Volume estimates for Kähler-Einstein metrics: the three dimensional case
aa r X i v : . [ m a t h . DG ] A p r Volume estimates for K¨ahler-Einstein metrics:the three dimensional case
X-X. Chen and S. K. DonaldsonJune 4, 2018
This is the first of a series of papers in which we obtain estimates for the volumeof certain subsets of K¨ahler-Einstein manifolds. These estimates form the mainanalytical input for an approach to general existence questions [9]. Let (
M, g )be any compact Riemannian manifold and r >
0. We let K r ⊂ M be the setof points x where | Riem | ≥ r − and write Z r for the r -neighbourhood of K r .Thus any point of the complement M \ Z r is the centre of a metric r -ball onwhich the curvature is bounded by r − . Re-scaling lengths by a factor r − , thisbecomes, in the re-scaled metric, a unit ball on which the curvature is boundedby 1. If in addition (as will be the case in our situation), we have control of thelocal injectivity radius we can say that, at the length scale r , the complement M \ Z r consists of “good points” with neighbourhoods of bounded geometry.Our aim is to derive estimates for the volume of the “bad” set Z r : the set wherethe geometry need not be standard at this scale.Let g be a K¨ahler-Einstein metric so Ric = λg for constant λ . Then one canfind in each complex dimension n constants a n , b n , depending only on n , so thatpointwise on the manifold, | Riem | dµ = ( a n c (Riem) + b n c (Riem)) ∧ ω n − , (1)where we write c i (Riem) for the standard integrand defining the Chern classesin Chern-Weil theory. (These constants do not depend on λ . As usual we write ω for the 2-form corresponding to g .) Thus on a compact manifold M Z M | Riem | = h a n c ∪ ω n − + b n c ∪ ω n − , M i . (2)The right hand side is a topological invariant, determined by the Chern classesof M and the K¨ahler class, which we will denote for brevity by E ( M ). Thisidentity shows that the curvature of the K¨ahler-Einstein metric cannot be verylarge on a set of large volume in M . More precisely we have an obvious estimatefor the volume of K r K r ) ≤ E ( M ) r . The goal of our work is to estimate the volume of Z r rather than K r . Weconsider K¨ahler-Einstein metrics with non-negative Ricci curvature and in thisfirst paper we restrict to complex dimension 3. We consider a compact K¨ahler-Einstein 3-fold ( M, g ) with Ric = λg , λ ≥
0. We suppose that the metricsatisfies the condition that Vol M ≥ κ Diam( M ) (3)for some κ >
0. The Bishop-Gromov comparison theorem implies that for allmetric balls B ( x, r ) with r ≤ Diam( M ),Vol B ( x, r ) ≥ κr . (4)Our main result is the following. Theorem 1
In this situation
Vol( Z r ) ≤ C (cid:0) E ( M ) r + b ( M ) r (cid:1) where C depends only on κ and b ( M ) is the second Betti number of M . Notice that the statement here is scale invariant (as of course it has to be).When c > κ determined by topological data. If wefix the scale by requiring that Ric = g then the content of the theorem is thebound Vol( Z ( r )) ≤ Cr . (5)We will also establish a small extension, which is probably not optimal. Theorem 2
With notation as in Theorem 1, there is a constants C ′ such thatfor all r there is a connected open subset Ω ′ ⊂ M \ Z r with Vol( M \ Ω ′ ) ≤ C ( E ( M ) r / + b ( M ) r )(In applications such as in [9] any bound O ( r µ ) with µ > n . When n = 2 the proofs are much easier because of the scaleinvariance of the L norm of the curvature in that dimension. In general for aball B ( x, r ) (with r ≤ Diam( M )) we define the “normalised energy” E ( x, r ) = r − n Z B ( x,r ) | Riem | dµ, (6)which is scale invariant. (When we want to emphasise the dependence on themetric we write E ( x, r, g ).) Notice that in our situation we have κr n ≤ Vol( B ( x, r )) ≤ ω n r n ω n is the volume of the unit ball in R n . So it is essentially the sameto normalise by the appropriate power of the volume of the ball. Normalisedenergy functionals of this kind appear in many other contexts in differentialgeometry, for example the theories of harmonic maps and Yang-Mills fields. Inthese two theories a crucial monotonicity property holds. This is the statementthat with a fixed centre the normalised energy is a decreasing function of r .If this monotonicity property held in our situation for K¨ahler-Einstein metrics,the proof of our theorem would be relatively straightforward. (Of course when n = 2 the monotonicity is obviously true.) The main work in this paper is toestablish a result which can be seen as an “approximate monotonicity” property.To state this cleanly, let us say that an open subset U ⊂ M carries homology if the inclusion map H ( ∂U, R ) → H ( U, R ) is not surjective. Note that if V ⊂ U carries homology then so does U and that if U , U . . . , U p ⊂ M aredisjoint domains which each carry homology then p cannot exceed the secondBetti number of M . Theorem 3
With the same hypotheses as Theorem 1, for each ǫ > there isa δ > such that if B ( x, r ) ⊂ M is a metric ball ( r ≤ Diam( M ) ) which doesnot carry homology and E ( x, r ) ≤ δ then for any y ∈ B ( x, r/ and r ′ ≤ r/ wehave E ( y, r ′ ) ≤ ǫ . The function δ ( ǫ ) depends only on the non-collapsing constant κ .In Section 2 of this paper we prove Theorem 3. The proof is an application ofthe extensive theory, due to Anderson, Cheeger, Colding and Tian, of Gromov-Hausdorff limits of Riemannian manifolds with lower bounds on Ricci curvature.In particular we make use of deep results of Cheeger, Colding and Tian oncodimension 4 singularities and of Cheeger and Colding on tangent cones atinfinity. Given Theorem 3, the deduction of Theorem 1, which we do in Section3, is fairly straightforward. We will first prove a “small energy” result, as follows. Proposition 1
With the same hypotheses as Theorem 1, there are δ , K suchthat if B ( x, r ) ⊂ M (with r ≤ Diam( M ) ) is a ball which does not carry homologyand E ( x, r ) ≤ δ then | Riem | ≤ Kr − on the interior ball B ( x, r/ . Theorem 1 follows from a straightforward covering argument. In Section 4we conclude with some remarks and discussion.In the sequel to this paper we will extend the results to all dimensions usinga rather different argument, making more use of the complex structure anddeveloping ideas of Tian in [13]. (Tian has informed us that, using these ideas,he obtained related results some while ago.) This argument also gives anotherapproach to the three-dimensional case here. But it appears to us worthwhileto write down both proofs.The authors have had this paper in draft form since early 2010. RecentlyCheeger and Naber have posted a preprint [6] establishing these volume esti-mates in all dimensions, and including results in a more general Riemanniangeometry setting. Their approach is somewhat different and it seems valuableto have this variety of arguments in the literature.3
Proof of Theorem 2
One foundation of our proof of Theorem 1 is a result of Cheeger, Colding andTian which states, roughly speaking, that the formation of codimension-4 sin-gularities requires a definite amount of energy. Let G ⊂ U (2) be a finite groupacting freely on S and n ≥
2. Consider the unit ball in the metric product C n − × (cid:0) C /G (cid:1) centred at (0 , V be a K¨ahler manifold of complex di-mension n with nonnegative Ricci curvature and B ′ be a unit ball centred at p ∈ V . Write d GH ( B, B ′ ) for the based Gromov-Hausdorff distance Proposition 2 ([5], Theorem 8.1) There are α n , η n > such that if d GH ( B, B ′ ) ≤ α n and G is non-trivial then Z B ′ | Riem | ≥ η n . Using this we now prove a result about metric cones in complex dimension3 with small energy.
Proposition 3
Let Y be a metric cone with vertex O . Suppose that the basedspace ( Y, O ) is the Gromov-Hausdorff limit of based K¨ahler-Einstein manifolds,(of complex dimension ) ( X n , g n , O n ) with Ric( g n ) = λ n g n where λ n ≥ and λ n → . Suppose that the X n satisfy a non-collapsing condition (4), with fixed κ > . Then there is a θ > such that if E ( O n , , g n ) ≤ θ for all n then Y issmooth away from the vertex. To see this we argue by contradiction. Suppose there is a sequence of suchexamples Y m , with fixed κ and with θ m →
0. Taking a subsequence, we cansuppose these have a based Gromov-Hausdorff limit Y ∞ and by a diagonal ar-gument this is the limit of a sequence of smooth based manifolds X n , as above,with E ( O n , , g n ) →
0. The non-collapsing condition means that, according tothe Cheeger, Colding, Tian theory, Y ∞ is a smooth Ricci-flat K¨ahler manifold,of real dimension 6, outside a set S ⊂ Y ∞ of Hausdorff dimension at most 2.Further, because of the complex structures present, if the dimension of S isstrictly less than 2 it must be 0 and this means that the singular set containsat most the vertex O , since it is invariant under the dilation action on the cone.So it suffices to show that dim S <
2. If, on the contrary, the dimension is 2then, again by the general theory, there is some point with a tangent cone ofthe form C × C /G for a nontrivial G . (In fact this is true for almost all pointsof the singular set, with respect to Hausdorff 2-measure.) By the invarianceunder the dilation action we can suppose this point is at distance 1, say, fromthe vertex. By the definition of tangent cone and of Gromov-Hausdorff con-vergence we can find a fixed small ρ such that for all large n there is a point p n ∈ X n such that the distance in X n from p n to O n is approximately 1 andthe Gromov-Hausdorff distance between the ρ -ball in X n centred at p n and the ρ -ball in the model space C × C /G is less than α ρ . But, rescaling the result4bove, this implies that the integral of | Riem | over this ρ -ball in X n is at least η ρ which gives a contradiction. Proposition 4
With the same hypotheses as in Proposition 2, for any σ, L wecan choose θ so that for all large n and points x in X n with L − ≤ d ( x, O n ) ≤ L we have | Riem | ≤ σd ( x, O n ) − . This follows from the same argument as above, and general theory.(By results ofAnderson [1], at points where the Gromov-Hausdorff limit is smooth the metricsconverge in C ∞ .) Proposition 5
With the same hypotheses as in Proposition 2, we can choose θ so that the cone Y must have the form C / Γ for some Γ ⊂ U (3) acting freelyon S . We know that Y has a K¨ahler metric with zero Ricci curvature, so thisfollows from the rigidity of C / Γ, among such cones. To give a direct argumentwe use the fact that the curvature tensor of a K¨ahler-Einstein metric satisfiesan identity of the form ∇ ∗ ∇ Riem = Riem ∗ Riem , (7)for a certain bilinear algebraic expression ∗ . With suitable normalisations thisgives a differential inequality∆ | Riem | ≥ −|
Riem | . (8)(We use the “analysts convention” for the sign of ∆.) Now apply this to ourcone Y and set f = | Riem | . Write r for the radial function on the cone. Clearly f is homogeneous of degree −
2, so (in an obvious notation) ∂f∂r = − r f ∂ f∂r = 6 r f. Consider the restriction of f to the cross-section r = 1 and a point p where f attains its maximal value m , say. If we write ∆ Σ for the Laplacian on thecross-section we have the usual formula∆ = r − ∂∂r (cid:18) r ∂∂r (cid:19) + r − ∆ Σ . By the homogeneity of f we have, at the point p ,∆ f = − f + ∆ Σ f, and ∆ Σ f ≤ p by the maximum principle. So we deduce that 4 m ≤ m andif m < m = 0. 5 .2 The main argument The second foundation of our proof is the existence, due to Cheeger and Colding[3], of “tangent cones at infinity”, under suitable hypotheses, as metric cones.It is convenient to first state an alternative form of Theorem 2.
Proposition 6
With the same hypotheses as in Theorem 2, for all ǫ > thereis an A ( ǫ ) such that if the ball B ( x, r ) ⊂ M does not carry homology and E ( x, r ) ≤ ǫA ( ǫ ) − then for all ρ ≤ r we have E ( x, ρ ) ≤ ǫ . To see that this statement implies Theorem 2, let B ( y, r ′ ) ⊂ B ( x, r ) be a ballof the kind considered there. The r/ B ( y, r/
2) centred at y lies in B ( x, r )and the corresponding normalised energy is at most 2 times E ( x, r ). It sufficesto take δ ( ǫ ) = ǫ (2 A ( ǫ )) − and apply Proposition 6 to B ( y, r/ ǫ is as small as we please and we will alwayssuppose that ǫ is less than the constant θ of Proposition 5. First fix ǫ, A andsuppose that the statement is false for these parameters, so we have a point x in a manifold M and radii ρ < r with E ( x, ρ ) ≥ ǫ, E ( x, r ) ≤ ǫA − . Thisimplies that ρ < A − r . Choose the largest possible value of ρ , so E ( x, s ) < ǫ for s ∈ ( ρ, r ] but E ( x, ρ ) = ǫ . Still keeping ǫ fixed we suppose that we have suchviolating examples for a sequence A n → ∞ , and data M n , g n , x n , ρ n , r n . Rescalethe (Riemannian) metrics by the factor ρ − / n : without loss of generality thereis a pointed Gromov-Hausdorff limit( M n , ρ − / n g n , x n ) → ( Z, O ) . Here Z may be singular. Let R m be a any sequence with R m → ∞ . Thenwe can rescale the metric on Z by factors R − m and the general theory of [3](using the noncollapsing condition (4)) tells us that a subsequence converges toa “tangent cone at infinity” Y which is a metric cone. The convergence is againin the sense of pointed Gromov-Hausdorff limits.Now go back to the smooth Riemannian manifolds M n and consider therescalings ( R m ρ n ) − / g n , indexed by n, m . If we choose a suitable function n ( m ) which increases sufficiently rapidly then the corresponding sequence ofRiemannian manifolds converges to the cone Y . Call these based manifold M m , ˜ g m , x m . We can also suppose n ( m ) increases so rapidly that A n ( m ) /R m tends to infinity with m . The choice of parameters means that E ( x m , s, ˜ g m ) <ǫ < θ if s ≤ A n /R m and s > R − m . In particular this holds for s = 2 and weare in the position considered in Theorem 2. Thus we deduce that the tangentcone Y is smooth (away from its vertex) and of the form C / Γ. It follows easilythat Z is smooth outside a compact set and that the curvature of Z satisfies abound | Riem | ≤ σr − Z , (9)(outside a compact set) where r Z denotes the distance in Z to the base pointand the constant σ can be made as small as we like by choosing ǫ small.The further result that we need is that the curvature actually decays faster.6 roposition 7 There is a γ > and K > such that outside a compact subsetof Z . | Riem | ≤ Kr − γZ . In fact our proof will establish the result for any γ <
4. The decay rate r − arises as that of the Green’s function in real dimension 6. We postpone theproof of Proposition 7, which is somewhat standard, and move on to completethe proof of Theorem 3, assuming this.Consider the general situation of a domain U ⊂ M with smooth boundary ∂U and a K¨ahler metric on M . Suppose that • The cohomology class of ω in H ( ∂U ) is zero. • U does not carry homology. • H ( ∂U ) = 0.The first two conditions imply that the class of ω in H ( U ) also vanishes,so we can write ω = da for a 1-form a on U . Now let p be the invariantpolynomial corresponding to the characteristic class a c + b c , as discussedin the Introduction. Thus p (Riem) is a closed 4-form on U and if the metric isK¨ahler-Einstein we have Z U | Riem | = Z U p (Riem) ∧ ω. Then Z U p (Riem) ∧ ω = Z ∂U p (Riem) ∧ a. (10)Further, the third condition implies that if ˜ a is any ∂U with d ˜ a = ω | ∂U then Z ∂U p (Riem) ∧ a = Z ∂U p (Riem) ∧ ˜ a. (11)Thus the integral of p (Riem) ∧ ω over U is determined by data on the boundaryand if the metric is K¨ahler-Einstein this coincides with the integral of | Riem | over U . In particular we get an inequality Z U | Riem | ≤ c max ∂U | Riem | Z ∂U | a | , (12)for some fixed constant c .To apply this, consider first the flat cone C / Γ and let Σ r be the cross-section at radius r . Then the real cohomology of Σ r vanishes in dimensions 1and 2 and we can obviously write ω | Σ r = da r with | a r | = O ( r ). In fact we can take a r to be one half the contraction of ω with the vector field r ∂∂r , so | a r | = r . Let V Γ r be the 5-volume of Σ r .7ow choose R so large that cK R − γ V Γ < ǫ/
2. By our discussion of thecone at infinity in Z we can choose R ≥ R such that near the level set r Z = R in Z there is a hypersurface Σ ′ with the property that the geometry of ω Z restricted to Σ ′ is very close to that of the cone metric restricted to Σ R , in anobvious sense. In particular we can suppose that ω Z | Σ ′ = da ′ where Z Σ ′ | a ′ | ≤ V Γ R , say. The bound on the curvature in Proposition 7 implies that c max Σ ′ | Riem | Z Σ ′ | a ′ | ≤ ǫ . (13)From now on R is fixed. We go back to the manifolds M n with rescaledmetrics ρ − / n g n converging to Z . We choose n so large that there is hypersurfaceΣ ′′ ⊂ M n on which the geometry is close to that of Σ ′ . Then Σ ′′ is the boundaryof an open set U ⊂ M n which does not carry homology by hypothesis. Bytaking n large we can make the boundary term in (10) as close as we like tothat estimated in (12),(13). So we can suppose that Z U | Riem | ≤ ǫ , say. But the U contains the unit ball centred at x n over which the integral of | Riem | is ǫ by construction. This gives the desired contradiction. Here we prove Proposition 7. The proof exploits the differential inequality∆ | Riem | ≥ −|
Riem | ≥ − σr − Z | Riem | . To explain the argument consider first a slightly different problem in which wework on a cone Y (of real dimension 6) with radial function r and we have asmooth positive function f on the set r ≥ f ≤ σr − f and f ≤ σr − .We have (∆ + σr ) r λ = ( λ ( λ + 4) + θ ) r λ − . (14)Let α = − −√ − σ, β = − √ − σ : the roots of the equation λ ( λ +4)+ θ = 0.We suppose σ is small, so α is close to − β is close to 0. Then any linearcombination g = Ar α + Br β satisfies the equation (∆+ σr ) g = 0. Fix R > m , m R be the maximumvalues of f on the cross-sections r = 1 , r = R respectively. We choose constants A, B so that g (1) ≥ m , g R ≥ m R . If we solve for the case of equality we get A = R β m − m R R β − R α , B = m R − R α m R β − R α . R β > R α it certainly suffices to take A = R β m R β − R α , B = m R R β − R α . Then set u = f − g so that u ≤ r = 1 or r = R and (∆+ σr − ) u ≥
0. Weclaim that u ≤ ≤ r ≤ R . For if we write u = hr − ,calculation gives(∆ + σr ) u = r − (∆ h − r − ∂h∂r − (4 − σ ) r − h ) (15)and (since 4 − σ >
0) we see that there can be no interior positive maximum of h . We see then that for 1 < r < R we have f ≤ Ar α + Br β where A, B are given by the formulae above. Using the information m ≤ θ, m R ≤ σR − we get Ar α ≤ cr α , Br β ≤ cR − β +2 r β , for some fixed c . Taking R very large compared to r we get f ≤ cr α , say.We want to adapt this argument to the function f = | Riem | on Z . Recallthat Z has base point O and r Z is the distance to O . We write A ( r , r ) = { z ∈ Z : r < r Z ( z ) < r } with r = ∞ allowed.There are several complications. One minor difficulty is that f may not besmooth, but this handled by standard approximation arguments. The second isthat the manifold is not exactly a cone, even at large distances, and the radiusfunction r Z need not be smooth. Lemma 1
For any τ > we can find an R τ > and a smooth function r onthe region A ( R τ , ∞ ) ⊂ Z such that • | r r Z − | ≤ τ, • ||∇ r | − | ≤ τ , • | ∆ r − | ≤ τ . We only sketch the proof. It is clear that we can choose R τ so that findsuch a function on any annulus A (2 p R τ , p +1 R τ ) for p = 1 , , . . . . This just usesthe convergence of the rescaled metric to the cone. If we extend these annulislightly we get a sequence of overlapping annuli and a function defined on each.To construct r we glue these together using cut-off functions. Notice that weonly have to glue adjacent terms so that the gluing errors do not accumulate.9ow think of σ and τ as fixed small numbers.( It will be clear from thediscussion below that one could calculate appropriate values explicitly: for ex-ample σ = τ = 1 /
100 will do) We want to adapt the preceding argument toprove Proposition 7. There is no loss in supposing that in fact R τ = 1 / | Riem | ≤ σ | Riem | on r ≥
1. Let α, β be the roots as above and choose α ′ , β ′ with α ′ slightly greater than α and β ′ slightly less than β . Then if τ issmall we will have (∆ + σr − ) r α ′ , (∆ + σr − ) r β ′ ≤ . (16)We want to choose A, B such that g ( r ) = Ar α ′ + Br β ′ has ∆ + σr − ≤ g (1) ≥ m , g ( R ) ≥ m R where m , M R are the maxima of f on r = 1 , r = R respectively. We take A = m R β ′ R β ′ − R α ′ , B = m R R β ′ − R α ′ . Then
A, B ≥ u = f − g and argue as before to show that u has no interior maximum. Wehave ∆ r − = r − (8 |∇ r | − ∆ r ) ≤ ( − δ ) r − , and the same argument goes through. We begin with a standard result.
Proposition 8
Given κ > there is a χ > such that if B is any unit ball ina K¨ahler-Einstein manifold (of real dimension ) and1. | Riem | = 1 at the center of B ;2. | Riem | ≤ throughout B ;3. Vol( B ) ≥ κ then R B | Riem | ≥ χ . One way to prove this is to apply the Moser iteration technique to | Riem | , usingthe differential inequality (8) and the fact that in this situation the Sobolevconstant is bounded. Another method is to use elliptic estimates in harmoniccoordinates. Proposition 9
Suppose M is a K¨ahler-Einstein manifold as considered in The-orem 1. There is an ǫ > such that U ⊂ M is any domain such that the nor-malised energy of any ball contained in U is less than ǫ then | Riem | ≤ d − ,where d denotes the distance to the boundary of U .
10o see this we let S be the maximum value of d − | Riem | over U and supposethat this is attained at p . If S > d ( q ) ≥ d ( p ) / q in theball of radius d ( p ) S − / centred at p . Rescale this ball to unit size and we arein the situation considered in the preceding proposition. If we take ǫ to be theconstant χ appearing there we get a contradiction, so in fact S ≤ ǫ as above and let δ ( ǫ ) be the valuegiven by Theorem 3. Suppose that B ( x, r ) is a ball of normalised energy lessthan δ and let U be the half-sized ball. Then Theorem 3 tells us that thenormalised energy of any ball in U is at most ǫ and we can apply the resultabove to see that | Riem | ≤ . r − in B ( x, r/ r and a K¨ahler-Einstein metric ( M, g ) asconsidered there we pick a maximal collection of point x a , a ∈ I in M such thatthe distance between any pair is at least r . Then the r -balls B ( x a , r ) cover M . Consider a ball B ( x a , r ). If this ball does not carry homology and itsnormalised energy is less than δ then by the result above we have | Riem | ≤ B ( x a , r ). Thus no point in Z r can lie in B ( x a , r ). Thus Z r is covered byballs B ( x a , r ) where either E ( x a , r ) > δ or B ( x a , r ) carries homology. Let I ′ ⊂ I denote the indices of the first kind and I ′′ ⊂ I those of the second.Suppose that N balls of the form B ( x a , r ) have a non-empty commonintersection. Let q be a point in the intersection, so the N centres x a all liein the 12 r ball centred at q . By construction the balls B ( x a , r/
2) are disjointand have volume at least ( κ/ ) r . Since the volume of B ( q, . r ) is boundedabove by a fixed multiple of r this gives a fixed bound on N , independent of r . Thus X a ∈ I ′ Z B ( x a , r ) | Riem | ≤ N Z M | Riem | . On the other hand, by definition, E ( x a , r ) ≥ δ for a ∈ I ′ so we see that thenumber of elements of I ′ is at most | I ′ | ≤ Nδ (12 r ) Z M | Riem | = C r − , say.By a similar argument there is a fixed upper bound N ′ on the number ofballs B ( x a , r ) which can meet any given one. It follows that the numberof these balls which carry homology is bounded by N ′ times the second Bettinumber of M . So | I ′′ | is bounded by a fixed number. ThenVol( Z r ) ≤ constant r ( | I ′ | + | I ′′ | ) ≤ C r + C r . connected open subset Ω ′ ⊂ M \ Z r so that the volume of the complement ofΩ ′ is bounded by a multiple of r / .To prove this we recall that in our situation there is a bound on the isom-perimetric constant, due to Croke [8]. If H ⊂ M is a rectifiable hypersurfacedividing M into two components M , M with Vol( M ) ≤ Vol( M ) thenVol( M ) / ≤ k Vol( H ) / , (17)for a fixed constant k . By the construction in the proof of Theorem 1 abovethe set Z r is contained in W which is a union of P balls of radius r with P ≤ C ( r − + 1). By the Bishop comparison theorem the 5-volume of theboundary of one of these balls is bounded by Cr . The boundary ∂W of W is arectifiable set of 5-volume at most the sum of the boundaries of the balls, thusVol( ∂W ) ≤ C ( r + r ). We can normalise so that the the volume of M is 1 andthen, without loss of generality suppose that r is so small that Vol( W ) ≤ / k Vol( ∂W ) ≤ /
10. Let Ω i be the connected components of M \ W . If acomponent Ω has volume greater than 1 / M ) thenby (14) its complement has volume less than k Vol( ∂W ) / == k C / r / andwe can take Ω ′ = Ω . So suppose that all components Ω i have volume less than1 /
2. Then it is clear that X i Vol( ∂ Ω i ) = Vol( ∂W )while X i Vol(Ω i ) ≥ / . The second equation implies that P i Vol(Ω i ) / ≥ /
10 and then (17) gives X i Vol( ∂ Ω i ) ≥ k − / , so Vol( ∂W ) ≥ k − / , contrary to our assumptions.
1. Proposition 5 and the ensuing arguments in 2.2 above are closely relatedto a result of Cheeger, Colding and Tian ([5], Theorem 9.26). Let X be acomplete, noncompact Ricci-flat K¨ahler manifold of complex dimension n with base point p . Suppose that Vol( B ( p, R ) ≥ κR n and R − n Z B ( p,R ) | Riem | → R → ∞ . Tian conjectured in [12] that in this situation X is an ALEmanifold, with tangent cone at infinity of the form C n / Γ. When n = 3this conjecture was established in the result quoted above. Our statementsare a little different since we establish a definite “small asymptotic energythreshold” which implies that the manifold is ALE.2. In our situation the tangent cone Y of Z at infinity is unique. In general,positive Ricci curvature does not imply uniqueness of tangent cones, evenwhen | Riem | ≤ cr − . See the discussion in [4] of examples, including anunpublished example due to Perelman.3. Theorem 3 becomes false if we omit the condition that the ball does notcarry homology. To see this one can consider for example the quotient M = T / Γ of a complex torus by a group Γ of order 3, acting with isolatedfixed points. Then M has a resolution M with c ( M ) = 0. According toJoyce [11] there is a family of ALE metrics on the resolution of C / Γ,parametrised by the K¨ahler class. For suitable K¨ahler classes on M , theCalabi-Yau metric is approximately given by gluing rescaled versions ofthese ALE metrics to the flat metric on N , just as in the familiar pictureof the Kummer construction for K3 surfaces. For any δ > M , a unit ball B ( x, ⊂ M onwhich the normalised energy is less than δ and an interior ball B ( x, ρ ) onwhich the normalised energy exceeds c , for some fixed c . The argumentsin our forthcoming paper will will show that this is essentially the onlyway in which approximate monotonicity can fail; see also [13].4. It is interesting to ask if similar results to those proved above can beestablished for constant scalar curvature and extremal K¨ahler metrics.For this it might be sensible to assume a bound on the Sobolev constant.Perhaps some of the techniques used in [14], [7] can be applied to thisproblem.5. Another question is whether a result like Theorem 3 holds for general6-dimensional Einstein metrics (with nonnegative Ricci curvature and anon-collapsing condition). It might be that a different topological sidecondition is appropriate.6. The function δ ( ǫ ) in Theorem 3 depends only on the collapsing constant κ . It would be interesting to determine the function effectively, but ourmethod cannot do this. One suspects that, in reality, it may be possibleto take δ ( ǫ ) not much smaller than ǫ , and that the constant C in Theorem1 can (in reality) be taken not too large. Some results on the numericalanalysis of K¨ahler-Einstein, and more generally extremal, metrics seemto give evidence for this suspicion [2], [10] but a theoretical derivation ofrealistic estimates seems a long way off.13 eferences [1] Anderson, M.T Convergence and rigidity of metrics under Ricci curvaturebounds
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