齐性爱度量的存在和不存在性

Invent. math. 84, 177 194 (1986) Irl verl tiorles mathematicae ~ Springer-Verlag 1986 Existence and non-existence of homogeneous Einstein metrics McKenzie Y. Wang 1 and Wolfgang Zil ler 2 . J Department of Mathematical Sciences, Mc Master University, Hamilton, Ontario, Canada, L8S 4K1 2 Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USA In this paper we first prove a general existence theorem for homogeneous Einstein metrics and then we exhibit some compact simply connected homo- geneous spaces which carry no homogeneous Einstein metric. A Riemannian metric g is called Einstein if R ic (g )=c . g for some constant c. If c>0 most known examples are homogeneous, see [Be]. Previous con- structions of homogeneous Einstein metrics were usually achieved by more or less explicit calculations on special families of homogeneous spaces, see, e.g. [Je 1,2], [DZ] , or [WZ 1]. On the other hand, the Einstein metrics of volume 1 on a compact manifold are precisely the critical points of the total scalar curvature functional T(g)= J S(g)dvolg on the space of R iemannian metrics of volume 1. This M suggests a var iat ional approach to finding Einstein metrics, which so far has not been successful. If G/H is a compact homogeneous space, we can restrict T to the subset of G-invariant metrics of volume 1. The critical points of the restriction of T are again precisely the G-invariant Einstein metrics of volume 1. In this paper we examine when T is bounded from above or below and/or proper. In part icular we prove the following Theorem. Let G be a connected compact Lie group and H a connected closed subgroup such that G/H is efJective. Then the functional T on the set of G- invariant metrics with t'olume 1 is bounded .from above and proper if[" H is a maximal connected subgroup of G. For such a G/H, T assumes its global maximum at a G-invariant metric which must be Einstein. By contrast, as we will see in w 2, it seldom happens that T is bounded from above but not proper or that T is bounded from below. * The first author acknowledges partial support from the Mathematical Sciences Research Institute at Berkeley and from the Natural Sciences and Engineering Research Council of Canada. The second author is partially supported by a grant from the Alfred P. Sloan Foundation and a grant from the National Science Foundation 178 M.Y. Wang and W. Ziller Combined with [Dy2], Theorem A yields immediately, without explicit calculation, numerous new homogeneous Einstein manifolds. For example, apart from a short finite list of exceptions, every irreducible representation ~z of a compact simple group H gives rise to a homogeneous Einstein manifold SO(n)/~z(H),SP(2)/~r(H),orSU(n)/~(H),dependingonwhether~isorthogo- hal, symplectic, or non-self-contragredient. Next, we exhibit some compact simply connected homogeneous spaces G/H which carry no G-invariant Einstein metric. These G/H have the property that every G-invariant metric is obtained from a fixed Riemannian submersion K/H--*G/H~G/K by re-scaling the metrics on the fibre and base. Hence the Einstein condition reduces to a quadratic equation in one variable which in some cases has no real roots. It is amusing to note that the trick of re-scaling the fibre in a Riemannian submersion has been frequently used in the past to produce homogeneous Einstein metrics. Our lowest dimensional non-existence example is the 12-dimensional ma- nifold SU(4)/SU(2) where SU(2)cSp(2)cSU(4) and SU(2) is the unique maxi- mal connected subgroup of Sp(2). We will show that no other Lie group acts transitively on SU(4)/SU(2). Hence it carries no homogeneous Einstein metric whatsoever. We do not know if any of these manifolds admit a non-homo- geneous Einstein metric. Recall that R. Hamilton showed [Ha] that if g is a metric of volume l on a compact 3-manifold M with Ric(g)>0, then there is a smooth 1-parameter family of metrics g, of volume 1 with go=g which is a solution of the natural evolution equation /}g?t '=-2 Ric(g,) d imM Moreover, gt converges as t -~ to a smooth Einstein metric, which must have constant sectional curvature since the dimension is 3. Hamilton's equation is a slightly modified version of the equation for the gradient flow of T. gt is a solution of the gradient flow of T if 1 sIg3g,). ~g---t=-2?~t Ric(gt) d imM If d imM>3, short-time existence and uniqueness of solutions of Hamilton's evolution equation still hold. However, our non-existence examples show that in general a solution curve gt will not converge, even if go has positive Ricci curvature and non-negative sectional curvature. This is seen as follows. Since Hamilton's equation is invariant under Diff(M), by uniqueness, Isom(go)clsom(gt) for all t>0 for which g, exists. If GcIsom(g0) acts tran- sitively on M, then gt will also be G-invariant and since S(gt)= T(gt), the flow of Hamilton's equation on the set of G-invariant metrics agrees with the gradient flow of T. Since the set of G-invariant metrics of volume 1 on M is a smooth finite dimensional manifold, a solution curve gt has the property that if tin,: is the maximal time for which g~ exists, then as t---~tma , either g, converges to a G-invariant Einstein metric or g, goes off to oo on the set of G-invariant Existence and non-existence of homogeneous Einstein melrics 179 metrics of volume 1. Hence any solution curve of Hamilton's equation with homogeneous initial metric on one of our non-existence examples will go off to oQ. Moreover, since every compact simply connected homogeneous space car- ries a normal homogeneous metric with positive Ricci curvature and non- negative sectional curvature, we may choose such a metric as initial metric. the same conclusion holds for any differential equation ~=F(gt ) Exactly (necessarily invariant under Diff(M)) that satisfies short time existence and uniqueness, and has the property that all stationary points are Einstein metrics. w 1. The scalar curvature functional Let G be a compact connected Lie group and H a closed subgroup such that G acts effectively on M=G/H, i.e. H contains no nontrivial normal subgroup of G. The assumption of effectiveness is not necessary but is convenient. Hence in giving actual examples of G/H we will not worry about effectiveness. We will also assume that H is connected, which is automatically the case if G/H is simply connected. For a Riemannian metric g on M we denote by S(g) its scalar curvature and by T(g) its total scalar curvature ~ S(g)dvolg. Then, on the set JCL of M Riemannian metrics on M with volume 1, the critical points of T are precisely the Einstein metrics on M, as follows from the first variation formula ([Hi] or [Bg] p. 289) where h is any symmetric 2-tensor on M such that 5( t rgh)dvolg=0 and n = dim M. M Let oggG be the set of G-invariant metrics of volume 1 on M. Note that on ,./t~, T(g)=S(g). The set of critical points of TI,/r are precisely the G- invariant Einstein metrics of volume l on M since at a critical point g of S TI.-/Ya we can choose h to be the G-invariant symmetric 2-tensor -g -R ic (g ) . n Let B be the negative of the Killing form of g. Then B(X,X)>O with equality iff Xe ~(g). We fix once and for all a biinvariant metric Q on g such that the induced normal homogeneous metric ge on G/H has volume 1. Next we consider the Ad(H)-invariant decomposition g=b@m with Q(b, m)=0. Then the set of G-invariant metrics on G/H can be identified with the set of Ad(H)-invariant inner products on m. Let ( , ) be an Ad(H)-invariant inner product on m and {e~} be a basis of m orthonormal with respect to ( , ). Then one has the following formula for the scalar curvature of ( , ) (see e.g. [Be], (7.39) or [Je2], p. 1130): 1 (1.1) S=�89 ~ B(e~, %)--~ 2 ([e, , ei~ ] .... [e~, e~],,) where [ , ],1 denotes the m-component. 180 M.Y. Wang and W. Z i l ler We now examine this formula more closely. Let m=m 1 | | o be a Q-orthogonal Ad(H)-invariant decomposition such that Ad(H) lm o =id and Ad(H)lmi is irreducible for i=1 . . . . . r. Such a decomposition is not unique if some of the representations of Ad(H) on n h are equivalent to each other. But the subspace m 0 and the numbers d~=dimm~ are independent of the chosen decomposition. The structure of ~#a can be described in terms of a fixed decomposition. By Schur's lemma, X{ G is diffeomorphic to the positive definite elements in R" x [ F[ HOmAdn(m i, mj)] x S2(mo) lO. Every Ad(H)-invariant inner product on m belongs to the family of Ad (H)- invariant diagonal metrics of some decomposition of m. This is seen as follows. For a given Ad(H)-invariant inner product ( , > on m, we first diagonalize ( , > with respect to Q to obtain a decomposition of nt into eigenspaces of ( , ), which are orthogonal with respect to both (2 and ( , ). These eigen- spaces are Ad(H)-invariant, and so can be further decomposed into irreduc- ible summands which are orthogonal with respect to Q and ( , ). Then ( , ) has the form (1.2) with respect to this decomposition, where the xi's are the eigenvalues of ( , > with respect to Q. Note that it can happen that the same Ad(H)-invariant inner product can be diagonal with respect to many different decompositions of m. For a fixed Q-orthogonal Ad(H)-invariant decomposition of m, the scalar curvature of the metrics of the form (1.2) has a nice expression. Let {e~} be a Q-orthonormal basis adapted to the decomposition of m, i.e., e~m i for some i, 7 - - ? and eO with bi=O iff m/c~(g), and ij [k ]=-O iff Q([mi, mj], mk)=O. ij (t.4) Remark. If for a fixed decomposition of 111 we have Ricg(mi, my)=0 whenever i@j for all metrics g of the form (1.2) with respect to the decom- position, then the first variation formula for T implies that the critical points for S in (1.3) on the set I:I x~d' = 1 are Einstein metrics on G/H. But in general if i= l m~ and m~ are equivalent Ad(H)-representations, then Ric(m~, tit j) may be non- zero, and hence the critical points of (1.3) will not necessarily be Einstein metrics. Nevertheless (1.3) will be sufficient for us to examine the global behavior of S on Jgc,- [k ] and the In Sect. 3 we will need the following relationship between ij Casimir operator C.,,,el ~ = -~ ad z~ o adz i, where z; is an orthonormal basis of i b with respect to QIb- Since m i is Ad(H)-irreducible C.,,,Qib=c i. Id with ci>0 and ci=O iff micro 0. [ k ] =di(bi - 2ci). (1.5) Lemma. ~, ij j , k P,.ooj: Z ij = y' 2 Q([e~, e~], e f j , k , e=em, ll, y = ~ ~ Q([e~, ets].,, [e~, eel,,) = ~ -tr. ,(Pr.,~ 2 e~em, = ~ (-tr..(ade=)2+2 tr~,(pr~ade.opr,.ade~)) = y' (B(e=, eD+2y'Q([e~, [% zi]], zi) ec~ ~ l rl t i = ~ (B(e=, e.)-2Q(C,.,.ol~(e~), e~) ememl =di(b i -2ci) . q.e.d. 182 M.Y. Wang and W. Ziller Notice that one can use (1.5) to collect terms in (1.3). The coefficient of l/x i in (1.3) is equal to 1 (d i b 1 1 1 k which is >0 and =0 iff m ic ro o and [m i ,mj]cmj for all j. We also need a formula for the scalar curvature of a Riemannian sub- mersion. Let ~: M---,B be a Riemannian submersion with totally geodesic fibres F. Let g denote the metric on M normalized so that vol(g)= 1. Let S(B) and S(F) respectively be the scalar curvature of the base and fibre. On M there is a natural family of metrics: gt=tglvZgJ~, where V and H are respectively the vertical and horizontal distributions. Then 1 S(g,) =t S(F) + S(B) - t I hA 1] 2 where IIAH is the norm of the O'Neill tensor computed with respect to g=gl . (See e.g. [BB], Lemma 14.) If we let f=d imF and n=dimM, then vol(g,) =t f/2' SO that ~,=t-f/"g, has volume 1 and (1.6) S(g"):tf/" ( i S(F)+ S(B) - t ]1AH2). Hence if S (F )>0 then S(~,)~ + ~ as t~0. In applications we will consider compact connected intermediate Lie groups K with H c K c G. Let 9 = i | m b and f = [3 | m I be Q-orthogonal de- compositions. For any Ad(H)-invariant metric ( , ) on m such that (my, rob) =0 and such that ( , ) I ra b is Ad(K)-invariant, the natural projection G/H~G/K becomes a Riemannian submersion with totally geodesic fibres if the metric on base and fibre are given respectively by ( , ) Ira b and ( , ) lm I. (See [BB], Prop. 2.) The induced family of metrics gt clearly lies in J~ . w 2. S bounded on d/~ (2.1) Theorem. S is bounded from below on d/[ G iff the universal cover of G/H is a product of several isotropy irreducible homogeneous spaces and a euclidean space R k, k>O. S is in addition proper iJf k=0. IJ" k=0, S has a unique critical point, which is a product of the unique Einstein metric on each Jactor, and S is bounded from below by a positive constant. If k> 1, S has a critical point iff G/H is a torus. Proof. Assume that S is bounded from below on dg G. Fix a decomposit ion of m. If ij "I=0 for some i,j, k with k~i and k=t=j, then (1.3) implies that S-+ -oc if x i~0, x j~0 and Xk~ + oC at suitable rates preserving the volume. Hence. Existence and non-existence of homogeneous Einstein metrics 183 k]=0 whenever using the symmetry of [ ] in all 3 indices, it follows that ij there are two distinct indices. Thus [m i, m~] =0 if i#:j and [mi, mi] < b �9 mi. If we let bi=[nh, mi]b, then the biinvariance of Q implies that g~=b~| are pairwise Q-orthogonal ideals of .q. The Q-orthogonal complement of �9 gi in g is thus an ideal contained in b, which is 0 by effectiveness. So g= | and the universal cover of G/H is the product tzI GJHi- Note that m o <~(g) and if i=1 tnlcnt0, then b~=0. It follows that b = @I)~ and b~ acts irreducibly on m> i=1 l0. The decomposition m=m 1 O. . .Qtn , Om o is unique since the Ad(H)-representations m> i>0, are pairwise inequivalent. Hence by (1.3) S on JgG has the form 1 r S= ~. dibi>o, 2 i=1 xi which is independent of the metric on m o = a(g). This completes the proof of the first assertion in (2.1). The next two assertions are immediate consequences of the first assertion and the form of S in the preceding paragraph. Finally, if k> 1, S has a critical point iff G/H is a torus since Ric is 0 on m 0 and positive on n h, i>0. q.e.d. Remark. The homogeneous spaces occurring in (2.1) are precisely the compact homogeneous spaces of normal type, i.e., every G-invariant metric on G/H is normal homogeneous, see [BB], Lemma 12. Note that Lemma 13 in [BB] is an immediate consequence of (1.3). (2.2) Theorem. S is bounded J?om above and proper on ~I~ iff H is a maximal connected subgroup of G, or, equivalently, b is a maximal subalgebra of 9. In this case S has a global maximum, which must be a G-invariant Einstein metric on G/H. Proof. Let us first assume that b is maximal in .q, Maximality has the following consequence: there is a constant a>O depending only on G/H such that for any non-empty proper subset l c {l ..... s} and for any decomposition of m [k ]>a. To see this, note first that for any there exist i , je I and k(~I with ij fixed decomposition of m and for any non-empty proper I c {1 . . . . . s} there [k ]>0s inceotherwise bQ~nhwould be a exists an i , j~I, k(~l such that ij i~r proper subalgebra of g properly containing b. Now for each non-empty subset k] where the infimum is taken all decom- /c{ l .... ,s} let a ,= in f~ ij over positions of m and the sum is taken over all i , j ,k with i, j s I and kr By the compactness of the set of decompositions of m and the continuity of the ij ' 184 M.Y. Wang and W. Ziller k] a I we see that a1>0, and hence for some i, je I , kr - - Since there are only finitely many I's, the existence of a follows, ij >=s 3" By the compactness of the set of decompositions of m, we can also find a constant b>0 such that for every decomposition bi~b, where bi is given by Blmi=b~Q[m i. Next we fix constants c~i s.t. 0=0~l<~2<.. .<~s=l and ~+1 >(1 + cq)/2. Let ~=min{2~i+l--C~i--1, l _< i _< s --1} > 0. For each fixed decomposition of m, we consider the family of metrics given by (1.2). If the metric is +Q, we write x~=e '~' with ~ v/Z=l, t>0, and ~ div i i=1 i=1 =0 since the volume is assumed to be 1. For a fixed (v~ .. . . . v~) let vmi n =min {vi}. Then there exists a constant c>0 which depends only on s and {di} i (and hence not on the decomposition) such that Vm~n<--C<0 for every (v 1 .. . . ,vs) satisfying ~ v/2=l and ~ divi=O. i=1 i= l Given (v~ .... , v.~) as above, we sub-divide the interval (Vmln, 0 ] into s--1 intervals (Cq+lVmi ., e~vmi,], i=1 .. . . . s - - l . Since at least one v~ is positive, at least one of these intervals does not contain any v~'s, say (C~o+tVmi ., ~ioVmi.]. Let I={ilv~a. Since {1 . . . . . s}. Hence there exist i , j ,k with i , j~l, kr and i j Vk>eioVmi n, V i, Vj -A implies that/31 (A) -A the eigenvalues of ( , ) with respect to Q lie between/~(A) and /32(A ). But the set of symmetric matrices with bounded eigenvalues is compact. Hence S is bounded from above and proper on ,//g~. Conversely, let G/H be such that S is bounded from above and proper on Jg~. If H is not a maximal connected subgroup, then there exists a connected subgroup K with H~K=G. If K is closed, then by (1.6) and boundedness from above, the metric induced by Q on K/H has zero sca