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APPEARED IN BULLETIN OF THE

arXiv:math/9210223v1 [math.DG] 1 Oct 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 27, Number 2, October 1992, Pages 288-291NEGATIVELY RICCI CURVED MANIFOLDSJoachim LohkampAbstract. In this paper we announce the following result:“Every manifold ofdimension ≥3 admits a complete negatively Ricci curved metric.” Furthermore wedescribe some sharper results and sketch proofs.I.

IntroductionThere are three well-known curvatures: sectional, Ricci, and scalar curvature.While the existence of complete negatively sectional curved metrics leads to manytopological implications for the underlying manifold, each manifold admits a com-plete metric of constant negative scalar curvature.Since Ricci curvature takes position between these two curvatures, it is reason-able to look for obstructions as well as for existence results for (complete) negativelyRicci curved metrics.This led to some long-standing conjectures in Riemannian geometry and appearsin lists of problems compiled by Yau [Y], problem 24, Kazdan [K], problem 9,Bourguignon [Bg], question 4.11, and others.A first notable result was proved by Gao and Yau (cf. [G, GY]).

They startedfrom Thurston’s Hyberbolic Dehn Surgery [T]: Every compact three-manifold Mcan be obtained from S3 by Dehn Surgery, along some link LM ⊂S3, whichcomplement S3\LM, admits a complete hyperbolic metric with finite volume. Gaoand Yau managed to perform this Dehn Surgery such that the Ricci curvature r(g)remains negative near LM and is just the hyperbolic metric outside, i.e., one gets ametric with r(g) < 0 on each compact three-manifold.

Finally they extended thisresult to three-manifolds of finite type (with complete metric).Subsequently, there was a paper, written by Brooks [Br], that addressed gen-eral existence theorems. He develops a technique of smoothing hyperbolic orbifoldsingularities of order K ≥12, which could be locally realized as the quotient of hy-perbolic space by an element of order K fixing a codimension 2 manifold N (whichis hyperbolic if dim N ≥2).On the other hand, Thurston theory yields such higher singular hyperbolic orb-ifold metrics on each compact three-manifold and Brooks combines these ideas toget: Every compact three-manifold admits a metric g with −a < r(g) < −b forconstants a > b > 0, which are independent of the chosen manifold.In contrast to that we will develop a completely different method of attack.

Start-ing from an almost arbitrary metric, we obtain our result by local deformations,which will be sketched in part III of this paper.1991 Mathematics Subject Classification. Primary 53C20, 57R99.Received by the editors November 6, 1991c⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2JOACHIM LOHKAMPII. New resultsTheorem 1.

There are constants a(n) > b(n) > 0 depending only on the dimensionn ≥3, such that each manifold M n admits a complete metric g with−a(n) < r(g) < −b(n) .Notice that even for n = 3 we give a proof that does not use Thurston theory.Bochner proved a classical result: The isometry group of a compact manifoldwith r(g) < 0 is finite. We will show that this “geometric restriction” is sharp.Theorem 2.

If M n, n ≥3, is a compact manifold and G is a subgroup of Diff(M),thenG = Isom(M, g)for some metric g with r(g) < 0 ⇔G is finite .Next, there are unexpected density and cut-offproperties of negatively Ricci-curved metrics.Gromov [Gr] introduced the so-called Hausdorffdistance dH between two metricspaces M1, M2, which can be viewed as the minimal distance or deviation betweenM1 and M2 for all possible isometric embeddings in any metric space M.Ofcourse, Riemannian manifolds can also be considered as metric spaces, and we getfor compact M n, n ≥3,Theorem 3. {(M n, g)|r(g) < 0} is dense in {(M n, g)|g arbitrary metric} withrespect to dH.Our final result is the most flexible one.It is a useful tool for constructingnegatively curved metrics.Theorem 4.

Let S ⊂M n, n ≥3, be a closed subset, S ⊂U an open neighborhood,and g0 a metric on S with r(g0) ≤0 (resp. r(g0) < 0) on U.

Then there is metricg on M with r(g) ≤0 (resp. r(g) < 0) on M, and g ≡g0 on S.Remark.

Theorem 1 implies that each manifold admits a complete metric withscalar curvature bounded by two negative constants.In this case, the Yamabeequation can be solved easily, and we get a complete metric of constant negativescalar curvature. This was proved before by Aubin [A] and Bland and Kalka [BK]in a different way.III.

Some ideas of the proofThe proof is by induction (with respect to dimension) and consists of two mainparts:(1) There is a metric g−n on Rn with r(g−n ) < 0 on B1(0) and g−n ≡gEucl. onRn\B1(0).

(2) Using this metric g−n a Besicovitch type covering argument yields the theo-rems for n-dimensional manifolds.In part (1) there are two cases where the proofs are completely different: n = 3and n ≥4.The case n = 3 starts from a metric g on R3 # S1 × S2, which is isometric to(R3\B1(0), gEucl.) outside a set U, with r(g) < 0 on U.

By some warped product

NEGATIVELY RICCI CURVED MANIFOLDS3trick, which also allows to extend Brooks results to general hyperbolic singularities(cf. part I.

), we obtain g−3 . In this short note we cannot go into further details.Instead (for the moment) we will use Gao and Yau’s result: S3 admits a metricwith r(g).

So we easily obtain g−3 by taking connected sums S3 # R3 = R3.The proof of case n ≥4 is more complicated: The main step consists in theconstruction of a metric g(n) on S1 × B2(0) ⊂S1 × Rn−1 with r(g(n)) < 0 onS1 × B1(0) and g(n) = gS1 + gEucl. outside.

To get this, one embeds S1 × S1 ×Bn−25(0) ⊂S1 ×S1×Rn−2 into S1 ×B1/2(0) in some special way and uses Theorem4 in dimension n −1 for S1 × Bn−25(0) and S1 × Bn−25(0) to construct two metricsthe combination of which yields (after some minor manipulations) g(n). Finally,one embeds S1 × B3(0) into Rn as tubular neighborhood of a large circle.

Somenot-too-hard deformation argument yields g−n .To give an idea of part (2), we sketch the existence proof for negatively Riccicurved metrics in a very special case: the n-dimensional flat torus T n = S1×· · ·×S1,each factor with length 2π·100. For each ̺ ∈]0, 1[ there is a discrete subset A̺ ⊂T nwith(i) d(a, b) > 5 · ̺ for a ̸= b ∈A̺;(ii) T n = Sa∈A̺ B5·̺(a);(iii) #{a ∈A̺|z ∈B10·̺(a)} ≤c(n), c(n) independent of z ∈T n and ̺ ∈]0, 1[.We define gA̺ := ̺2 · f ∗a,̺(g−n ) on B2·̺(a), a ∈A̺, and gA̺ ≡gT n elsewherewith fa,̺ : B2·̺(a) →Rn, fa,̺ ≡(1/̺) · Ia ◦exp−1a , where Ia : TaT n →Rn issome linear isometry (gA̺ depends on the choice of Ia; for Theorem 2 one choosesI−1f(a) ◦Ia ≡Dfa for each f ∈G ⊂Diff(M)).

We are now ready to consider thefollowing metric:g(A̺, d, s) :=Ya∈A̺exp(2F ̺d,s · h̺(10 · ̺ −d(a, idT n))) · gA̺where F ̺d,s ≡s · exp(−d · ̺/ idR), h̺ ≡h((1/̺) · idR) with h ∈C∞(R, [0, 1]), h ≡0on R≤1/2, h ≡1 on R≥3/4, and d(·, ·) is the usual distance on T n.Then there are d0, s0 > 0 independent of ̺ such that r(g(A̺, d, s)) < 0 for eachd > d0, s ∈]0, s0[. Of course, for the flat torus T n, ̺ is a useless parameter, butthink of the general case: A Besicovitch type argument gives similar coverings foreach manifold M n (with c(n) independent of M n) if a suitable start metric is chosen.If ̺ is chosen very small, the background metric on the given manifold appearsquite “flat relative to B̺(a)”.

Again, there are ̺, d, s such that an analogouslydefined metric g(A̺, d, s) is negatively Ricci curved, but ̺, d, and s are no longerindependent.AcknowledgmentThis paper is an abstract of the author’s doctoral thesis. The author thanksProfessor J¨urgen Jost for his friendly support.References[A]T. Aubin, M´etriques riemanniennes et courbure, J. Differential Geom.

4 (1970), 383–424.[BK]J. Bland and M. Kalka, Negative scalar curvature metrics on noncompact manifolds, Trans.Amer.

Math. Soc.

316 (1989), 433–446.

4JOACHIM LOHKAMP[Bg]J. P. Bourguignon, Ricci curvature and Einstein metrics, Global Differential Geometry,Lecture Notes in Math., vol. 838, Springer, New York, 1981, pp.

42–63.[Br1]R. Brooks, A construction of metrics of negative Ricci curvature, J. Differential Geom.

29(1989), 85–94.[G]L. Z Gao, The construction of negatively Ricci curved manifolds, Math.

Ann. 271 (1985),185–208.[GY]L.

Z. Gao and S. T. Yau, The existence of negatively Ricci curved on three manifolds,Invent. Math.

85 (1986), 637–652.[Gr]M. Gromov, Structures m´etriques pour les vari´et´es riemanniennes, Editions CEDIC, Paris,1981.[K]J.

L. Kazdan, Prescribing the curvature of a Riemannian manifold, CBMS Regional Conf.Ser. in Math., vol.

57, Conf. Board Math.

Sci., Washington, DC, 1985.[T]W. Thurston, The geometry and topology of three manifolds, Princeton Lecture Notes, vol.57, Princeton, NJ, 1980.[Y]S.

T. Yau, Seminar on differential geometry, problem section, Ann. of Math.

Stud., vol.102, Princeton Univ. Press, Princeton, NJ, 1982.Ruhr-Universit¨at Bochum, Mathematisches Institut, D-4630 Bochum, Germany

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