat manifold, then the Schouten tensor is a Codazzi tensor.3. PROOF OF THEOREM 1.1Let (Mn, g(t)), n ≥ 4, be a complete ancient solution to the Ricci flow. We assume thatalong the flow, the Weyl tensor remains identically zero. As it was observed in [12], thiscondition implies a strong rigidity on the eigenvalues of the Ricci tensor. More precisely, onehas the following result. Draft4 GIOVANNI CATINO, CARLO MANTEGAZZA, AND LORENZO MAZZIERILemma 3.1 ([12, Corollary 1.2]). Let (Mn, g), n ≥ 4, be a solution to the Ricci flow such that theWeyl tensor remains identically zero at every time. Then, at every point, either the Ricci tensor isproportional to the metric or it has an eigenvalue of multiplicity (n − 1) and another of multiplicity1.By the results in [14] and [33], which generalize the well–known Hamilton–Ivey curvatureestimate, we know that every complete ancient solution g(t) to the Ricci flow whose Weyltensor is identically zero for all times, is forced to have nonnegative curvature operator forevery time t. Moreover, by Hamilton’s strong maximum principle for systems in [21], wehave that either the metric has strictly positive curvature operator or it splits a line. By The-orem 1.167 in [6] a Riemannian product (R ×Nn−1, ds × h) is locally conformally flat if andonly if the manifold (Nn−1, h) has constant curvature, hence, either one of the following pos-sibilities holds: (Mn, g(t)) is flat or it is a quotient of a rescaling of R× Sn−1 or it has positivecurvature operator. Since the first two cases satisfy the conclusion of the theorem, from nowon we assume that (Mn, g(t)) is a complete, locally conformally flat, ancient solution to theRicci flow with positive curvature operator.As we have seen in the previous section, the relation (2.4) implies that the Cotton tensoris identically zero, hence the Schouten tensorSij = Rij −12(n − 1)R gijis a Codazzi tensor. Moreover, from Lemma 3.1, we know that, at every point, either themetric is Einstein or the Ricci tensor (and so the Schouten tensor) has two distinct eigenval-ues of multiplicity 1 and (n− 1), respectively.Now, it was proved by Bando [5] that solutions to the Ricci flow are real analytic. To beprecise, Bando showed that any Ricci flow solutions is real analytic if Mn is compact or ifit is complete with uniformly bounded curvature. This result was recently improved byKotschwar [23], who showed a local version of Bando’s result. It follows that if the metricis Einstein in some open subset of Mn, then it is Einstein everywhere, and by conformallyflatness and positivity of the curvature the manifold (Mn, g) must be isometric to a quotientof Sn. Thus, either (Mn, g(t)) has constant positive sectional curvature or the Schouten ten-sor has an eigenvalue of multiplicity 1 and a different one of multiplicity (n − 1) at everypoint of some open dense subset U of Mn. In the latter case, we apply to such open set Uthe following classification result of Riemannian manifolds admitting a Codazzi tensor withtwo distinct eigenvalues.Lemma 3.2 ([13, Theorem 2.1]). Let T be a Codazzi tensor on the Riemannian manifold (U, g),with n ≥ 3. Suppose that at every point of U , the tensor T has exactly two distinct eigenvalues ρ andσ of multiplicity 1 and n− 1, respectively. Finally, we let W = {p ∈ U∣∣ dσ(p) 6= 0}. Then, we havethat(1) The closed set W = W ∪ ∂W with the metric g|W is locally isometric to the warped productof some (n − 1)–dimensional Riemannian manifold on an interval of R and σ is constantalong the ”leaves” of the warped product.(2) The boundary of W , if present, is given by the disjoint union of connected totally geodesichypersurfaces where σ is constant.(3) Each connected component of the complement of W in U , if present, has σ constant and it isfoliated by totally geodesic hypersurfaces. DraftLOCALLY CONFORMALLY FLAT ANCIENT RICCI FLOWS 5We are going to to show that, under our assumptions, case (3) cannot occurs and W = U .In fact, if a connected component of the complement of W in U is present, actually, the proofof this lemma (see also [6, Proposition 16.11]) shows that the totally geodesic foliation of(U, g(t)) is integrable. Then, since the manifold (U, g(t)) has nonnegative sectional curvature,it follows from [1, Corollary 2], that such component must split a flat factor. This is clearly incontradiction with the positivity of the curvature, hence it must be U = W in this lemma and(U, g(t)) is locally a warped product of some (n− 1)–dimensional manifold on an interval ofR.Since (U, g(t)) is locally conformally flat with positive curvature operator, we have that the(n − 1)–dimensional fibers of the warped product are isometric to Sn−1 and the metric isrotationally symmetric. By the density of U in Mn, this conclusion clearly holds for thewhole (Mn, g). This concludes the proof of Theorem 1.1.Remark 3.3. We would like to notice that, the same argument shows that the conclusion ofTheorem 1.1 still holds if one consider a Ricci flow solution (Mn, g(t)), n ≥ 4, defined onsome time interval I ⊆ R, which is locally conformally flat with nonnegative curvatureoperator for every t ∈ I .4. PROOF OF THEOREM 1.2Now we turn our attention to the classification of locally conformally flat Ricci solitons.Let (Mn, g,X) be a complete, locally conformally flat shrinking or steady Ricci soliton. Inparticular, it generates a self–similar ancient solution g(t) to Ricci flow (see [34]) which islocally conformally flat at every time t. Hence, Theorem 1.1 implies that the manifold is ro-tationally symmetric with nonnegative curvature operator. As we observed in Remark 3.3,the conclusion still holds if we consider an expanding Ricci soliton with nonnegative curva-ture operator. To prove Theorem 1.2 we then apply the following result.Lemma 4.1 ([12, Proposition 2.6]). Let (Mn, g,X) be a complete, locally warped, locally con-formally flat Ricci soliton with nonnegative Ricci tensor, then it is a gradient Ricci soliton with apotential function f : Mn → R (hence, X = ∇f ) depending only on the r variable of the warpinginterval.Acknowledgments . The authors are partially supported by the Italian projects FIRB–IDEAS“Analysis and Beyond” and GNAMPA projects “Flussi geometrici e soluzioni autosimilari” and“Equazioni di evoluzione geometriche e strutture di tipo Einstein”.REFERENCES1. K. Abe, Applications of a Riccati type differential equation to Riemannian manifolds with totally geodesic distributions,Tohoku Math. J. 25 (1973), 425–444.2. P. Baird, A class of three–dimensional Ricci solitons, Geom. Topol. 13 (2009), 979–1015.3. P. Baird and L. Danielo, Three–dimensional Ricci solitons which project to surfaces, J. Reine Angew. Math. 608(2007), 65–91.4. I. Bakas, S. 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Soc. 137 (2009), no. 8, 2755–2759.(Giovanni Catino) DIPARTIMENTO DI MATEMATICA, POLITECNICO DI MILANO, PIAZZA LEONARDO DAVINCI 32, MILANO, ITALY, 20133E-mail address, G. Catino: giovanni.catino@polimi.it(Carlo Mantegazza) SCUOLA NORMALE SUPERIORE DI PISA, PIAZZA DEI CAVALIERI 7, PISA, ITALY, 56126E-mail address, C. Mantegazza: c.mantegazza@sns.it(Lorenzo Mazzieri) SCUOLA NORMALE SUPERIORE DI PISA, PIAZZA DEI CAVALIERI 7, PISA, ITALY, 56126E-mail address, L. Mazzieri: l.mazzieri@sns.it

Locally conformally flat ancient Ricci flows

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