Complete and

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Proc.Nati. Acad.Sci. USA

Vol. 76, No. 3, pp. 1024-1025,March 1979 Mathematics

Complete manifolds with nonnegative scalar curvature and the positive action conjecture in general relativity

(minimal hypersurface)

RICHARDM. SCHOEN* AND SHING-TUNG YAUt

*DepartmentofMathematics,University ofCalifornia,Berkeley, California 94720; and tDepartment of Mathematics, Stanford University, Stanford, California 94305

CommunicatedbyShiing-shen Chern, November27, 1978

ABSTRACT We find some integrability conditions for low-dimensional manifolds to admit metrics with nonnegative scalar curvature. In particular, we solve the positive action conjecture ingeneralrelativityintheaffirmative.

This is a continuation of our previous paper (1). We generalize most of the results there tohigher-dimensional manifolds. Thus, the basicproblem that we are considering is to find integrability conditions for manifolds to admit metrics with nonnegative scalar curvature.

Forsimplicity, we shall assume that all manifolds that we are going to consider are orientable. We define a class of compact manifolds inductively in the following manner. A three-dimensional manifold M is said to be of class C ifxr1(M)has no subgroup isomorphictothefundamental group of a compact surface of genus>1. Ingeneral, we say that ann-dimensional compact manifold M with n > 4 isof class C if either the A- genusof M (see refs. 2 and 3) is zero or, for every finite covering space of M, any codimensional one homologyclass can be represented byacompacthypersurfaceofclass C.

The basic theorem that we want to announce here is the following:

THEOREM 1. Let M be acompact manifold with dimension

<7 andnotofclass C. Then,unless Misflat, Mdoesnot admit anymetricwith nonnegativescalar curvature.

Inorder to show that there are a lot of manifolds that are not ofclass C, we mention the following class of manifolds. Suppose there is asequence of manifoldsM,

Mn 1,.

.

.,

M4 and M3 sothat eachMihasdimensioniand there is a continuous map from

Mi

ontoMi that pulls the fundamental class

[Mi-I]

back tobea nonzeroclassinHi-l(M2,R).If M3 admits anembedded incompressiblesurface of genus >1,then one can prove that each

Mi

isnot of class C for i > 3. Hence we have the following:

COROLLARY. Theonly metrics withnonnegative scalar curvature onthen-dimensional torus with n<7 aremetrics withzerocurvature.

There is ageneralization of Theorem1 to thenoncompact case. We mentionhere a case that iscalled to be the positive actionconjectureingeneral relativity. Ithas considerableim- portanceinS.Hawking's theoryof quantum gravity.

LetMbe a noncompact n-dimensional manifold with finite number of ends so that each end isdiffeomorphictoRnminus

acompact set. Suppose oneach end, the metric tensor gij ofM can be written as (1 +

m/rn-2)6,j

pluslower-order terms, in whichr istheeuclidean distance function. Then we call mto be the action of this end.

THEOREM2.LetMbe anasymptotically flat manifold with dimension <7. Suppose the scalar curvatureof M is nonnegative. Then the actionof each end ofMisnonnegative. Itis zerofor some end only ifMisflat.

Thedetailsoftheproof of Theorems1 and 2 will appear elsewhere. The basic proof follows our previous note. We pro- duce aminimalhypersurfaceof the manifold and try to pro- duce a contradiction ifthe conclusion is not correct. However, some new ideas have to be introduced here because the di- mensionof ourminimal hypersurface is greater than two. In- deed, one can make serious mistakes by claiming that our previousprocedureinproducing a minimal hypersurface works without essential change. This is because in applying the second variational formula

ff2[Ric(n)

+ IA 11

2]

<SIVf

1

2

forthe minimalhypersurface H, the functionfhas to be re- stricted ifone is notcarefulinproducingtheminimalhyper- surface. This restriction onf is so severethat our previous argument willnotwork.As a matteroffact,wefind outthat, whether the action of the manifold is positive or not, we can always produceaminimalhypersurfaceasinourprevious note when dimM > 4.This shows that any attempt withoutaltering theplanofourpreviousnotewill endupinafailure. Hence in ourproof, we areforcedto use aspecialkind of variational approachtoproducetheminimalhypersurface. Afterthat, we have to use nontrivial arguments to show that this minimal hypersurface isasymptotically flat with vanishing mass. As before,the second variational formula still shows that

f(f2R + IVfI2) > O,

in which R is the scalar curvature of the hypersurface.

Hence

f(f2R

+

8IVfI2)>O.

Whenourclassof functionfislarge enough,^{we can use}the aboveinequalitytoconformallydeform themetriconthehy- persurfacetoonewithzeroscalarcurvatureandnegativemass.

1024 Thepublication costsof this article were defrayed in part by page chargepayment.This articlemusttherefore beherebymarked"ad- vertisement"inaccordance with18 U.S. C. §1734solelytoindicate this fact.

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Mathematics: Schoen andYau

This violatesour previous theorem (1). The dimensionalre- strictioncomesfrom the singularity of the minimal hypersurface. Webelieve thatit is notessential.

Note Added in Proof. Concerning the existence of metrics with positivescalar curvature, we are able to prove that the connected sum of twomanifolds with positive scalar curvature also admits such a metric. Infact, we can connect such manifolds along the sphere bundle of a higher codimensional submanifold. Inparticular, when we do surgery on amanifoldwith positive scalar curvaturealongasphere

Proc.Nati.Acad.Sci. USA 76(1979) 1025

with codimension <3, we still obtain a manifold with positive scalar curvature. In many cases, this last fact reduces the classification of manifolds of positive scalar curvature to a topologicalproblem.

This researchwassupportedinpartby National Science Foundation GrantMCS75-04763.

1. Schoen, R. & Yau, S. T. (1978) Proc. Natl. Acad. Sci. USA 75, 2567.

2. Atiyah,M. F. &Singer,I.M.(1968)Ann.Math.87,546-604.

3. Lichnerowicz, A. (1963) C. R. Hebd. SeancesAcad. Sci. 257, 7-9.