RIGIDITY OF QUASI-EINSTEIN METRICS

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1 RIGIDITY OF QUASI-EINSTEIN METRICS JEFFREY CASE, YU-JEN SHU, AND GUOFANG WEI Abstract. We call a etric quasi-einstein if the -Bakry-Eery Ricci tensor is a constant ultiple of the etric tensor. This is a generalization of Einstein etrics, which contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein etrics. We study properties of quasi-einstein etrics and prove several rigidity results. We also give a splitting theore for soe Kähler quasi-einstein etrics.. Introduction Einstein etrics and their generalizations are iportant both in atheatics and physics. A particular exaple is fro the study of sooth etric easure spaces. Recall a sooth etric easure space is a triple (M n, g, e f dvol g, where M is a coplete n-diensional Rieannian anifold with etric g, f is a sooth real valued function on M, and dvol g is the Rieannian volue density on M. A natural extension of the Ricci tensor to sooth etric easure spaces is the -Bakry-Eery Ricci tensor (. Ric f = Ric + Hessf df df for 0 <. When f is constant, this is the usual Ricci tensor. We call a triple (M, g, f (a Rieannian anifold (M, g with a function f on M (-quasi-einstein if it satisfies the equation (.2 Ric f = Ric + Hessf df df = λg for soe λ R. This equation is especially interesting in that when = it is exactly the gradient Ricci soliton equation; when is a positive integer, it corresponds to warped product Einstein etrics (see Section 2 for detail; when f is constant, it gives the Einstein equation. We call a quasi-einstein etric trivial when f is constant (the rigid case. Many geoetric and topological properties of anifolds with Ricci curvature bounded below can be extended to anifolds with -Bakry-Eery Ricci tensor bounded fro below when is finite or is infinite and f is bounded, see the survey article [8] and the references there for details. Quasi-Einstein etrics for finite and for = share soe coon properties. It is well-known now that copact solitons with λ 0 are trivial [8]. The sae result is proven in [0] for quasi-einstein etrics on copact anifolds with finite. Copact shrinking Ricci solitons have positive scalar curvature [8, 5]. Here we show Partially supported by NSF grant DMS

2 2 JEFFREY CASE, YU-JEN SHU, AND GUOFANG WEI Proposition.. A quasi-einstein etric with < and λ > 0 has positive scalar curvature. In diension 2 and 3 copact Ricci solitons are trivial [6, 8]. More generally copact shrinking Ricci solitons with zero Weyl tensor are trivial [5, 5, 3]. We prove a siilar result in diension 2 for finite. Theore.2. All 2-diensional quasi-einstein etrics on copact anifolds are trivial. In fact, fro the correspondence with warped product etrics, 2-diensional quasi-einstein etrics (with finite can be classified, see [, Theore 9.9]. (The proof was not published though. In Section 3 we also extend several properties for Ricci solitons ( = to quasi-einstein etrics (general. On the other hand we show Kähler quasi-einstein etrics behave very differently when is finite and is infinite. Theore.3. Let (M n, g be an n-diensional coplete siply-connected Rieannian anifold with a Kähler quasi-einstein etric for finite. Then M = M M 2 is a Rieannian product, and f can be considered as a function of M 2, where M is an (n-2-diensional Einstein anifold with Einstein constant λ, and M 2 is a 2-diensional quasi-einstein anifold. Cobine this with Theore.2 we iediately get Corollary.4. There are no nontrivial Kähler quasi-einstein etrics with finite on copact anifolds. Note that all known exaples of (nontrivial copact shrinking soliton are Kähler, see the survey article [3]. Ricci solitons play a very iportant role in the theory of Ricci flow and are extensively studied recently. Warped product Einstein etrics have considerable interest in physics and any Einstein etrics are constructed in this for, especially on noncopact anifolds []. It was asked in [] whether one could find Einstein etrics with nonconstant warping function on copact anifolds. Fro Corollary.4 only non-kähler ones are possible. Indeed in [2] warped product Einstein etrics are constructed on a class of S 2 bundles over Kähler-Einstein bases warped with S for 2, giving copact nontrivial quasi-einstein etrics for positive integers n 4, 2. When =, there are no nontrivial quasi-einstein etrics on copact anifolds, see Reark 3.5 and Proposition 2.. When n = 3, 2 it reains open. In Section 4 we also give a characterization of quasi-einstein etrics with finite which are Einstein at the sae tie. 2. Warped Product Einstein Metrics In this section we show that when is a positive integer the quasi-einstein etrics (.2 correspond to soe warped product Einstein etrics, ainly due to the work of [0]. Recall that given two Rieannian anifolds (M n, g M, (F, g F and a positive sooth function u on M, the warped product etric on M F is defined by (2.3 g = g M + u 2 g F.

3 RIGIDITY OF QUASI-EINSTEIN METRICS 3 We denote it as M u F. Warped product is very useful in constructing various etrics. When 0 < <, consider u = e f. Then we have Therefore (.2 can be rewritten as u = e f f, u Hess u = Hessf + df df. (2.4 Ric u Hess u = λg. Hence we can use equation (2.4 to study (.2 when is finite and vice verse. Taking trace of (2.4 we have (2.5 u = u (R λn. Since u > 0 this iediately gives the following result which is siilar to the = (soliton case. Proposition 2.. A copact quasi-einstein etric with constant scalar curvature is trivial. In [0] it is shown that a Rieannian anifold (M,g satisfies (2.4 if and only if the warped product etric M u F is Einstein, where F is an -diensional Einstein anifold with Einstein constant µ satisfies µ = u u + ( + λu 2. (In [0] it is only stated for copact Rieannian anifold, while copactness is redundant. Also the Laplacian there and here have different sign. Therefore we have the following nice characterization of the quasi-einstein etrics as the base etrics of warped product Einstein etrics. Theore 2.2. (M, g satisfies the quasi-einstein equation (.2 if and only if the warped product etric M F is Einstein, where F is an -diensional e f Einstein anifold with Einstein constant µ satisfying (2.6 µe 2 f = λ ( f f Forulas and Rigidity for Quasi-Einstein Metrics In this section we generalize the calculations in [4] for Ricci solitons to the etrics satisfying the quasi-einstein equation (.2. Recall the following general forulas, see e.g. [4, Lea 2.] for a proof. Lea 3.. For a function f in a Rieannian anifold (3.7 2(div Hessf( f = 2 f 2 Hessf 2 + Ric( f, f + f, f, (3.8 div f = Ric f + f.

4 4 JEFFREY CASE, YU-JEN SHU, AND GUOFANG WEI The trace for of (.2 (3.9 R + f f 2 = λn, where R is the scalar curvature, will be used later. Using these forulas and the contracted second Bianchi identity (3.0 R = 2 div Ric, we can show the following forulas for quasi-einstein etrics, which generalize soe of the forulas in Section 2 of [4]. Lea 3.2. If Ric f (3. (3.2 = λg, then 2 f 2 = Hessf 2 Ric( f, f + 2 f 2 f, 2 R = Ric( f + (R (n λ f, 2 R f R = tr (Ric (λi Ric (R nλ(r (n λ = Ric 2 (3.3 n Rg + n n(n (R nλ(r n + n λ. Proof. Fro (3.7 we have (3.4 2 f 2 = 2(div Hessf( f + Hessf 2 Ric( f, f f, f. By taking the divergence of (.2, we have (3.5 div Ric + div Hessf f df ( f f = 0, where X is the dual -for of the vector field X. Using (3.0 we get (3.6 2 div Hessf( f = R, f + 2 f f Hessf( f, f. Now taking the covariant derivative of (3.9 yields (3.7 R + f f 2 = 0. Plug this into (3.6 and then plug (3.6 into (3.4 we get 2 f 2 = f f 2, f + 2 f f Hessf( f, f + Hessf 2 Ric( f, f f, f which is (3.. = Hessf 2 Ric( f, f + 2 f 2 f,

5 RIGIDITY OF QUASI-EINSTEIN METRICS 5 For (3.2, using (3.0, (3.5, (3.8, and (3.7, we get R = 2 div Ric = 2 div Hessf + 2 f f + 2 f f = 2Ric( f 2 f + 2 f f + 2 f f = 2Ric( f + 2 R 2 f f f + 2 f f. Solving for R and noting that f 2 = 2 f f, we get Fro (.2, we have that R = 2Ric( f 2 f f + 2 f f. f f = (λ + f 2 f Ric( f, so by using this substitution and (3.9 for f, we arrive at (3.2, 2 R = Ric( f + (R (n λ f. Taking the divergent of the above equation we have (3.8 Now 2 R = div (Ric( f + div ((R (n λ f. div (Ric( f = div Ric, f + tr (Ric Hessf = ( (Ric 2 R, f + tr df df + λg Ric (3.9 = 2 R, f + Ric( f, f + tr (Ric (λg Ric = R, f + ( 2 2 R, f (R (n λ f 2 +tr (Ric (λg Ric, where the last equation coes fro (3.2. Also (3.20 div ((R (n λ f = (R (n λ f + R, f. Plugging (3.9 and (3.20 into (3.8 and using (3.9 we arrive at 2 R f R = Let λ i be the eigenvalues of the Ricci tensor, we get tr (Ric (λi Ric (R nλ(r (n λ. tr (Ric (λi Ric = λ i (λ λ i = Ric 2 n Rg + R (λ n R, which yields (3.3.

6 6 JEFFREY CASE, YU-JEN SHU, AND GUOFANG WEI As in [4], these forulas give iportant inforation about quasi-einstein etrics. Cobining the first equation (3. in Lea 3.2 with the axial principle we have Proposition 3.3. If a copact Rieannian anifold satisfying (.2 and Ric( f, f 2 f 2 f then the function f is constant so it is Einstein. Equation (3.2 gives Proposition 3.4. When, a quasi-einstein etric has constant scalar curvature if and only if Ric( f = (R (n λ f. Reark 3.5. When =, the constant µ in (2.6 is zero, cobining with (3.9, we get R = (n λ. The scalar curvature is always constant. Equation (3.3 gives the following results. Proposition 3.6. If a Rieannian anifold M satisfies (.2 with and a λ > 0 and M is copact then the scalar curvature is bounded below by n(n (3.2 R + n λ. Equality holds if and only if =. b λ = 0, the scalar curvature is constant and >, then M is Ricci flat. c λ < 0 and the scalar curvature is constant, then n(n nλ R + n λ and when >, R equals either of the extree values iff M is Einstein. Reark 3.7. When is finite, a anifold with quasi-einstein etric and λ > 0 is autoatically copact [6]. Reark 3.8. Let =, we recover the well know result [8] that copact shrinking Ricci soliton has positive scalar curvature, and soe results in [4] about gradient Ricci solitons with constant scalar curvature. Proof. a Since M is copact, applying the equation (3.3 to a inial point of R, we have + n n(n (R in nλ(r in n + n λ Ric 2 n Rg 0. So n(n + n λ R in nλ which gives (3.2. b c Since R is constant, fro (3.3 + n n(n (R nλ(r n + n λ = Ric 2 n Rg 0.

7 RIGIDITY OF QUASI-EINSTEIN METRICS 7 So if λ = 0, >, then Ric = nrg and R = 0, thus it is Ricci flat. If λ < 0, R [nλ, n(n +n λ]. 4. Two Diensional Quasi-Einstein Metrics First we recall a characterization of warped product etrics found in [4] (see also [5]. Theore 4. (Cheeger-Colding. A Rieannian anifold (M n, g is a warped product (a, b u N n if and only if there is a nontrivial function h such that Hess h = kg for soe function k : M R. (u = h up to a ultiplicative constant Fro this we can give a characterization of quasi-einstein etrics which are Einstein. Proposition 4.2. A coplete finite quasi-einstein etric (M n, g, u is Einstein if and only if u is constant or M is diffeoorphic to R n with the warped product structure R a e ar N n, where N n is Ricci flat, a is a constant (see below for its value. Proof. If g is Einstein, then Ric = µg for soe constant µ. Fro (2.4 we have (4.22 Hess u = µ λ u g for soe u > 0. If M is copact, then u (thus f is constant. So if u is not constant, then M is noncopact and λ 0, µ 0 and µ > λ. So λ < 0, λ < µ 0 and u is a strictly convex function. Therefore M n is diffeoorphic to R n. By (4.22 and Theore 4., µ λ u = ce r, where c is soe constant. And M is R N n with the warped product etric g = dr 2 + a 2 e 2ar g 0, µ λ where a =. Since g is Einstein we get µ = (n µ λ < 0 and g 0 is Ricci flat. Reark 4.3. The Taub-NUT etric [7] is a Ricci flat etric on R 4 which is not flat. Since a 2-diensional Rieannian anifold satisfies Ric = R 2 g, we get an iediate corollary of Theore 4.. Corollary 4.4. A two diensional quasi-einstein etric 2.4 is a warped product etric. Now we will prove Theore.2. Proof. Since M is copact, by [0], we only need to prove the theore when λ > 0. Fro (3.2 we have (4.23 R 2 + λ. So up to a cover we ay assue M is diffeoorphic to S 2. Since M is 2-diensional, we have Ric = R 2 g. Thus (3.2 becoes (4.24 R = + ( R 2 + λ f.

8 8 JEFFREY CASE, YU-JEN SHU, AND GUOFANG WEI ( Now let u = e f, then fro (2.4 Hess u = u R 2 λ g. In particular, u is conforal. By the Kazdan-Warner identity [9], we have R, u dv = 0. Thus M R, f e f dv = 0. Using (4.24, since R 2 +λ, we get f = R = Kähler Quasi-Einstein Metrics There are any nontrivial exaples of shrinking Kähler-Ricci solitons [2, 7]. In contrast, we will show here that Kähler quasi-einstein etrics with finite are very rigid. Proof of Theore.3 First, since on a Kähler anifold, the etric and Ricci tensor are both copatible with the coplex structure J, fro (2.4 we have Hess u(ju, JV = Hess u(u, V for all vector fields U, V. That iplies (5.25 J X u = JX u, and the φ defined by φ(u, V = Hess u(ju, V is an (,-for. Note that 2Hess u = L u g. By (5.25 L u JX = JL u X, so 2φ = L u ω, where ω is the Kähler for. Since L X d = dl X and ω is closed we have φ is closed. Furtherore, since the Ricci for is closed, fro (2.4 we get that the (,-for φ u is also closed, so du φ = 0. Now (du φ(u, V, W = Uu g( JV u, W + V u g( JW u, U + W u g( JU u, V. Let U, V u, W = u, we have Hess u(ju, V = 0 for all U, V u. Hence (5.26 X u J u for all X u. Let U = u, V = J u, W u we get u u u. Now we consider the 2-diensional distribution T that is spanned by u and J u at those points where u is nonzero. We will show that T = Span{ u, J u} is invariant under parallel transport, i.e. if γ is a path in M, and U is a parallel field along γ, then ( g(u, u u g(u, J uj u (5.27 γ + = 0. Since the covariant derivative is linear in γ, we can prove this in three cases: ( when γ u, J u: so γ u and Jγ u, by (5.26 γ u = 0. Since the coplex structure J is parallel, γ J u = J γ u = 0. By assuption γ U = 0, hence (5.27 follows. (2 when γ = u: Using u u u, we have

9 RIGIDITY OF QUASI-EINSTEIN METRICS 9 γ ( g(u, u u + = γ ( g(u, u = g(u, u u g(u, J uj u g(u, u u + γ u + γ ( g(u, J u u 2g( u, u ug(u, u u 4 u + J u + g(u, u + g(u, uj u J u 2g( u, u ug(u, J u u 4 J u + ( u Hess u( u, u = Hess u(u, u g(u, u + J u ( Hessu( u, u Hess u(ju, u + g(ju, u ( u = Hess u g(u, u U u, u J u Hess u = 0, ( JU g(u, J u γ J u g( u, u u u g(u, J u g(j u, u J u J u g(ju, u u, u where the last equality follows fro (5.26 and that U g(u, u u u u. 2 (3 γ = J u: Using J X u = JX u it reduces to the previous case. Now we have an orthogonal decoposition of the tangent bundle T M = T T that is invariant under parallel transport. By DeRha s decoposition theore on a siply-connected anifold [, Page 87], M is a Rieannian product, and all the clais in the theore follow. Proof of Corollary.4 Since the anifold M is copact, by [0], we can assue λ > 0. Then, by [6], the universal cover M is also copact. Now the result follows fro Theore.3 and.2. References [] A.L. Besse, Einstein anifolds, Springer-Verlag, 987. [2] H. Cao, Existence of gradient Kähler-Ricci solitons, Elliptic and Parabolic Methods in Geoetry, A K Peters, Wellesley (996, pp. 6. [3] Cao, Huai-Dong, Geoetry of Ricci solitons. Chinese Ann. Math. Ser. B 27 (2006, no. 2, 2 42 [4] J. Cheeger, T. Colding, Lower bounds on Ricci curvature and the alost rigidity of warped products. Ann. of Math. (2 44 (996, no., [5] M. Einenti, G. Nave, C. Mantegazza, Ricci solitons-the equation point of view, arxiv:ath/ [6] R. Hailton, The Ricci flow on surfaces, Contep. Math. 7 AMS, Providence RI (988, [7] Hawking, S. W. Gravitational instantons. Phys. Lett. A 60 (977, no. 2, [8] T. Ivey, Ricci soliotns on copact three-anifolds, Diff. Geo. and its Appl. 3 (993, [9] J. Kazdan and F. Warner, Existence and conforal deforation of etrics with prescribed Gaussian and scalar curvature, Ann. of Math., Vol. 0 (975, [0] D.-S. Ki and Y.H Ki, Copact Einstein warped product spaces with nonpositive scalar curvature, Proc. Aer. Math. Soc. 3 ( [] S. Kobayashi and K. Noizu, Foundations of differential geoetry. Vol I. Interscience Publishers, a division of John Wiley & Sons, New York-Lond on 963 xi+329 pp.

10 0 JEFFREY CASE, YU-JEN SHU, AND GUOFANG WEI [2] L, H., Page, Don N., Pope, C. N. New inhoogeneous Einstein etrics on sphere bundles over Einstein-Khler anifolds. Phys. Lett. B 593 (2004, no. -4, [3] L. Ni and N. Wallach, On a classification of the gradient shrinking solitons, arxiv:ath.dg/ [4] P. Peterson and W. Wylie, Rigidity of Ricci Solitons, arxiv:ath.dg/ v. [5] P. Peterson and W. Wylie, On the classification of gradient Ricci solitons, arxiv:ath.dg/ v2. [6] Z. Qian, Estiates for weighted volues and applications, Quart. J. Math. Oxford Ser. (2 48, No. 90 (997 pp [7] X. Wang, X. Zhu, Kähler Ricci solitons on toric anifolds with positive first Chern class, Advances in Matheatics 88, (2004, pp [8] G. Wei and W. Wylie, Coparison Geoetry for the Sooth Metric Measure Spaces, ICCM, E-ail address: casej@ath.ucsb.edu E-ail address: yjshu@ath.ucsb.edu E-ail address: wei@ath.ucsb.edu Departent of Matheatics, UCSB, Santa Barbara, CA 9306