On some asymptotically flat manifolds with non maximal volume growth
aa r X i v : . [ m a t h . DG ] S e p ON SOME ASYMPTOTICALLY FLAT MANIFOLDS WITH NONMAXIMAL VOLUME GROWTH.
VINCENT MINERBE
Abstract.
Asymptotically flat manifolds with Euclidean volume growth are known tobe ALE. In this paper, we consider a class of asymptotically flat manifolds with slowervolume growth and prove that their asymptotic geometry is that of a fibration over an ALEmanifold. In particular, we show that gravitational instantons with cubic volume growthare ALF.
Introduction.
The aim of this paper is to understand the geometry at infinity of some asymptoticallyflat manifolds, that is complete noncompact Riemannian manifolds (
M, g ) whose curvaturetensor Rm g fastly goes to zero at infinity: | Rm | g = O ( r − − ǫ )where r is the distance function to some point and ǫ is a positive number. Such manifoldsare known to have finite topological type [A2]: there is a compact subset K of M such that M \ K has the topology of ∂K × R ∗ + . In contrast, any (connected) manifold carries a completemetric with quadratic curvature decay ( | Rm | g = O ( r − ), see [LS]).In this paper, we are interested in the geometry at infinity of some asymptotically flatmanifolds: we want to understand the metric g in the unbounded part M \ K .A basic fact is that an asymptotically flat manifold has at most Euclidean volume growth:there is a constant B such that ∀ x ∈ M, ∀ t ≥ , vol B ( x, t ) ≤ Bt n . A fundamental geometric result was proved in [BKN]: an asymptotically flat manifold withmaximal volume growth, that is ∀ x ∈ M, ∀ t ≥ , vol B ( x, t ) ≥ At n , is indeed Asymptotically Locally Euclidean (“ALE” for short): there is a compact set K in M , a ball B in R n , a finite subgroup G of O ( n ) and a diffeomorphism φ between R n \ B and M \ K such that φ ∗ g tends to the standard metric g R n at infinity. It is also proved in [BKN]that a complete Ricci flat manifold with maximal volume growth and curvature in L n ( dvol )is ALE.In the paper [BKN], S. Bando, A. Kasue and H. Nakajima raise the following naturalquestion: can one understand the geometry at infinity of asymptotically flat manifolds whosevolume growth is not maximal ?Some motivation comes from theoretical physics. Quantum gravity and string theorymake use of the so called “gravitational instantons”: from a mathematical point of view, agravitational instanton is a complete noncompact hyperk¨ahler 4-manifold with decaying cur-vature at infinity. In dimension four, “hyperk¨ahler” means Ricci flat and K¨ahler. Note that Date : May 30, 2018. in the initial definition, by S. Hawking [Haw], gravitational instantons were not supposedto be K¨ahler, including for instance the Riemannian Schwarzschild metric. The curvatureshould satisfy a “finite action” assumption: typically, we want the curvature tensor to bein L . These manifolds have recently raised a lot of interest, both from mathematicians(for instance, [HHM], [EJ]) and physicists. Works by [BKN] and [K1], [K2] enable to clas-sify the geometry of gravitational instantons with maximal volume growth: these are ALEmanifolds, obtained as resolutions of quotients of C by a finite subgroup of SU (2). In casethe volume growth is not maximal, S. Cherkis and A. Kapustin have made a classificationconjecture, inspired from string theory ([EJ]). Essentially, gravitational instantons shouldbe asymptotic to fibrations over a Euclidean base and the fibers should be circles (“ALF”case, for “Asymptotically Locally Flat”), tori (“ALG” case, because E,F,... G) or compactorientable flat 3-manifolds (there are 6 possibilities, this is the “ ALH ” case).Let us state our main theorem. Here and in the sequel, we will denote by r the distance tosome fixed point o , without mentionning it. We will also use the measure dµ = r n vol B ( o,r ) dvol .It was shown in [Min] that this measure has interesting properties on manifolds with non-negative Ricci curvature. Note that in maximal volume growth, it is equivalent to theRiemannian measure dvol . Theorem — Let ( M , g ) be a connected complete hyperk¨ahler manifold with curvaturein L ( dµ ) and whose volume growth obeys ∀ x ∈ M, ∀ t ≥ , At ν ≤ vol B ( x, t ) ≤ Bt ν with < A ≤ B and ≤ ν < . Then ν = 3 and there is a compact set K in M , a ball B in R , a finite subgroup G of O (3) and a circle fibration π : M \ K −→ ( R \ B ) /G . Moreover,the metric g can be written g = π ∗ ˜ g + η + O ( r − ) , where η measures the projection along the fibers and ˜ g is ALE on R , with ˜ g = g R + O ( r − τ ) for every τ < . Up to finite covering, the topology at infinity (i.e. modulo a compact set) is thereforeeither that of R × S (trivial fibration over R ) or that of R (Hopf fibration). Examplesinvolving the Hopf fibration are provided by S. Hawkings’ multi-Taub-NUT metrics, whichwe will describe in the text.Our assumption on the curvature may seem a bit strange at first glance. It should benoticed it is a priori weaker than Rm = O ( r − − ǫ ). So we are in the realm of asymptoticallyflat manifolds. In the appendix, we will show that indeed, our integral assumption impliesstronger estimates. Under our hypotheses, a little analysis provides ∇ k Rm = O ( r − − k ), forany k in N !Our volume growth assumption is uniform: the constants A and B are assumed to hold atany point x . This is not anecdotic. By looking at flat examples, we will see the importanceof this uniformity. This feature is not present in the maximal volume growth case, wherethe uniform estimate ∃ A, B ∈ R ∗ + , ∀ x ∈ M, ∀ t ≥ , At n ≤ vol B ( x, t ) ≤ Bt n is equivalent to ∃ A, B ∈ R ∗ + , ∃ x ∈ M, ∀ t ≥ , At n ≤ vol B ( x, t ) ≤ Bt n . Our result is related to a theorem by A. Petrunin and W. Tuschmann [PT]. They haveshown that if an asymptotically flat 4-manifold has a simply connected end, then this endadmits a tangent cone at infinity that is isometric to R , R or R × R + . The R case is the N ASYMPTOTICALLY FLAT MANIFOLDS WITH NON MAXIMAL VOLUME GROWTH. 3
ALE case and the case R × R + conjecturally never occurs. So we are left with R , and this isconsistent with our theorem (which does not require simply-connectedness at infinity, by theway). In greater dimension n , [PT] says the tangent cone at infinity is R n as soon as the endis simply connected. Indeed, our theorem admits a version (see 3.26 below) in any dimensionand without the hyperk¨ahler assumption; it requires estimates on the covariant derivatives ofthe curvature, but also an unpleasant assumption on the holonomy of short loops at infinity(it should be sufficiently close to the identity). Under these strong assumptions, a volumegrowth comparable to that of R n − in dimension n ≥ R n − , which forbids simply connectedends. In this point of view, the fact that dimension 4 is special in [PT] comes from theexistence of the Hopf fibration over S .The idea of the proof is purely Riemannian. The point is the geometry at infinity collapses,the injectivity radius remains bounded while the curvature gets very small, so Cheeger-Fukaya-Gromov theory [CG],[CFG] applies. The fibers of the circle fibration will come fromsuitable regularizations of short loops based at each point.The structure of this paper is the following.In a first section, we will consider examples, with three goals: first, we want to explain ourvolume growth assumption through the study of flat manifolds; second, these flat exampleswill also provide some ideas about the techniques we will develop later; third, we will describethe Taub-NUT metric in detail, so as to provide the reader with a concrete example to thinkof.In a second section, we will try to analyse some relations between three Riemanniannotions: curvature, injectivity radius, volume growth. We will introduce the “fundamentalpseudo-group”. This object, due to M. Gromov [GLP], encodes the Riemannian geometryat a fixed scale. It is our basic tool and its study will explain for instance the volume growthself-improvement phenomenon in our theorem (from 3 ≤ ν < ν = 3).In the third section, we completely describe the fundamental pseudo-group at a convenientscale, for gravitational instantons. This enables us to build the fibration at infinity, locallyfirst, and then globally, by a gluing technique. Then we make a number of estimates toobtain the description of the geometry at infinity that we announced in the theorem. Thispart requires a good control on the covariant derivatives of the curvature tensor and thedistance functions. This is provided by the appendices. Acknowledgements.
I wish to thank Gilles Carron for drawing my attention to thegeometry of asymptotically flat manifolds and for many discussions. This work benefitedfrom the French ANR grant GeomEinstein.
VINCENT MINERBE
Contents
Introduction. 11. Examples. 41.1. Flat plane bundles over the circle. 41.2. The Taub-NUT metric. 72. Injectivity radius and volume growth. 92.1. An upper bound on the injectivity radius. 92.2. The fundamental pseudo-group. 92.3. Fundamental pseudo-group and volume. 153. Collapsing at infinity. 173.1. Local structure at infinity. 173.2. Holonomy in gravitational instantons. 213.3. An estimate on the holonomy at infinity. 223.4. Local Gromov-Hausdorff approximations. 253.5. Local fibrations. 283.6. Local fibration gluing. 353.7. The circle fibration geometry. 373.8. What have we proved ? 42Appendix A. Curvature decay. 43Appendix B. Distance and curvature. 46References 481.
Examples.
Flat plane bundles over the circle.
To have a clear picture in mind, it is useful tounderstand flat manifolds obtained as quotients of the Euclidean space R by the action ofa screw operation ρ . Let us suppose this rigid motion is the composition of a rotation ofangle θ and of a translation of a distance 1 along the rotation axis. The quotient manifoldis always diffeomorphic to R × S , but its Riemannian structure depends on θ : one obtainsa flat plane bundle over the circle whose holonomy is the rotation of angle θ . These verysimple examples conceals interesting features, which shed light on the link between injectivityradius, volume growth and holonomy. In this paragraph, we stick to dimension 3 for thesake of simplicity, but what we will observe remains relevant in higher dimension.When the holonomy is trivial, i.e. θ = 0, the Riemannian manifold is nothing but thestandard R × S . The volume growth is uniformly comparable to that of the Euclidean R : ∃ A, B ∈ R ∗ + , ∀ x ∈ M, ∀ t ≥ , At ≤ vol B ( x, t ) ≤ Bt . The injectivity radius is 1 / θ = 2 π · p/q , for some coprime numbers p , q . A covering of order q brings us back to the trivial case. The volume growth is thus uniformly comparable to thatof R . What about the injectivity radius ? Because of the cylindric symmetry, it dependsonly on the distance to the ”soul”, that is the image of the screw axis: let us denote byinj( t ) the injectivity radius at distance t from the soul. This defines a continuous functionadmitting uniform upper and lower bounds, but non constant in general. The soul is alwaysa closed geodesic, so that inj(0) = 1 /
2. But as t increases, it becomes necessary to compare N ASYMPTOTICALLY FLAT MANIFOLDS WITH NON MAXIMAL VOLUME GROWTH. 5 the lengths l k ( t ) of the geodesic loops obtained as images of the segments [ x, ρ k ( x )], with x at distance t from the axis. We can give a formula:(1) l k ( t ) = q k + 4 t sin ( kθ/ . The injectivity radius is given by 2 inj( t ) = inf k l k ( t ). In a neighbourhood of 0, 2 inj equals l ; then 2 inj may coincide with l k for different indices k . If k < q is fixed, since sin kθ doesnot vanish, the function t l k ( t ) grows linearly and tends to infinity. The function l q isconstant at q and l q ≤ l k for k ≥ q . Thus, outside a compact set, the injectivity radius isconstant at q/ l k ( t ) ≍ t ). xxt θ = π Figure 1.
The holonomy angle is θ = π . On the left, a geodesic loop basedat x with length l ( t ) = 3. On the right, a geodesic loop based at x withlength l ( t ) = √ t .When θ is an irrational multiple of 2 π , the picture is much different. In particular, theinjectivity radius is never bounded. Proposition — The injectivity radius is bounded if and only if θ is a rational multipleof π .Proof. The ”only if” part is settled, so we assume the function t inj( t ) is bounded by somenumber C . For every t , there is an integer k ( t ) such that 2 inj( t ) = l k ( t ) . Formula (1) impliesthe function t k ( t ) is bounded by C . Since its values are integers, there is a sequence ( t n )going to infinity and an integer k such that k ( t n ) = k for every index n . Then (1) yields ∀ n ∈ N , l k ( t n ) = k + 4 t n sin ( kθ/ ≤ C . Since t n goes to infinity, this requires sin kθ = 0: there is an integer m such that kθ/ mπ ,i.e. θ/ π = m/k . (cid:3) What about volume growth ? The volume of balls centered in some given point growsquadratically: ∀ x, ∃ B x , ∀ t ≥ , vol B ( x, t ) ≤ B x t . In the “rational” case, the estimate is even uniform with respect to the center x of the ball:(2) ∃ B, ∀ x, ∀ t ≥ , vol B ( x, t ) ≤ Bt . VINCENT MINERBE
In the “irrational” case, this strictly subeuclidean estimate is never uniform. Why ? Theproposition above provides a sequence of points x n such that r n := inj( x n ) goes to infinity.Given a lift ˆ x n of x n in R , the ball B (ˆ x n , r n ) is the lift of B ( x n , r n ) and its volume is πr n .If we assume two points v and w of B (ˆ x n , r n ) lift the same point y of B ( x n , r n ), there is bydefinition an integer number k such that ρ k ( v ) = w ; since ρ is an isometry of R , we get (cid:12)(cid:12)(cid:12) ρ k (ˆ x n ) − ˆ x n (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) ρ k (ˆ x n ) − ρ k ( v ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ρ k ( v ) − ˆ x n (cid:12)(cid:12)(cid:12) = | ˆ x n − v | + | w − ˆ x n | < r n = 2 inj( x n ) , which contradicts the definition of inj( x n ) (the segment [ ρ k (ˆ x n ) , ˆ x n ] would go down as atoo short geodesic loop at x n ). Therefore B (ˆ x n , r n ) and B ( x n , r n ) are isometric, hencevol B ( x n , r n ) = πr n , which prevents an estimate like (2).For concreteness, we wish to conclude this paragraph with quantitative estimates on theinjectivity radius in the irrational case. It is indeed a Diophantine approximation problem. Proposition — Assume θ/ (2 π ) is an irrational but algebraic number. Given α in ]0 , / , there are positive numbers T , C and A depending only on θ and α and such that forevery t > T , if x is at distance t from the soul, then inj( t ) ≥ Ct α and vol B ( x, Ct α ) = At α . Proof.
Roth theorem provides a positive constant C = C ( θ, α ) such that for every ( k, m ) in Z ∗ × I r : (cid:12)(cid:12)(cid:12)(cid:12) θ π − mk (cid:12)(cid:12)(cid:12)(cid:12) ≥ Ck /α . Without loss of generality, we can suppose C ≤ /
2. It follows that for every positive integer k , Ck − /α ≤ π | kθ − πm | ≤ (cid:12)(cid:12)(cid:12) e ikθ − (cid:12)(cid:12)(cid:12) , so that we obtain for every positive number t and for every integer k in [1 , t α ]:(3) (cid:12)(cid:12)(cid:12) te ikθ − t (cid:12)(cid:12)(cid:12) ≥ Ct α . Consider a ball B := B ( x, Ct α ) centered in a point x at distance t from the soul. Let ˆ x be alift of x in R and let ˆ B be the ball centered in ˆ x and with radius Ct α in R . To prove bothestimates, it is sufficient to ensure this ball covers B only once. Suppose v and w are twodistinct lifts in ˆ B of a single element y of B : w = ρ k v for some k . Denote by s the commondistance of v and w to the soul. The following holds: | w − v | = k + (cid:12)(cid:12) e ikθ − (cid:12)(cid:12) s . Since w and v belong to the ball ˆ B , we get k ≤ | w − v | ≤ Ct α ≤ t α . Apply (3) to obtain: | w − v | ≥ (cid:12)(cid:12)(cid:12) e ikθ − (cid:12)(cid:12)(cid:12) s ≥ Ct α − s. With | w − v | ≤ Ct α , we find:(4) s ≤ t . Now, as v belongs to ˆ B , we can write | v − ˆ x | ≤ Ct α , which combines with the triangleinequality to yield:(5) s ≥ t − Ct α . Comparing (4) and (5), we arrive at t ≤ (2 C ) − α . If we suppose t ≥ T := 2(2 C ) − α , itensures ˆ B covers B only once, hence the result. (cid:3) This estimate on the injectivity radius is (quasi) optimal:
N ASYMPTOTICALLY FLAT MANIFOLDS WITH NON MAXIMAL VOLUME GROWTH. 7
Proposition — Whatever θ is, the growth of the injectivity radius is bounded fromabove by ∀ t ≥ , inj( t ) ≤ √ π √ t. Proof.
Fix t ≥
1. The pigeonhole principle yields an integer k in [1 , √ t ] such that2 | sin kθ/ | = (cid:12)(cid:12)(cid:12) e ikθ − (cid:12)(cid:12)(cid:12) ≤ π √ t . We deduce: 2 inj( t ) ≤ l k ( t ) = r k + 4 t sin kθ ≤ p π √ t. (cid:3) When θ/ (2 π ) admits good rational approximations, an almost rational behaviour can berecovered, with a slowly growing injectivity radius. For instance, if θ/ (2 π ) is the Liouvillenumber ∞ X n =1 − n ! , then lim inf t −→∞ (cid:0) t − a inj( t ) (cid:1) = 0 for every a > The Taub-NUT metric.
The Taub-NUT metric is the basic non trivial example ofALF gravitational instanton. This Riemannian metric over R was introduced by StephenHawking in [Haw]. To describe it, we mostly follow [Leb].Thanks to the Hopf fibration (Chern number: − R \ { } = R ∗ + × S asthe total space of a principal circle bundle π over R ∗ + × S = R \ { } . Denote by g R thestandard metric on R and by r the distance to 0 in R . Let V be the harmonic functionwhich is defined on R \ { } by: V = 1 + 12 r . In polar coordinates, the volume form on R reads r dr ∧ Ω, so that the definition dV ∧ ∗ dV = | dV | r dr ∧ Ωyields ∗ dV = − Ω. The Chern class c of the U (1)-bundle π is the cohomology class of − Ω4 π . Chern-Weil theory asserts a 2-form α with values in u = i R id C is the curvature of aconnection on π if and only if i π Tr α represents the cohomology class c . The identity i π Tr( ∗ dV ⊗ i id) = − Ω4 π therefore yields a connection with curvature ∗ dV ⊗ i id C on π . Let ω ⊗ i id C be the 1-formof this connection. Lifts of objects on the base will be endowed with a hat: for instance,ˆ V = V ◦ π .On R \ { } , the Taub-NUT metric is given by the formula g = ˆ V ˆ g R + ˆ V − ω . Setting ρ = √ r , we obtain the behaviour of the metric near 0: g ∼ = dρ + ρ ˆ g S ρ ω = dρ + ρ ˆ g S . It can thus be extended as a complete metric on R by adding one point, sent on the originof R by π . The additional point can be seen as the unique fixed point for the action of S . VINCENT MINERBE
By construction, the metric is S -invariant, fiber length tends to a (nonzero) constant atinfinity, while the induced metric on the base is asymptotically Euclidean(it is at distance O ( r − ) from the Euclidean metric). Thus there are positive constants A and B such that ∀ R ≥ , AR ≤ vol B ( x, R ) ≤ BR . Moreover, the Taub-NUT metric is hyperk¨ahler. K¨ahler structures can be described in thefollowing way. Let ( x, y, z ) be the coordinates on R and choose a local gauge: ω = dt + θ forsome vertical coordinate t and a 1-form θ such that dθ = ∗ dV . An almost complex structure J x can be defined by requiring the following action on the cotangent bundle: J x (cid:16)p ˆ V d ˆ x (cid:17) = 1 p ˆ V ω et J x (cid:16)p ˆ V d ˆ y (cid:17) = √ V d ˆ z. It is shown in [Leb] that ( g, J x ) is indeed a K¨ahler structure ( J x being parallel outside 0, itsmoothly extends on the whole R ). A circular permutation of the roles of x , y and z yieldsthree K¨ahler structure ( g, J x ), ( g, J y ), ( g, J z ) satisfying J x J y = J z , hence the hyperk¨ahlerstructure. [Leb] even shows that these complex structures are biholomorphic to that of C .Note the Taub-NUT metric can also be obtained as a hyperk¨ahler quotient ([Bes]).In an orthonormal and left invariant trivialization ( σ , σ , σ ) of T ∗ S , the metric can bewritten g = V ( r ) dr + 4 r V ( r )( σ + σ ) + 1 V ( r ) σ (we forget the hats). Let H be the solution of H ′ = 1 p V ( H )with H (0) = 0 and set r = H ( t ). Note that H ′ ∼ H ∼ t at infinity. The equationabove becomes: g = dt + (cid:18) HH ′ (cid:19) ( σ + σ ) + H ′ σ . Using [Unn], it is possible to compute the curvature of such a metric. It decays at a cubicrate: | Rm | = O ( r − ) . This ansatz produces a whole family of examples: the ”multi-Taub-NUT” metrics or A N − ALF instantons. These are obtained as total spaces of a circle bundle π over R minus somepoints p , . . . , p N , endowed with the metricˆ V ˆ g R + ˆ V − ω , where V is the function defined on R \ { p , · · · , p N } by V ( x ) = 1 + N X i =1 | x − p i | and where ω is the form of a connexion with curvature ∗ dV ⊗ i id C . As above, a completionby N points is possible. The circle bundle restricts on large sphere as a circle bundle ofChern number − N . The metric is again hyperk¨ahler and has cubic curvature decay. Theunderlying complex manifold is C / Z N . The geometry at infinity is that of the Taub-NUTmetric, modulo an action of Z N , which is the fundamental group of the end.Other examples are built in [ChH]: the geometry at infinity of these D k ALF gravitationalinstantons is essentially that of a quotient of a multi-Taub-NUT metric by the action of areflection on the base.
N ASYMPTOTICALLY FLAT MANIFOLDS WITH NON MAXIMAL VOLUME GROWTH. 9 Injectivity radius and volume growth.
An upper bound on the injectivity radius.
Proposition — There is a universal constant C ( n ) such that on any complete Riemannian manifold ( M n , g ) satisfying (6) inf t> lim sup x −→∞ vol B ( x, t ) t n < C ( n ) , the injectivity radius is bounded from above, outside a compact set. The assumption (6) means there is a positive number T and a compact subset K of M such that:(7) ∀ x ∈ M \ K, vol B ( x, T ) < C ( n ) T n . We think of a situation where there is a function ω going to zero at infinity and such thatfor any point x , vol B ( x, t ) ≤ ω ( t ) t n . The point is we require a uniform strictly subeuclideanvolume growth. Even in the flat case, we have seen that a uniform estimate moderates thegeometry much more than a centered strictly subeuclidean volume growth. Proof.
The constant C ( n ) is given by Croke inequality [Cro]:(8) ∀ t ≤ inj( x ) , ∀ x ∈ M, vol B ( x, t ) ≥ C ( n ) t n . Let x be a point outside the compact K given by (7). If inj( x ) is greater than the number T in (7), (8) yields: C ( n ) T n ≤ vol B ( x, T ) < C ( n ) T n , which is absurd. The injectivity radius at x is thus bounded from above by T . (cid:3) Cheeger-Fukaya-Gromov theory applies naturally in this setting: it describes the geometryof Riemannian manifolds with small curvature and injectivity radius bounded from above[CG]. Let us quote the
Corollary — Let ( M n , g ) be a complete Riemannian manifold whose curvature goesto zero at infinity and satisfying (6). Outside a compact set, M carries a F -structure ofpositive rank whose orbits have bounded diameter. It means we already know there is some kind of structure at infinity on these manifolds.Our aim is to make it more precise, under additional assumptions.2.2.
The fundamental pseudo-group.
The notion of ”fundamental pseudo-group” wasintroduced by M. Gromov in the outstanding [GLP]. It is very natural tool in the study ofmanifolds with small curvature and bounded injectivity radius. Let us give some details.Let M be a complete Riemannian manifold and let x be a point in M . We assume thecurvature is bounded by Λ (Λ ≥
0) on the ball B ( x, ρ ), with Λ ρ < π/
4. In particular,the exponential map in x is a local diffeomorphism on the ball ˆ B (0 , ρ ) centered in 0 and ofradius 2 ρ in T x M . The metric g on B ( x, ρ ) thus lifts as a metric ˆ g := exp ∗ x g on ˆ B (0 , ρ ).We will denote by Exp the exponential map corresponding to ˆ g .An important fact is proved in [GLP]: any two points in ˆ B (0 , ρ ) are connected by aunique geodesic which is therefore minimizing; moreover, balls are strictly convex in thisdomain.When the injectivity radius at x is greater than 2 ρ , the Riemannian manifolds ( B ( x, ρ ) , g )and ( ˆ B (0 , ρ ) , ˆ g ) are isometric. But if it is small, there are short geodesic loops based at x and x admits differents lifts in ˆ B (0 , ρ ). The fundamental pseudo-group Γ( x, ρ ) in x and at scale ρ measures the injectivity defect of the exponential map over ˆ B (0 , ρ ) [GLP] : Γ( x, ρ ) isthe pseudo-group consisting of all the continuous maps τ from ˆ B (0 , ρ ) to T x M which satisfyexp x ◦ τ = exp x and send 0 in ˆ B (0 , ρ ). Since exp x is a local isometry, these maps send geodesics ontogeodesics hence preserve distances : they are isometries onto their image. In particular, theyare smooth sections of the exponential map at x .Given a lift v of x in ˆ B (0 , ρ ) (i.e. exp x ( v ) = p ), consider the map τ v := Exp v ◦ ( T v exp x ) − . ( T v exp x ) − maps a point w in ˆ B (0 , ρ ) on the initial speed vector of the geodesic lifting t exp x tw from v . Thus τ v ( w ) is nothing but the tip of this geodesic. In particular, τ v ( w )lifts exp x w , i.e. exp x ( τ v ( w )) = exp x w , and τ v (0) belongs to ˆ B (0 , ρ ). So τ v is an element ofΓ( x, ρ ) (cf. figure 2). T x M exp x v = τ v (0) M wτ v ( w ) x x w Figure 2. τ v ( w ) is obtained in the following way. Push the segment [0 , w ]from T x M to M thanks to exp x anf lift the resulting geodesic from v to obtaina new geodesic in T x M whose tip is τ v ( w ).Reciprocally, if τ is an element of Γ( x, ρ ) mapping 0 to v ∈ ˆ B (0 , ρ ), then one can see that τ = τ v . Indeed, given w in ˆ B (0 , ρ ), t τ ( tw ) is the lift of t exp x tw from v , so that theargument above yields τ ( w ) = τ v ( w ).There is therefore a one-to-one correspondance between elements of Γ( x, ρ ) and orientedgeodesic loops based at x with length bounded by ρ . Since exp x is a local diffeomorphism,Γ( x, ρ ) is in particular finite. Thus, given x , (Γ( x, ρ )) <ρ<π/ (4Λ) is a nondecreasing finitepseudo-group family. Example . Consider a flat plane bundle over S , with rational holonomy ρ (cf. section1): the screw angle θ is π times p/q , with coprime p and q . For large ρ and x fartherthan ρ/ sin( π/q ) from the soul (when q = 1 , there is no condition), the fundamental pseudo-group Γ( x, ρ ) is generated by the unique geodesic loop with length q . It therefore consists oftranslations only. In particular, it does not contain ρ , except in the trivial case ρ = id . Ingeneral, many geodesic loops are forgotten, for they are too long. N ASYMPTOTICALLY FLAT MANIFOLDS WITH NON MAXIMAL VOLUME GROWTH. 11
Every nontrivial element of Γ( x, ρ ) acts without fixed points. Let us show it. We assumea point w is fixed by some τ v in Γ( x, ρ ). First, w is the tip of the geodesic γ : t tw ,defined on [0 ,
1] and lifting t exp x tw from 0. Second, w = τ v ( w ) is also the tip of thegeodesic γ : t τ v ( tw ) lifting the same geodesic t exp x tw , but from v . Differentiatingat t = 1 the equality exp x ◦ γ ( t ) = exp x ◦ γ ( t ) , we obtain T w exp x ( γ ′ (1)) = T w exp x ( γ ′ (1)) and thus γ ′ (1) = γ ′ (1). Geodesics γ and γ must then coincide, which implies 0 = γ (0) = γ (0) = v , hence τ v = id.In the pseudo-group Γ( x, ρ ), every element has a well-defined inverse. To see this, given τ = τ v in Γ( x, ρ ), consider the geodesic loop σ : t exp x tv and define ˜ v := − σ ′ (1). Themaps τ ˜ v and τ v are well defined on the ball ˆ B (0 , ρ ), so τ ˜ v ◦ τ v is well defined on ˆ B (0 , ρ ). Itis a section of exp x and fixes 0: τ ˜ v ◦ τ v (0) = τ ˜ v ( v ) is the tip of the geodesic lifting σ from ˜ v ;by construction, it is 0. So τ ˜ v ◦ τ v is the identity. In the same way, one checks that τ v ◦ τ ˜ v isthe identity. In this sense, τ ˜ v is the inverse of τ v .Given a geodesic loop σ with length bounded by ρ , let us call “sub-pseudo-group generatedby σ in Γ( x, ρ )” the pseudo-group Γ σ ( x, ρ ) which we describe now : it contains an element τ v of Γ( x, ρ ) if and only if v is the tip of a piecewise geodesic segment staying in ˆ B (0 , ρ ) andobtained by lifting several times σ from 0. If τ is an element of Γ( x, ρ ) which correspondsto a loop σ , we will also write Γ τ ( x, ρ ) for the sub-pseudogroup generated by τ in Γ( x, ρ ). If k is the largest integer such that τ i (0) belongs to the ball ˆ B (0 , ρ ) for every natural number i ≤ k , then: Γ τ ( x, ρ ) = Γ σ ( x, ρ ) = (cid:8) τ i / − k ≤ i ≤ k (cid:9) . If 2 ρ ≤ ρ ′ < π , then the orbit space of the points of the ball ˆ B (0 , ρ ) under the actionof Γ( x, ρ ′ ), ˆ B (0 , ρ ) / Γ( x, ρ ′ ), is isometric to B ( x, ρ ), through the factorization of exp x . Theonly thing to check is the injectivity. Given two lifts w , w ∈ ˆ B (0 , ρ ) of the same point y ∈ B ( x, ρ ), let us prove they are in the same orbit for Γ( x, ρ ′ ). Consider the uniquegeodesic γ from w to 0, push it by exp x and lift the resulting geodesic from w to obtaina geodesic γ , from w to some point v (cf. figure 3). Then v is a lift of x in ˆ B (0 , ρ ′ ) (bytriangle inequality) and τ v maps w to w , hence the result.We will need to estimate the number N x ( y, ρ ) of lifts of a given point y in the ball ˆ B (0 , ρ )of T x M . Lifting one shortest geodesic loop from 0 =: v , we arrive at some point v . Liftingthe same loop from v , we arrive at a new point v , etc. This construction yields a sequenceof lifts v k of x which eventually goes out of ˆ B (0 , ρ ): otherwise, since there cannot exist anaccumulation point, the sequence would be periodic; τ v would then fix the centre of theunique ball with minimal radius which contains all the points v k , which is not possible, since τ v is nontrivial hence has no fixed point (the uniqueness of the ball stems from the strictconvexity of the balls, cf. [G1]). Of course, one can do the same thing with the reverseorientation of the same loop. Since the distance between two points v k is at least 2 inj( x ),this yields at least ρ/ inj( x ) lifts of x in ˆ B (0 , ρ ): | Γ( x, ρ ) | = N x ( x, ρ ) ≥ ρ/ inj( x ) . Lifting one shortest geodesic between x and some point y from the lifts of x and estimatingthe distance between the tip and 0 with the triangle inequality (cf. figure 9), we get:(9) N x ( y, ρ ) ≥ N x ( x, ρ − d ( x, y )) = | Γ( x, ρ − d ( x, y )) | ≥ ρ − d ( x, y )inj( x ) . M w v T x M ˆ B (0 , ρ ) ˆ B (0 , ρ )exp x γ yγ x w Figure 3. τ v ( w ) = w .For d ( x, y ) ≤ ρ/
2, this yields:(10) ρ x ) vol B ( x, ρ/ ≤ | Γ( x, ρ/ | vol B ( x, ρ/ ≤ vol ˆ B (0 , ρ ) . M xT x M exp x ˆ B (0 , ρ ) v v v − yv = 0 v − Figure 4.
Take a minimal geodesic between x and y and lift it from everypoint in the fiber of x to obtain points in the fiber of y .For ρ ≤ ρ ′ < π , the set F ( x, ρ, ρ ′ ) := n w ∈ ˆ B (0 , ρ ) . ∀ γ ∈ Γ( x, ρ ′ ) , d (0 , γ ( w )) ≥ d (0 , w ) o is a fundamental domain for the action of Γ( x, ρ ′ ) on the ball ˆ B (0 , ρ ). Finiteness ensures eachorbit intersects F . Furthermore, if τ belongs to Γ( x, ρ ′ ), the set F ( x, ρ, ρ ′ ) ∩ τ ( F ( x, ρ, ρ ′ ))consists of points whose distances to 0 and τ (0) are equal, hence has zero measure: by N ASYMPTOTICALLY FLAT MANIFOLDS WITH NON MAXIMAL VOLUME GROWTH. 13 finiteness again, up to a set with zero measure, F ( x, ρ, ρ ′ ) contains a unique element of eachorbit. For the same reason, if τ belongs to Γ( x, ρ ′ ), the set F τ ( x, ρ, ρ ′ ) := n w ∈ ˆ B (0 , ρ ) . ∀ γ ∈ Γ τ ( x, ρ ′ ) , d (0 , γ ( w )) ≥ d (0 , w ) o is a fundamental domain for the action of the sub-pseudo-group Γ τ ( x, ρ ′ ). From our discus-sion follows an important fact: if 2 ρ ≤ ρ ′ < π , thenvol F ( x, ρ, ρ ′ ) = vol B ( x, ρ ) . We will need to control the shape of these fundamental domains.
Lemma — Fix ρ ≤ ρ ′ < π and consider a nontrivial element τ in Γ( x, ρ ′ ) . Denote by I τ ( x, ρ ) the set of points w in ˆ B (0 , ρ ) such that g x ( w, τ (0)) ≤ | τ (0) | ρ | τ (0) | and g x (cid:0) w, τ − (0) (cid:1) ≤ | τ (0) | ρ | τ (0) | . Then F τ ( x, ρ, ρ ′ ) is a subset of I τ ( x, ρ ) . A picture is given by figure 5 (it represents the plane containing 0, τ (0) and τ − (0)). τ (0) τ − (0) I τ ( x, ρ ) Figure 5.
The domain I τ ( x, ρ ). Proof.
Consider a point w in F τ ( x, ρ, ρ ′ ), set v = τ (0) and denote by θ ∈ [0 , π ] the anglebetween v and w . We first assume g x ( w, v ) >
0, that is θ < π/
2. Since any two pointsin ˆ B (0 , ρ ) are connected by a unique geodesic which is therefore minimizing, we can applyToponogov theorem to all triangles. In particular, in the triangle 0 vw , we findcosh(Λ d ( v, w )) ≤ cosh(Λ | w | ) cosh(Λ | v | ) − sinh(Λ | w | ) sinh(Λ | v | ) cos θ. Observing | w | = d (0 , w ) ≤ d (0 , τ − ( w )) = d ( v, w ) (figure 6 shows what we expect), we getcosh(Λ | w | ) ≤ cosh(Λ d ( v, w )) ≤ cosh(Λ | w | ) cosh(Λ | v | ) − sinh(Λ | w | ) sinh(Λ | v | ) cos θ, vθw Figure 6. d (0 , w ) ≤ d ( v, w ) implies w is on the left of the dotted line, up toan error term.hence tanh(Λ | w | ) cos θ ≤ cosh(Λ | v | ) − | v | ) . With g x ( v, w ) = | v | | w | cos θ , it follows that(11) g x ( v, w ) | v | ≤ Λ | w | tanh Λ | w | cosh(Λ | v | ) − | v | sinh(Λ | v | ) . Taylor formulas yield Λ | w | tanh Λ | w | ≤ ρ . and cosh Λ | v | − | v | sinh Λ | v | ≤
12 + Λ ρ . Combining this with Λ ρ ≤ π <
1, one can see that (11) implies g x ( v, w ) | v | ≤
12 + Λ ρ . Assuming g x ( w, τ − (0)) >
0, we can work in the same way (with v = τ − (0)) so as tocomplete the proof. (cid:3) To understand the action of the elements in the fundamental pseudo-group, the followinglemma is useful: it approximates them by affine transformations.
Lemma — Consider a complete Riemannian manifold ( M, g ) and a point x in M suchthat the curvature is bounded by Λ , Λ ≥ , on the ball B ( x, ρ ) , ρ > , with Λ ρ < π/ . Let v be a lift of x in ˆ B (0 , ρ ) ⊂ T x M . Define • the translation t v with vector v in the affine space T x M , • the parallel transport p v along t exp x tv , from t = 0 to t = 1 . • the map τ v = Exp v ◦ ( T v exp x ) − ,where Exp denotes the exponential map of ( T x M, exp ∗ x g ) . Then for every point w in ˆ B (0 , ρ −| v | ) , d ( τ v ( w ) , t v ◦ p − v ( w )) ≤ Λ | v | | w | ( | v | + | w | ) . N ASYMPTOTICALLY FLAT MANIFOLDS WITH NON MAXIMAL VOLUME GROWTH. 15
Proof.
Proposition 6 . V is defined byExp V = v and if W belongs to T T x M , then(12) d (Exp v ◦ ˆ p v ( W ) , Exp ( V + W )) ≤
13 Λ | V | | W | sinh(Λ( | V | + | W | )) sin ∠ ( V, W ) , where ˆ p v is the parallel transport along t Exp tV , from t = 0 to t = 1. Set w = Exp W .We stress the fact that Exp = T exp x is nothing but the canonical identification betweenthe tangent space T T x M to the vector space T x M and the vector space itself, T x M . Inparticular, Exp ( V + W ) = v + w = t v ( w ). Since exp x is a local isometry, we haveˆ p v = ( T v exp x ) − ◦ p v ◦ T exp x , so that Exp v ◦ ˆ p v ( W ) = τ v ◦ p v ( w ). Withsinh(Λ( | V | + | W | )) ≤ Λ( | V | + | W | ) cosh(1) ≤ | V | + | W | ) , it follows from (12) that: d ( τ v ◦ p v ( w ) , t v ( w )) ≤ Λ | v | | w | ( | v | + | w | ) . Changing w into p − v ( w ), we obtain the result. (cid:3) Fundamental pseudo-group and volume.
Back to the injectivity radius.
Our discussion of the fundamental pseudo-group enablesus to recover a result of [CGT].
Proposition — Let ( M n , g ) be a completeRiemannian manifold. Assume the existence of Λ ≥ and V > such that for every point x in M , | Rm x | ≤ Λ and vol B ( x, ≥ V. Then the injectivity radius admits a positive lower bound I = I ( n, Λ , V ) .Proof. Set ρ = min(1 , π ) and assume there is a point x in M and a geodesic loop based at x with length bounded by ρ . Apply (10) to find ρ x ) vol B ( x, ρ/ ≤ vol ˆ B (0 , ρ ) . Bishop theorem estimates the right-hand side by ω n cosh(Λ ρ ) n − ρ n , where ω n is the volumeof the unit ball in R n . We thus obtain inj( x ) ≥ C ( n, Λ) vol B ( x, ρ/
2) for some C ( n, Λ) > C ′ ( n, Λ) > B ( x, ≤ C ′ ( n, Λ) − vol B ( x, ρ/ , we are left with inj( x ) ≥ C ( n, Λ) C ′ ( n, Λ) vol B ( x, ≥ C ( n, Λ) C ′ ( n, Λ) V . (cid:3) Combining propositions 2.1 and 2.5, we obtain
Corollary — Let ( M n , g ) be a complete Riemannianmanifold with bounded curvature. Suppose: ∀ x ∈ M, V ≤ vol B ( x, t ) ≤ ω ( t ) t n for some positive number V and some function ω going to zero at infinity. Then there arepositive numbers I , I such that for any point x in M : I ≤ inj( x ) ≤ I . Self-improvement of volume estimates.
Here and in the sequel, we will always distin-guish a point o in our complete non-compact Riemannian mainfolds. The distance functionto o will always be denoted by r o or r . We will always assume our manifolds are smooth andconnected. Proposition — Let ( M n , g ) be a complete non-compact Riemannian manifold withfaster than quadratic curvature decay, i.e. | Rm | = O ( r − − ǫ ) for some ǫ > . If there exists a function ω which goes to zero at infinity and satisfies ∀ x ∈ M, ∀ t ≥ , vol B ( x, t ) ≤ ω ( t ) t n , then there is in fact a number B such that ∀ x ∈ M, ∀ t ≥ , vol B ( x, t ) ≤ Bt n − . Indeed, under faster than quadratic curvature decay assumption, if C ( n ) denotes theconstant in 2.1, the property inf t> lim sup x −→∞ vol B ( x, t ) t n < C ( n ) , automatically implies lim sup t −→∞ sup x ∈ M vol B ( x, t ) t n − < ∞ . Proof.
Proposition 2.1 yields an upper bound I on the injectivity radius. Our assumptionon the curvature implies that, given a point x in M \ B ( o, R ), with large enough R , onecan apply (10) with 2 I ≤ ρ = 2 t ≤ r ( x ) / t inj( x ) vol B ( x, t ) ≤ vol ˆ B (0 , t ) . Thanks to the curvature decay, if R is large enough, Bishop theorem bounds the right-handside by ω n cosh(1) n − (2 t ) n ; with proposition 2.1, it follows that for I ≤ t ≤ r ( x ) / B ( x, t ) ≤ ω n cosh(1) n − n I t n − . We have found some number B such that for every x outside the ball B ( o, R ) and for every t in [ I , r ( x ) / B ( x, t ) ≤ B t n − . From lemma 3.6 in [LT], which refers to the construction in the fourth paragraph of [A2], wecan find a number N such that for any natural number k , setting R k = R k , the annulus A k := B ( o, R k ) \ B ( o, R k ) is covered by a family of balls ( B ( x k,i , R k / ≤ i ≤ N centered in A k . Since the volume of the balls B ( x k,i , R k /
2) is bounded by B ( R k / n − , we deduce theexistence of a constant B such that for every t ≥ I ,vol B ( o, t ) ≤ B ⌈ log ( t/R ) ⌉ X k =0 (2 k ) n − , and thus, for some new constant B , we have(14) ∀ t ≥ I , vol B ( o, t ) ≤ B t n − . Now, for every x in M \ B ( o, R ) and every t ≥ r ( x ) /
4, we can writevol B ( x, t ) ≤ vol B ( o, t + r ( x )) vol B ( o, t ) ≤ n − B t n − . N ASYMPTOTICALLY FLAT MANIFOLDS WITH NON MAXIMAL VOLUME GROWTH. 17
And when x belongs to B ( o, R ), for t ≥ I , we observevol B ( x, t ) ≤ vol B ( o, t + R ) ≤ vol B ( o, (1 + R / t ) ≤ B (1 + R / n − t n − . Therefore there is a constant B such that for every x in M and every t ≥ I , the volume ofthe ball B ( x, t ) is bounded by Bt n − . The result immediately follows. (cid:3) When the Ricci curvature is nonnegative, the assumption on the curvature can be relaxed.
Proposition — Let ( M n , g ) be a complete non-compact Riemannian manifold withnonnegative Ricci curvature and quadratic curvature decay, i.e. | Rm | = O ( r − ) . If there exists a function ω which goes to zero at infinity and satisfies ∀ x ∈ M, ∀ t ≥ , vol B ( x, t ) ≤ ω ( t ) t n , then there is in fact a number B such that ∀ x ∈ M, ∀ t ≥ , vol B ( x, t ) ≤ Bt n − . Proof.
The previous proof can easily be adapted. (13) holds for I ≤ t ≤ δr ( x ), with a small δ >
0. The existence of the covering leading to (14) stems from Bishop-Gromov theorem(the x k,i are given by a maximal R k / (cid:3) This threshold effect shows that the first collapsing situation to study is that of a “codi-mension 1” collapse, where the volume of balls with radius t is (uniformly) comparable to t n − . This explains the gap between ALE and ALF gravitational instantons, under a uniformupper bound on the volume growth: there is no gravitational instanton with intermediatevolume growth, between vol B ( x, t ) ≍ t and vol B ( x, t ) ≍ t .3. Collapsing at infinity.
Local structure at infinity.
We turn to codimension 1 collapsing at infinity. It turnsout that the holonomy of short geodesic loops plays an important role. In order to obtaina nice structure, we will make a strong assumption on it. The next paragraph will explainwhy gravitational instantons satisfy this assumption.
Proposition — Let ( M n , g ) be a complete Rie-mannian manifold such that ∀ x ∈ M, ∀ t ≥ , At n − ≤ vol B ( x, t ) ≤ Bt n − with < A ≤ B . We assume there is constant c > such that | Rm | ≤ c r − and such that if γ is a geodesic loop based at x and with length L ≤ c − r ( x ) , then theholonomy H of γ satisfies | H − id | ≤ cLr ( x ) . Then there exists a compact set K in M such that for every x in M \ K , there is a uniquegeodesic loop σ x of minimal length x ) . Besides there are geometric constants L and κ > such that the fundamental pseudo-group Γ( x, κr ( x )) has at most Lr ( x ) elements, allof which are obtained by successive lifts of σ . Definition — σ x is the “fundamental loop at x ”. Proof.
Let us work around a point x far away from o , say with r ( x ) > I c . Recall(2.6) yields constants I , I such that 0 < I ≤ inj ≤ I . The fundamental pseudo-groupΓ := Γ( x, r ( x )4 c ) contains the sub-pseudo-group Γ σ := Γ σ ( x, r ( x )4 c ) corresponding to the loop σ of minimal length 2 inj( x ). Denote by τ = τ v one of the two elements of Γ that correspondto σ : | v | = 2 inj( x ). (10) implies for ρ = r ( x )2 c : | Γ | vol B (cid:18) x, r ( x )4 c (cid:19) ≤ vol ˆ B (cid:18) , r ( x )2 c (cid:19) . Bishop theorem bounds (from above) the Riemannian volume of ˆ B (0 , r ( x )2 c ) by (cosh c ) n timesits Euclidean volume. With the lower bound on the volume growth, we thus obtain: | Γ | A (cid:18) r ( x )4 c (cid:19) n − ≤ (cosh c ) n ω n (cid:18) r ( x )2 c (cid:19) n , where ω n denotes the volume of the unit ball in R n . We deduce the estimate | Γ | ≤ Lr ( x )with L := 2 n − ω n (cosh c ) n Ac .
Now, consider an oriented geodesic loop γ , based at x and with length inferior to r ( x )4 c . Name τ z the corresponding element of Γ := Γ( x, r ( x )4 c ). H z will denote the holonomy of the oppositeorientation of γ . By assumption, | H z − id | ≤ c | z | r ( x ) . The vector z = τ z (0) is the initial speed of the geodesic γ , parameterized by [0 ,
1] in thechosen orientation. In the same way, τ − z (0) is the initial speed vector of γ , parameterizedby [0 , − z is obtained as the parallel transportof τ − z (0) along γ : H z ( τ − z (0)) = − z . From the estimate above stems:(15) (cid:12)(cid:12) τ − z (0) + z (cid:12)(cid:12) ≤ c | z | r ( x ) . Given a small λ , say λ = c , we consider a point w in the domain I τ z ( x, λr ( x )) (see thedefinition in 2.3). It satisfies g x ( w, τ − z (0)) ≤ | z | c λ | z | . With g x ( w, z ) = − g x ( w, τ − z (0)) + g x ( w, τ − z (0) + z ) ≥ − g x ( w, τ − z (0)) − | w | (cid:12)(cid:12) τ − z (0) + z (cid:12)(cid:12) , we find g x ( w, z ) ≥ − | z | − c λ | z | − λc | z | , that is g x ( w, z ) ≥ − | z | (cid:0) c λ + 2 λc (cid:1) . With lemma 2.3, this ensures: F τ z (cid:18) x, λr ( x ) , r ( x )4 (cid:19) ⊂ ( w ∈ ˆ B (0 , λr ( x )) . | g x ( w, z ) | ≤ | z | (cid:0) c λ + 2 λc (cid:1)) . N ASYMPTOTICALLY FLAT MANIFOLDS WITH NON MAXIMAL VOLUME GROWTH. 19
And with λ = c , this leads to(16) F τ z (cid:18) x, λr ( x ) , r ( x )4 (cid:19) ⊂ ( w ∈ ˆ B (0 , λr ( x )) . | g x ( w, z ) | ≤ | z | ) . Let τ ′ be an element of Γ \ Γ σ such that v ′ := τ ′ (0) has minimal norm. Suppose | v ′ | < λr ( x ).Then, the minimality of | v ′ | combined with (16) yields (cid:12)(cid:12) g x (cid:0) v ′ , v (cid:1)(cid:12)(cid:12) ≤ | v | . If θ ∈ [0 , π ] is the angle between v and v ′ , this means: | v ′ | | cos θ | ≤ . | v | . Since | v | ≤ | v ′ | ,we deduce | cos θ | ≤ .
75, hence sin θ ≥ .
5. Applying (16) to τ and τ ′ , we also get F (cid:18) x, λr ( x ) , r ( x )4 c (cid:19) ⊂ F τ v (cid:18) x, λr ( x ) , r ( x )4 c (cid:19) ∩ F τ v ′ (cid:18) x, λr ( x ) , r ( x )4 c (cid:19) ⊂ n w ∈ ˆ B (0 , λr ( x )) . | g x ( w, v ) | ≤ | v | , (cid:12)(cid:12) g x (cid:0) w, v ′ (cid:1)(cid:12)(cid:12) ≤ (cid:12)(cid:12) v ′ (cid:12)(cid:12) o . θ vv ′ τ − v ′ (0) τ − v (0) 0 θ Figure 7.
The fundamental domain is inside the dotted line.The Riemannian volume of F ( x, λr ( x ) , r ( x ) / (4 c )) equals that of B ( x, λr ( x )), so it is notsmaller than Aλ n − r ( x ) n − . The Euclidean volume of n w ∈ ˆ B (0 , λr ( x )) . | g x ( w, v ) | ≤ | v | and (cid:12)(cid:12) g x (cid:0) w, v ′ (cid:1)(cid:12)(cid:12) ≤ (cid:12)(cid:12) v ′ (cid:12)(cid:12) o is not greater than 4 | v | | v ′ | (2 λr ( x )) n − / sin θ ≤ n +2 λ n − I | v ′ | r ( x ) n − . Comparison yields Aλ n − r ( x ) n − ≤ n +2 (cosh c ) n λ n − I (cid:12)(cid:12) v ′ (cid:12)(cid:12) r ( x ) n − , that is (cid:12)(cid:12) v ′ (cid:12)(cid:12) ≥ λA n +2 I (cosh c ) n r ( x ) . Given a positive number κ which is smaller than λ and λA n +2 I (cosh c ) n , we conclude that forany x outside some compact set, the pseudo-group Γ( x, κr ( x )) only consists of iterates of τ (in Γ( x, r ( x )4 c )). Suppose there are two geodesic loops with minimal length 2 inj( x ) at x . They correspondto distinct points v and v ′ in T x M . Write F (cid:18) x, λr ( x ) , r ( x )4 (cid:19) ⊂ F τ v (cid:18) x, λr ( x ) , r ( x )4 (cid:19) ∩ F τ v ′ (cid:18) x, λr ( x ) , r ( x )4 (cid:19) ⊂ n w ∈ ˆ B (0 , λr ( x )) . | g x ( w, v ) | ≤ | v | , (cid:12)(cid:12) g x (cid:0) w, v ′ (cid:1)(cid:12)(cid:12) ≤ | v | o . As above, we find Aλ n − r ( x ) n − ≤ n (cosh c ) n λ n − | v | (cid:12)(cid:12) v ′ (cid:12)(cid:12) r ( x ) n − / sin θ, where θ ∈ [0 , π ] is the angle between the vectors v and v ′ . Here, | v | = | v ′ | ≤ I , so Aλr ( x ) sin θ ≤ n +2 I (cosh c ) n . The minimality of | v | and distance comparison yield | v | ≤ d ( v, v ′ ) ≤ cosh(0 . (cid:12)(cid:12) v − v ′ (cid:12)(cid:12) hence cos θ ≤ .
51. For the same reason, we find | v | ≤ d ( τ − v (0) , v ′ ) ≤ cosh(0 . (cid:12)(cid:12) τ − v (0) − v ′ (cid:12)(cid:12) . With (15), which gives (cid:12)(cid:12) τ − v (0) + v (cid:12)(cid:12) ≤ . | v | , we deduce (cid:12)(cid:12) v + v ′ (cid:12)(cid:12) ≥ (cid:12)(cid:12) τ − v (0) − v ′ (cid:12)(cid:12) − (cid:12)(cid:12) τ − v (0) + v (cid:12)(cid:12) ≥ . | v | hence cos θ ≥ − .
52, then | cos θ | ≤ .
52, and sin θ ≥ .
8. Eventually, we obtain0 . Aλr ( x ) ≤ n +2 I (cosh c ) n , which cannot hold if x is far enough from o . This proves the uniqueness of the shortestgeodesic loop. (cid:3) Uniqueness implies smoothness:
Lemma — In the setting of proposition 3.1, there are smooth local parameterizations forthe family of loops ( σ x ) x . More precisely, given an orientation of σ x , we can lift it to T x M through exp x ; denoting the tip of the resulting segment by v , if w is in neighborhood of in T x M , then the fundamental loop at exp x w is the image by exp x of the unique geodesicconnecting w to τ v ( w ) .Proof. We first prove continuity. Let y be in M (outside the compact set K ) and let ( y n )be a sequence converging to y . Let V n be a sequence of of initial unit speed vector for σ y n .Compactness ensures V n can be assumed to converge to V . Let α be the geodesic emanatingfrom y with initial speed V . For every index n , we have exp y n (inj( y n ) V n ) = σ y n (inj( y n )) = y n .Continuity of the injectivity radius ([GLP]) allows to take a limit: α (inj( y )) = exp y inj( y ) V = y . Uniqueness implies α parameters σ y . This yields the continuity of ( σ x ) x . Now, given w in a neighborhood of 0 in T x M , consider the e ( w ) of the lift of σ exp x w . The map e is acontinuous section of exp x and e (0) = τ v (0) : e = τ v . The result follows. (cid:3) Now we turn to gravitational instantons: we can control their holonomy and thus applythe previous proposition.
N ASYMPTOTICALLY FLAT MANIFOLDS WITH NON MAXIMAL VOLUME GROWTH. 21
Holonomy in gravitational instantons.
Lemma — Let ( M , g ) be a complete hyperk¨ahler manifold with: inj( x ) ≥ I > and | Rm | ≤ Qr − . Then there is some positive c = c ( I , Q ) such that the holonomy H of geodesic loops basedat x and with length L ≤ r ( x ) /c satisfies | H − id | ≤ cr ( x ) . Proof.
Consider a point x (far from o ) and a geodesic loop based at x , with L ≤ r ( x ) / τ v ∈ Γ( x, r ( x ) /
4) be a corresponding element. Thanks to (2.4), we know that for everypoint w in T x M such that | w | ≤ r ( x ) / d ( τ v ( w ) , t v ◦ p − v ( w )) ≤ Qr ( x ) − | v | | w | ( | v | + | w | )and therefore d ( τ v ( w ) , t v ◦ p − v ( w )) ≤ QLr ( x ) − . Set H = p − v : d ( τ v ( w ) , Hw + v ) ≤ QLr ( x ) − . Since we are working on a hyperk¨ahler 4-manifold, the holonomy group is included in SU (2),so that in some orthonormal basis of T x M , seen as complex 2-space, H reads H = (cid:18) e iθ e − iθ (cid:19) with an angle θ in ] − π, π ]. Suppose θ is not zero (otherwise the statement is trivial). Theequation Hw + v = w admits a solution: w = (cid:18) v − e iθ v − e − iθ (cid:19) where v and v denote the coordinates of v . If | w | ≤ r ( x ) /
4, we obtain d ( τ v ( w ) , w ) ≤ QLr ( x ) − . The lower bound on the injectivity radius yields d ( τ v ( w ) , w ) ≥ I . So we find L ≥ I r ( x ) Q .
As a consequence, if
L < I r ( x ) Q , then | w | = L | − e iθ | > r ( x )4 , that is | H − id | = (cid:12)(cid:12) − e iθ (cid:12)(cid:12) ≤ Lr ( x ) − . (cid:3) As a result, we obtain the
Proposition — Let ( M , g ) be a complete hyperk¨ahler manifold with Z M | Rm | rdvol < ∞ and ∀ x ∈ M, ∀ t ≥ , At ≤ vol B ( x, t ) ≤ Bt with < A ≤ B . Then there exists a compact set K in M such that for every x in M \ K ,there is a unique geodesic loop σ x of minimal length x ) . Besides there are geometricconstants L and κ > such that the fundamental pseudo-group Γ( x, κr ( x )) has at most Lr ( x ) elements, all of which are obtained by successive lifts of σ .Proof. Since M is hyperk¨ahler, it is Ricci flat. So [Min] applies (see appendix A): | Rm | = O ( r − ). So we use proposition 3.1, thanks to lemma 3.4. (cid:3) Remark . From now on, we will remain in the setting of four dimensional hyperk¨ahlermanifolds. It should nonetheless be noticed that the only reason for this is lemma 3.4. If theconclusion of this lemma is assumed and if we suppose convenient estimates on the covariantderivatives of the curvature tensor, then we can work in any dimension (see 3.26 below).
An estimate on the holonomy at infinity.
To go on, we will need a better estimateof the holonomy of short loops. This is the goal of this paragraph. First, let us state an easylemma, adapted from [BK].
Lemma — Let γ : [0 , L ] −→ N be a curve in a Riemannianmanifold N and let t α t be a family of loops, parameterized by ≤ s ≤ l with α t (0) = α t ( l ) = γ ( t ) . We denote by p γ ( t ) the parallel transportation along γ , from γ (0) to γ ( t ) . Weconsider a vector field ( s, t ) X ( s, t ) along the family α and we suppose it is parallel alongeach loop α t ( ∇ s X ( s, t ) = 0 ) and along γ ( ∇ t X (0 , t ) = 0 ). Then: (cid:12)(cid:12) p γ ( L ) − X ( l, L ) − X ( l, (cid:12)(cid:12) ≤ Z L Z l | Rm( ∂ s σ t , ∂ t σ t ) X ( s, t ) | dsdt. γ (0) γ ( t ) α t α L α γ ( L ) Figure 8.
A one parameter family of loops.Consider a complete hyperk¨ahler manifold ( M , g ) with ∀ x ∈ M, ∀ t ≥ , At ≤ vol B ( x, t ) ≤ Bt
3N ASYMPTOTICALLY FLAT MANIFOLDS WITH NON MAXIMAL VOLUME GROWTH. 23 (0 < A ≤ B ) and Z M | Rm | rdvol < ∞ or equivalently | Rm | = O ( r − ) . We choose a unit ray γ : R + −→ M starting from o and we denote by p γ ( t ) the paralleltransportation along γ , from γ (0) to γ ( t ). For large t , we can define the holonomy endomor-phism H γ ( t ) of the fundamental loop σ γ ( t ) : here, there is an implicit choice of orientationfor the loops σ γ ( t ) , which we can assume continuous. This yields an element of O ( T γ ( t ) M ).Holonomy comparison lemma 3.6 asserts that for large t ≤ t : (cid:12)(cid:12) p γ ( t ) − H γ ( t ) p γ ( t ) − p γ ( t ) − H γ ( t ) p γ ( t ) (cid:12)(cid:12) ≤ Z ∞ t Z | Rm | ( c ( t, s )) | ∂ s c ( t, s ) ∧ ∂ t c ( t, s )) | dsdt, where, for every fixed t , c ( t, . ) parameterizes σ γ ( t ) by [0 , γ ( t )). Lemma — | ∂ s c ∧ ∂ t c ) | is uniformly bounded.Proof. The upper bound on the injectivity radius bounds | ∂ s c | . We need to bound thecomposant of ∂ t c that is orthogonal to ∂ s c . We concentrate on a neighborhhood of somepoint x along γ . For convenience, we change the parameterization so that x = γ (0). We alsolift the problem to T x M =: E , endowed with the lifted metric ˆ g . If v = γ ′ (0), γ lifts as acurve ˆ γ parameterized by t tv . The lift ˆ c of c consists of the geodesics ˆ c ( t, . ) connecting tv to τ ( tv ); τ is the element of the fundamental pseudo-group corresponding to σ x , for thechosen orientation. Observe ˆ c ( t, s ) = Exp tv sX ( t )where X ( t ) ∈ T tv E is defined by Exp tv X ( t ) = τ ( tv ) . The vector field J defined along ˆ c (0 , . ) by J ( s ) = ∂ t ˆ c (0 , s ) = ddt (cid:12)(cid:12)(cid:12) t =0 Exp tv sX ( t )is a Jacobi field with initial data J (0) = ddt (cid:12)(cid:12)(cid:12) t =0 tv = v (we identify E to T E thanks to Exp , in the natural way) and J ′ (0) = ( ∇ s ∂ t ˆ c ) (cid:12)(cid:12)(cid:12) ( t,s )=(0 , = ( ∇ t ∂ s ˆ c ) (cid:12)(cid:12)(cid:12) ( t,s )=(0 , = ( ∇ t ∂ s Exp tv sX ( t )) (cid:12)(cid:12)(cid:12) ( t,s )=(0 , = ( ∇ t X ) (0) . Suppose the curvature is bounded by Λ , Λ >
0, in the area under consideration and applylemma 6.3.7 of [BK]: the part ˜ J of J that is orthogonal to ˆ c (0 , . ) satisfies (cid:12)(cid:12)(cid:12) ˜ J ( s ) − p ( sv ) ˜ J (0) − sp ( sv ) ˜ J ′ (0) (cid:12)(cid:12)(cid:12) ≤ a ( s ) where p ( . ) is the radial parallel transportation and where a solves a ′′ − Λ a = Λ (cid:16)(cid:12)(cid:12)(cid:12) ˜ J (0) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ˜ J ′ (0) (cid:12)(cid:12)(cid:12)(cid:17) with a (0) = a ′ (0) = 0, i.e. a ( s ) = (cid:16)(cid:12)(cid:12)(cid:12) ˜ J (0) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ˜ J ′ (0) (cid:12)(cid:12)(cid:12)(cid:17) (cosh(Λ s ) − . Since here 0 ≤ s ≤ <<
1, we only need a bound on (cid:12)(cid:12)(cid:12) ˜ J (0) (cid:12)(cid:12)(cid:12) and (cid:12)(cid:12)(cid:12) ˜ J ′ (0) (cid:12)(cid:12)(cid:12) to control ˜ J and end the proof. Since J (0) = v has unit length, we are left to bound ˜ J ′ (0).We consider the family of vectors Y in T E ∼ = E that is defined by X ( t ) = p ( tv ) Y ( t ) . With ∇ t X ( t ) = p ( tv ) ddt p ( tv ) − X ( t ) = p ( tv ) Y ′ ( t ) , we see that J ′ (0) is exactly Y ′ (0). Let f be the map from E × T E ∼ = E to E that is definedby f ( w, W ) = Exp w p ( w ) W. The equality f ( tv, Y ( t )) = τ ( tv ) . can be differentiated into(17) ∂ f (0 ,Y (0)) v + ∂ f (0 ,Y (0)) Y ′ (0) = ( Dτ ) v. Lemma 6.6 in [BK] ensures ∂ f (0 ,Y (0)) is Λ -close to the identity. Besides, τ is an isometryfor ˆ g , so ( Dτ ) is uniformly bounded. Finally, ∂ f (0 ,Y (0)) v is the value at time 1 of the Jacobifield K along s sY (0) corresponding to the geodesic variation H ( t, s ) Exp tv sp ( tv ) Y (0) . As the initial data for K ( s ) = ∂ t H (0 , s ) are K (0) = v and K ′ (0) = 0, we obtain (corollary6.3.8 of [BK]) a bound on K and thus on ∂ f (0 ,Y (0)) v . This yields a bound on Y ′ (0) (thanksto 17) and we are done. (cid:3) This lemma and the curvature decay lead to the estimate (cid:12)(cid:12) p γ ( t ) − H γ ( t ) p γ ( t ) − p γ ( t ) − H γ ( t ) p γ ( t ) (cid:12)(cid:12) ≤ C Z ∞ t t − dt ≤ C t − . Now recall the holonomy of the loops under consideration goes to the identity at infinity.Setting t =: t and letting t go to infinity, we find(18) (cid:12)(cid:12) p γ ( t ) − H γ ( t ) p γ ( t ) − id (cid:12)(cid:12) ≤ C t − . Since M has zero (hence nonnegative) Ricci curvature and cubic volume growth, it followsfrom Cheeger Gromoll theorem that M has only one end. Relying on faster than quadraticcurvature decay, [Kas] then ensures large spheres S ( o, t ) are connected with intrinsic diameterbounded by Cs . Thus every point x in S ( o, t ) is connected to γ ( t ) by some curve β withlength at most Ct et and remaining outside B ( o, t/ (cid:12)(cid:12)(cid:12) p − β H x p β − H γ ( t ) (cid:12)(cid:12)(cid:12) ≤ Ct − , N ASYMPTOTICALLY FLAT MANIFOLDS WITH NON MAXIMAL VOLUME GROWTH. 25 where p β is the parallel transportation along β and H x is the holonomy endomorphismcorresponding to a consistent orientation of σ x . It follows that(20) | H x − id | ≤ Cr ( x ) − . So we have managed to improve our estimate on the holonomy of fundamental loops.
Lemma — Let ( M , g ) be a complete hyperk¨ahler manifold with Z M | Rm | rdvol < ∞ and ∀ x ∈ M, ∀ t ≥ , At ≤ vol B ( x, t ) ≤ Bt ( < A ≤ B ). Then the holonomy H x of the fundamental loops σ x satisfies | H x − id | ≤ Cr ( x ) − . Local Gromov-Hausdorff approximations.
A first way to describe the local geom-etry consists in saying that, “seen from far away”, it is close to a simpler geometry. We willshow that the local geometry of the codimension 1 collapsings that we are looking at is closeto the Euclidean geometry in the immediately inferior dimension, for the Gromov-Hausdorfftopology.
Remark . Cubic curvature decay is important in the following lemma. A (not so) heuristicreason is the following. Forget geodesic loops and look at the metric in the exponential chartat x . If the curvature is bounded by Λ in the area under consideration, comparison asserts (cid:18) sin Λ r Λ r (cid:19) g x ≤ exp ∗ x g ≤ (cid:18) sinh Λ r Λ r (cid:19) g x on a scale r << Λ − . The corresponding distances thus obey sin Λ r Λ r d g x ≤ d exp ∗ x g ≤ sinh Λ r Λ r d g x , hence (cid:12)(cid:12) d exp ∗ x g − d g x (cid:12)(cid:12) ≤ C Λ r d g x . If we want to control the difference between these distances by some constant on the scale r ,we therefore need a bound on Λ r r , which means the curvature ( Λ ) should be bounded by r − . The following lemma ensures the elements of the fundamental group are almost transla-tions.
Lemma — Let ( M , g ) be a complete hyperk¨ahler manifold with Z M | Rm | rdvol < ∞ and ∀ x ∈ M, ∀ t ≥ , At ≤ vol B ( x, t ) ≤ Bt ( < A ≤ B ). Then there exists a compact set K in M and geometric constants J , L , κ > such that for every point x in M \ K and every τ in Γ( x, κr ( x )) , one has ∀ w ∈ ˆ B (0 , κr ( x )) , | τ ( w ) − t kv x ( w ) | ≤ J where v x is a lift of the tip of σ x and k is a natural number bounded by Lr ( x ) . Proof.
Proposition 3.1 asserts we can write τ = τ kv x , where v x is a lift of a tip of σ x and k isa natural number bounded by Lr ( x ). Lemma 2.4 ensures that for every w in ˆ B (0 , r ( x ) / (cid:12)(cid:12) τ v x ( w ) − v x − p − v x ( w ) (cid:12)(cid:12) ≤ Cr ( x ) − | v x | | w | ( | v x | + | w | ) . Thanks to cubic curvature decay, (20) yields: (cid:12)(cid:12) p − v x ( w ) − w (cid:12)(cid:12) ≤ Cr ( x ) − | w | . Combining these estimates, we obtain: | τ v x ( w ) − t v x ( w ) | = | τ v x ( w ) − v x − w | ≤ Cr ( x ) − | w | . For every natural number i ≤ k , we set e i = τ iv x − t iv x and observe the formula e i +1 − e i = e ◦ τ iv x . With (cid:12)(cid:12) τ iv x ( w ) (cid:12)(cid:12) = d ( τ iv x ( w ) ,
0) = d ( τ − iv x (0) , w ) ≤ (cid:12)(cid:12) τ − iv x (0) (cid:12)(cid:12) + | w | , we find that for every w in ˆ B (0 , κr ( x )): | e i +1 ( w ) − e i ( w ) | ≤ Cr ( x ) − (cid:12)(cid:12) τ iv x ( w ) (cid:12)(cid:12) ≤ Cr ( x ) − . By induction, it follows that | e k ( w ) | ≤ Ckr ( x ) − and since k ≤ Lr ( x ), we are led to: | τ ( w ) − t kv x ( w ) | = (cid:12)(cid:12)(cid:12) τ kv x ( w ) − kv x − w (cid:12)(cid:12)(cid:12) = | e k ( w ) | ≤ C. (cid:3) Proposition — Let ( M , g ) be a complete hy-perk¨ahler manifold with Z M | Rm | rdvol < ∞ and ∀ x ∈ M, ∀ t ≥ , At ≤ vol B ( x, t ) ≤ Bt ( < A ≤ B ). Then there exists a compact set K in M and geometric constants I , κ > such that every point x in M \ K has a neighborhood Ω whose Gromov-Hausdorff distance tothe ball of radius κr ( x ) in R is bounded by I .Proof. Choose a lift of σ x in T x M and denote by v x its tip. We call H the hyperplaneorthogonal to v x and write v v H for the Euclidean orthogonal projection onto H (for g x ).If y is a point in B ( x, κr ( x ) / h ( y ) as affine center of mass of the points v H obtained from lifts v of y in ˆ B (0 , κr ( x ) / h from B ( x, κr ( x ) /
2) to H ∼ = R (we endow H of the Euclidean structure induced by g x = | . | ).We consider the ball B centered in 0 and with radius 0 . κr ( x ) in H : 0 . κ will be the κ of the statement. Let us set Ω := h − ( B ). We want to see that h : Ω −→ B is the promisedGromov-Hausdorff approximation. We need to check that this map h has I -dense image andthat for all points y and z in Ω: | d ( y, z ) − | h ( y ) − h ( z ) || ≤ I. Firstly, since v is in B , lemma 3.9 ensures that for every τ = τ kv x in Γ( x, κr ( x )), we have | τ ( v ) − v − kv x | ≤ J and thus, using Pythagore theorem, | τ ( v ) H − v | ≤ J. N ASYMPTOTICALLY FLAT MANIFOLDS WITH NON MAXIMAL VOLUME GROWTH. 27
Passing to the center of mass, we get | h (exp x v ) − v | ≤ J. If d ( v, H \ B ) > J , this proves h (exp x v ) belongs to B and therefore exp x v belongs to Ω; asa result, d ( v, h (Ω)) ≤ J . As { v ∈ B / d ( v, H \ B ) > J } is J -dense in B , we have shown that h (Ω) is 2 J -dense in B .Secondly, consider two points y and z in Ω. Lift them into v and w ( ∈ B ( x, κr ( x ) / d ( v, w ) = d ( y, z ). As above, we get | h ( y ) − v H | ≤ J and | h ( z ) − w H | ≤ J , hence || h ( y ) − h ( z ) | − | v H − w H || ≤ J. In particular, we obtain | h ( y ) − h ( z ) | ≤ | v H − w H | + 2 J ≤ | v − w | + 2 J. Comparison yields | v − w | ≤ Cr ( x ) − r ( x )sin Cr ( x ) − r ( x ) d ( v, w ) ≤ (cid:0) Cr ( x ) − (cid:1) d ( v, w )hence | v − w | ≤ d ( v, w ) + Cr ( x ) − d ( v, w ) ≤ d ( v, w ) + C. We deduce | h ( y ) − h ( z ) | ≤ d ( v, w ) + 2 J + C = d ( y, z ) + 2 J + C. Now, consider lifts v ′ and w ′ at minimal distance from H and observe lemma 2.3 yields: | v ′ − v ′ H | ≤ C et | w ′ − w ′ H | ≤ C . We deduce: (cid:12)(cid:12) v ′ − w ′ (cid:12)(cid:12) ≤ (cid:12)(cid:12) v ′ H − w ′ H (cid:12)(cid:12) + C. The distance between y and z is nothing but the infimum of the distances between their lifts,so d ( y, z ) ≤ d ( v ′ , w ′ ). As above, comparison ensures: d ( v ′ , w ′ ) ≤ (cid:12)(cid:12) v ′ − w ′ (cid:12)(cid:12) + C. These three inequalities give altogether: d ( y, z ) ≤ (cid:12)(cid:12) v ′ H − w ′ H (cid:12)(cid:12) + C And since (cid:12)(cid:12) | h ( y ) − h ( z ) | − (cid:12)(cid:12) v ′ H − w ′ H (cid:12)(cid:12)(cid:12)(cid:12) ≤ J, we arrive at d ( y, z ) ≤ | h ( y ) − h ( z ) | + 2 J + C. We have proved | d ( y, z ) − | h ( y ) − h ( z ) || ≤ I, hence the result. (cid:3) The following step consists in regularizing local Gromov-Hausdorff approximations toobtain local fibrations which accurately describe the local geometry at infinity.
Local fibrations.
The local Gromov-Hausdorff approximation that we built above hasno reason to be regular. We will now smooth it into a fibration. The technical device issimply a convolution, as in [Fuk] and [CFG]. We basically need theorem 2.6 in [CFG]. Thetrouble is this general result will have to be refined, by using fully the cubic decay of thecurvature and the symmetry properties of the special Gromov-Hausdorff approximation wesmooth. This technique requires a control on the covariant derivatives of the curvature, butit is heartening to know that this is given for free on gravitational instantons (see theoremA.6 in appendix A).We say f is a C -almost-Riemannian submersion if f is a submersion such that for everyhorizontal vector v (i.e. orthogonal to fibers), e − C | v | ≤ | df x ( v ) | ≤ e C | v | . Proposition — Let ( M , g ) be a complete hyperk¨ahler manifold with Z M | Rm | rdvol < ∞ and ∀ x ∈ M, ∀ t ≥ , At ≤ vol B ( x, t ) ≤ Bt ( < A ≤ B ). Then there exists a compact set K in M and geometric constants κ > , C > such that for every point x in M \ K , there is a circle fibration f x : Ω x −→ B x definedon a neighborhood Ω x of x and with values in the Euclidean ball B x with radius κr ( x ) in R .Moreover, • f x is a Cr ( x ) − -almost-Riemannian submersion, • its fibers are submanifolds diffeomorphic to S , with length pinched between C − and C , • (cid:12)(cid:12) ∇ f x (cid:12)(cid:12) ≤ Cr ( x ) − , • ∀ i ≥ , (cid:12)(cid:12) ∇ i f x (cid:12)(cid:12) = O ( r ( x ) − i ) .Proof. In the proof of 3.10, we introduced a function h from the ball B ( x, κr ( x )) to thehyperplane H , orthogonal to the tip v x of a lift of σ x in T x M ; this hyperplane H is identifiedto the Euclidean space R through the metric induced by g x .Let us choose a smooth nonincreasing function χ from R + to R + , equal to 1 on [0 , / /
3. We also fix a scale ǫ := 0 . κr ( x ) and set χ ǫ ( t ) = χ (2 t/ǫ ). Note theestimates:(21) (cid:12)(cid:12)(cid:12) χ ( k ) ǫ (cid:12)(cid:12)(cid:12) ≤ C k ǫ − k . We consider the function defined on B ( x, κr ( x )) by: f ( y ) := R T y M h (exp y v ) χ ǫ ( d (0 , v ) / dvol ( v ) R T y M χ ǫ ( d (0 , v ) / dvol ( v ) . Here, dvol and d are taken with respect to exp ∗ y g . If w is a lift of y in T x M , we can changevariables thanks to the isometry τ w := Exp w ◦ ( T w exp x ) − between ( T y M, exp ∗ y g ) and ( T x M, exp ∗ x g ). For every point v in T x M , we introduce thefunction ρ v := d ( v,. ) and set ˆ f := f ◦ exp x , ˆ h := h ◦ exp x . We then get the formula: f ( y ) = ˆ f ( w ) = R T x M ˆ h ( v ) χ ǫ ( ρ v ( w )) dvol ( v ) R T x M χ ǫ ( ρ v ( w )) dvol ( v ) . N ASYMPTOTICALLY FLAT MANIFOLDS WITH NON MAXIMAL VOLUME GROWTH. 29
The point is we can now work on a fixed Euclidean space, ( T x M, g x ). The Riemannianmeasure dvol can be compared to Lebesgue measure dv : on a scale ǫ , if the curvature isbounded by Λ , we have (cid:18) sin Λ ǫ Λ ǫ (cid:19) dv ≤ dvol ≤ (cid:18) sinh Λ ǫ Λ ǫ (cid:19) dv. Cubic curvature decay implies Λ is of order ǫ − , so that we find(22) − Cǫ − dv ≤ dvol − dv ≤ Cǫ − dv. Distance comparison yields in the same way: | d ( v, w ) − | v − w || ≤ C Λ ǫ d ( v, w ) ≤ C, hence(23) (cid:12)(cid:12)(cid:12) ρ v ( w ) − | v − w | / (cid:12)(cid:12)(cid:12) ≤ Cǫ.
Eventually, the proof of 3.10 shows ˆ h is close to a Euclidean projection onto H :(24) (cid:12)(cid:12)(cid:12) ˆ h ( v ) − v H (cid:12)(cid:12)(cid:12) ≤ C. We can write Z ˆ h ( v ) χ ǫ ( ρ v ( w )) dvol ( v ) = Z ˆ h ( v ) χ ǫ ( ρ v ( w ))( dvol ( v ) − dv )+ Z ˆ h ( v ) (cid:16) χ ǫ ( ρ v ( w )) − χ ǫ ( | v − w | / (cid:17) dv + Z (ˆ h ( v ) − v H ) χ ǫ ( | v − w | / dv + Z v H χ ǫ ( | v − w | / dv. The support of v χ ǫ ( ρ v ( w )) is included in a ball whose radius is of order ǫ : ˆ h will thereforetake its values in a ball with radius of order ǫ . With (22), we can then bound the first termof the right-hand side by Cǫ · ǫ − · ǫ = Cǫ . With (21) and (23), we bound the second termby Cǫ · ǫ − · ǫ · ǫ = Cǫ . Eventually, (24) controls the third term by Cǫ . We get: Z ˆ h ( v ) χ ǫ ( ρ v ( w )) dvol ( v ) = Z v H χ ǫ ( | v − w | / dv + O ( ǫ ) , where O ( ǫ ) stands for an error term of magnitude ǫ .Thanks to (22), (21) and (23), we obtain in the same way: Z T x M χ ǫ ( ρ v ( w )) dvol ( v ) = Z χ ǫ ( | v − w | / dv + O ( ǫ ) . Observing Z v H χ ǫ ( | v − w | / dv = O ( ǫ )and C − ǫ ≤ Z χ ǫ ( | v − w | / dv ≤ Cǫ , we deduce ˆ f ( w ) = R v H χ ǫ ( | v − w | / dv R χ ǫ ( | v − w | / dv + O (1) . The change of variables z = v − w yields:ˆ f ( w ) − w H = R z H χ ǫ ( | z | / dz R χ ǫ ( | z | / dz | {z } =0 by parity + O (1) , hence(25) ˆ f ( w ) = w H + O (1) . The differential of ˆ f reads d ˆ f w = R (ˆ h ( v ) − ˆ f ( w )) χ ′ ǫ ( ρ v ( w ))( dρ v ) w dvol ( v ) R χ ǫ ( ρ v ( w )) dvol ( v ) . The same kind of approximations, based on (10), (21), (23), (B.3), (24) and (25) imply(26) d ˆ f w = − R z H χ ′ ǫ ( | z | / z, . ) dz R χ ǫ ( | z | / dz + O ( ǫ − ) . Let us choose an orthonormal basis ( e , . . . , e ) of T x M , with e ⊥ H . If i = j , parity shows Z z i χ ′ ǫ ( | z | / z j dz = 0 . On the contrary, an integration by parts ensures that for every α ≥ Z ∞−∞ z i χ ′ ǫ ( z i / α ) dz i = − Z ∞−∞ χ ǫ ( z i / α ) dz i , so that − Z z i χ ′ ǫ ( | z | / dz = Z χ ǫ ( | z | / dz. This means precisely: − R z H χ ′ ǫ ( | z | / z, . ) dz R χ ǫ ( | z | / dz = X i =1 e i ⊗ ( e i , . ) . And one can recognize the Euclidean projection onto H . We deduce d ˆ f w = X i =1 e i ⊗ ( e i , . ) + O ( ǫ − )With (39), this proves ˆ f is a Cǫ − -almost-Riemannian submersion. Since exp is a localisometry, f is also a Cǫ − -almost-Riemannian submersion.The Hessian reads: ∇ ˆ f w = R (ˆ h ( v ) − ˆ f ( w )) (cid:0) χ ′′ ǫ ( ρ v ( w ))( dρ v ) w ⊗ ( dρ v ) w + χ ′ ǫ ( ρ v ( w ))( ∇ ρ v ) w (cid:1) dvol ( v ) R χ ǫ ( ρ v ( w )) dvol ( v ) − d ˆ f w ⊗ R χ ′ ǫ ( ρ v ( w ))( dρ v ) w ) dvol ( v ) R χ ǫ ( ρ v ( w )) dvol ( v ) . N ASYMPTOTICALLY FLAT MANIFOLDS WITH NON MAXIMAL VOLUME GROWTH. 31
Again, with (10), (21), (23), (B.3), (24) and (25), we arrive at ∇ ˆ f w = R z H (cid:16) χ ′′ ǫ ( | z | / z, . ) ⊗ ( z, . ) + χ ′ ǫ ( | z | / ., . ) (cid:17) dz R χ ǫ ( | z | / dz − d ˆ f w ⊗ R χ ′ ǫ ( | z | / z, . ) dz R χ ǫ ( | z | / dz + O ( ǫ − ) . To begin with, parity ensures Z χ ′ ǫ ( | z | / z, . ) dz = 0 and Z z H χ ′ ǫ ( | z | / ., . ) dz = 0 . The i th component of the integral Z z H χ ′′ ǫ ( | z | / z, . ) ⊗ ( z, . )can be written as a sum of terms (cid:18)Z z i z j z k χ ′′ ǫ ( | z | / dz . . . dz (cid:19) ( e j , . ) ⊗ ( e k , . )which vanish for a parity reason. Therefore: ∇ ˆ f w = O ( ǫ − ) . The proof of theorem 2.6 in [CFG] yields the remaining properties of f x := f . Essentially, f is a fibration because it is C -close to a fibration. The connexity of the fibers follows fromthe bound on the Hessian of f . The length of the fibers is controlled by the assumption onthe volume growth (since f is a almost-Riemannian submersion). (cid:3) We will need to relate neighboring fibrations (this somewhat corresponds to proposition5 . Lemma — The setting is the same as in proposition3.11. Given two points x and x ′ in M \ K , with d ( x, x ′ ) ≤ κr ( x ) , if Ω x,x ′ = Ω x ∩ Ω x ′ (notationsin 3.11), then there is a Cr ( x ) − -almost-isometry φ x,x ′ between f x ′ (Ω x,x ′ ) and f x (Ω x,x ′ ) , forwhich moreover • (cid:12)(cid:12) f x − φ x,x ′ ◦ f x ′ (cid:12)(cid:12) ≤ C , • (cid:12)(cid:12) Df x − Dφ x,x ′ ◦ Df x ′ (cid:12)(cid:12) ≤ Cr ( x ) − , • (cid:12)(cid:12) D φ x,x ′ (cid:12)(cid:12) ≤ Cr ( x ) − , • ∀ i ≥ , (cid:12)(cid:12) D i φ x,x ′ (cid:12)(cid:12) = O ( r ( x ) − i ) .Proof. We use the same notations as in the previous proof, adding subscripts to precise thepoint under consideration, and we work in T x M . Choose a lift u of y at minimal distancefrom o and set τ u := Exp u ◦ ( T u exp x ) − the corresponding isometry (between large balls in T x ′ M and T x M ). We consider the map φ x,x ′ := f x ◦ exp x ′ | f x ′ (Ω x,x ′ ) . In order to bring everything back into T x M , we write φ x,x ′ ◦ f x ′ ◦ exp x = f x ◦ exp x ′ ◦ f x ′ ◦ exp x . The relation exp x ◦ τ u = exp x ′ leads to the reformulation φ x,x ′ ◦ f x ′ ◦ exp x = f x ◦ exp x ◦ τ u ◦ f x ′ ◦ exp x ′ ◦ τ − u that is(27) φ x,x ′ ◦ f x ′ ◦ exp x = ˆ f x ◦ ˜ f x ′ with ˆ f x = f x ◦ exp x and ˜ f x ′ = τ u ◦ f x ′ ◦ exp x ′ ◦ τ − u . We need to understand this lattest map. Vuτ u ( H x ′ ) u + H x T x M T x ′ MH x τ u ( T exp x ) − v x T exp x ′ ) − v x ′ H x ′ Since τ u is an isometry between the metrics exp ∗ x ′ g and exp ∗ x g and since H x ′ is the unionof all the geodesics starting from 0 and with a unit speed orthogonal to ( T exp x ′ ) − ( v x ′ ), τ u ( H x ′ ) is the hypersurface generated by the geodesics starting from u with a unit speedorthogonal to V := ( dτ u ) ◦ ( T exp x ′ ) − ( v x ′ ). v x ′ is by definition one of the lifts of x ′ byexp x ′ which are not 0 but at minimal distance from 0 (in T x ′ M ). So τ u ( v ′ x ) is one of thetwo lifts of x ′ by exp x which are not τ u (0) = u but at minimal distance from τ u (0) = u (in T x M ). We have seen in lemma 3.3 that such a point τ u ( v ′ x ) is τ v x ( u ) or τ − v x ( u ). To fix ideas,assume we are in the first case: τ u ( v ′ x ) = τ v x ( u ).The exponential map of T x ′ M (at 0) maps ( T exp x ′ ) − ( v x ′ ) to v x ′ , so V = ( dτ u ) ◦ ( T exp x ′ ) − ( v x ′ ) is the vector which is mapped by the exponential map of T x M (at τ u (0) = u )to τ u ( v ′ x ) = τ v x ( u ): Exp u V = τ v x ( u ). Consider the geodesic γ ( t ) := Exp u tV . Taylor formula γ (1) − γ (0) − ˙ γ (0) = Z (1 − t )¨ γ ( t ) dt and the estimate | ¨ γ | ≤ Cr ( x ) − | V | ≤ Cr ( x ) − , steming from lemma B.2 and the bound onthe injectivity radius (2.1), together imply | τ v x ( u ) − u − V | ≤ Cr ( x ) − . With the estimate | τ v x ( u ) − u − v x | ≤ Cr ( x ) − , we deduce(28) | V − v x | ≤ Cr ( x ) − . The angle between vectors V and v x is thus bounded by Cr ( x ) − , so that, with ˆ U :=ˆ B (0 , κr ( x )) ∩ ˆ B ( u, κr ( x ′ )), the affine hyperplanes pieces ( u + V ⊥ ) ∩ ˆ U and ( u + v ⊥ x ) ∩ ˆ U remain at bounded distance.Considering the geodesic γ ( t ) = Exp u tW , with W ⊥ V and | W | ≤ Cr ( x ), we obtain in thesame way (thanks to lemma B.2): | Exp u W − u − W | ≤ Cr ( x ) − r ( x ) = C. N ASYMPTOTICALLY FLAT MANIFOLDS WITH NON MAXIMAL VOLUME GROWTH. 33
This means the affine hyperplane piece ( u + V ⊥ ) ∩ ˆ U and the hypersurface piece τ u ( B x ′ ) ∩ ˆ U =Exp u V ⊥ ∩ ˆ U remain at bounded distance.And we conclude τ u ( B x ′ ) ∩ ˆ U and ( u + v ⊥ x ) ∩ ˆ U remain C -close, namely the map Ψ definedfrom τ u ( B x ′ ) ∩ ˆ U to ( u + v ⊥ x ) ∩ ˆ U by ψ : Exp u W u + W is C -close to the identity.In the preceding proof, we saw that f x ′ ◦ exp x ′ was C -close to the orthogonal projection(for g x ′ ) onto H x ′ . Now, τ u is an isometry between the metrics exp ∗ x ′ g and exp ∗ x g , whichare respectively Cr ( x ) − -close to g x ′ and g x . Thus for every point w in the area underconsideration, (cid:12)(cid:12)(cid:12) ˜ f x ′ ( w ) − w (cid:12)(cid:12)(cid:12) is C -close to the distance (for g x ) between w and τ u ( H x ′ ), sothat (cid:12)(cid:12)(cid:12) ψ ◦ ˜ f x ′ ( w ) − w (cid:12)(cid:12)(cid:12) is C -close to the distance (for g x ) between w and ( u + v ⊥ x ): ψ ◦ ˜ f x ′ andthus ˜ f x ′ are C -close to the orthogonal projection onto ( u + v ⊥ x ), which is nothing but thecomposition of the orthogonal projection onto H x and of the translation with vector u − u H x : (cid:12)(cid:12)(cid:12) ˜ f x ′ ( w ) − w H x − ( u − u H x ) (cid:12)(cid:12)(cid:12) ≤ C. We deduce (cid:12)(cid:12)(cid:12) ˜ f x ′ ( w ) − ˆ f x ( w ) − ( u − u H x ) (cid:12)(cid:12)(cid:12) ≤ C and, composing with ˆ f x , we find (cid:12)(cid:12)(cid:12) ˆ f x ◦ ˜ f x ′ ( w ) − ˆ f x ( w ) (cid:12)(cid:12)(cid:12) ≤ C. Recalling formula (27), we obtain (cid:12)(cid:12) φ x,x ′ ◦ f x ′ ◦ exp x − f x ◦ exp x (cid:12)(cid:12) ≤ C, and, with the surjectivity of exp x , this yields (cid:12)(cid:12) φ x,x ′ ◦ f x ′ − f x (cid:12)(cid:12) ≤ C. Relation (27) also implies(29) D ( φ x,x ′ ◦ f x ′ exp x ) = D ˆ f x ◦ D ˜ f x ′ . Let z be a point in ˆ U and set z ′ = τ − u ( z ) ∈ T x ′ M . The preceding proof has shown that D z ˆ f x is Cr ( x ) − -close to the orthogonal projection in the direction of H x . In the same way, D z ′ ( f x ′ exp x ′ ) is Cr ( x ) − -close to the orthogonal projection in the direction of H x ′ , i.e. inthe direction orthogonal to v x ′ . Conjugating by Dτ u , we find that D z ˜ f x ′ is Cr ( x ) − close tothe projection in the direction orthogonal to D z τ u ( v x ′ ).Let Z ′ be the initial speed of the geodesic connecting z ′ to τ v x ′ ( z ′ ) in unit time. Theargument leading to (28) yields (cid:12)(cid:12) Z ′ − v x ′ (cid:12)(cid:12) ≤ Cr ( x ) − . If we set Z := D z τ u Z ′ , we thus have | Z − D z τ u ( v x ′ ) | ≤ Cr ( x ) − . Now Z is the initial speed of the geodesic connecting z to τ v x ( z ) (or τ − v x ( z )) in unit time.So again: | Z − v x | ≤ Cr ( x ) − , so that | v x − D z τ u ( v x ′ ) | ≤ Cr ( x ) − . Finally, D z ˜ f x ′ is Cr ( x ) − -close to the projection in the direction of the hyperplane H x ,orthogonal to v x : (cid:12)(cid:12)(cid:12) D ( φ x,x ′ ◦ f x ′ ◦ exp x ) − D ˆ f x (cid:12)(cid:12)(cid:12) ≤ Cr ( x ) − , hence(30) (cid:12)(cid:12) Dφ x,x ′ ◦ Df x ′ − Df x (cid:12)(cid:12) ≤ Cr ( x ) − . Let W be a vector tangent to f x ′ (Ω x,x ′ ) and let W ′ be its horizontal lift for f x ′ : Df x ′ W ′ = W . As D ˜ f x ′ and D ˆ f x are Cr ( x ) − -close, an horizontal vector for f x ′ is Cr ( x ) − -close to ahorizontal vector for f x . And since f x and f x ′ are Cr ( x ) − -almost-Riemannian submersions,we get (cid:12)(cid:12)(cid:12)(cid:12) Df x ( W ′ ) (cid:12)(cid:12) − (cid:12)(cid:12) W ′ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cr ( x ) − (cid:12)(cid:12) W ′ (cid:12)(cid:12) and (cid:12)(cid:12) | W | − (cid:12)(cid:12) W ′ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cr ( x ) − (cid:12)(cid:12) W ′ (cid:12)(cid:12) . Writing (cid:12)(cid:12)(cid:12)(cid:12) Dφ x,x ′ W (cid:12)(cid:12) − | W | (cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) Dφ x,x ′ ( Df x ′ W ′ ) (cid:12)(cid:12) − (cid:12)(cid:12) Df x W ′ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) Df x W ′ (cid:12)(cid:12) − (cid:12)(cid:12) W ′ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) W ′ (cid:12)(cid:12) − | W | (cid:12)(cid:12) ≤ (cid:12)(cid:12) Dφ x,x ′ ( Df x ′ W ′ ) − Df x W ′ (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) Df x W ′ (cid:12)(cid:12) − (cid:12)(cid:12) W ′ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) W ′ (cid:12)(cid:12) − | W | (cid:12)(cid:12) and using (30), we obtain (cid:12)(cid:12)(cid:12)(cid:12) Dφ x,x ′ ( W ) (cid:12)(cid:12) − | W | (cid:12)(cid:12) ≤ Cr ( x ) − | W | , which proves φ x,x ′ is a Cr ( x ) − -quasi-isometry.Higher order estimates stem from those on f x and f x ′ , thanks to formula (27): the boundson the curvature covariant derivatives ensure exp ∗ x g (resp. exp ∗ x ′ g )) is close to the flat g x (resp. g x ′ ) in C ∞ topology, so that the estimates for one or the other are equivalent; thusexp x , exp x ′ and τ u can be treated like isometries. (cid:3) We will also need the following lemma. Indeed, it stems from the previous one.
Lemma — The setting is the same as in lemma 3.12. Weconsider three points x , x ′ and x ′′ in M \ K , whose respective distances are bounded by κr ( x ) .Then, wherever it makes sense, we have • (cid:12)(cid:12) φ x,x ′′ − φ x,x ′ ◦ φ x ′ ,x ′′ (cid:12)(cid:12) ≤ C , • (cid:12)(cid:12) Dφ x,x ′′ − Dφ x,x ′ ◦ Dφ x ′ ,x ′′ (cid:12)(cid:12) ≤ Cr ( x ) − . Proof.
On the intersection of Ω x , Ω x ′ and Ω x ′′ , we can write (cid:12)(cid:12) f x − φ x,x ′ ◦ f x ′ (cid:12)(cid:12) ≤ C et (cid:12)(cid:12) f x ′ − φ x ′ ,x ′′ ◦ f x ′′ (cid:12)(cid:12) ≤ C. Since φ x,x ′ is a quasi-isometry, it follows that: (cid:12)(cid:12) f x − φ x,x ′ ◦ φ x ′ ,x ′′ ◦ f x ′′ (cid:12)(cid:12) ≤ (cid:12)(cid:12) f x − φ x,x ′ ◦ f x ′ (cid:12)(cid:12) + (cid:12)(cid:12) φ x,x ′ ◦ ( f x ′ − φ x ′ ,x ′′ ◦ f x ′′ ) (cid:12)(cid:12) ≤ C. Using the estimate (cid:12)(cid:12) f x − φ x,x ′′ ◦ f x ′′ (cid:12)(cid:12) ≤ C, we obtain by triangle inequality: (cid:12)(cid:12) ( φ x,x ′′ − φ x,x ′ ◦ φ x ′ ,x ′′ ) ◦ f x ′′ (cid:12)(cid:12) ≤ C. From the surjectivity of f x ′′ , we see that: f x ′′ (Ω x,x ′′ ∩ f x ′′ (Ω x ′ ,x ′′ ∩ φ − x ′ ,x ′′ f x ′ (Ω x,x ′ ) . Since f x ′′ is a submersion, the same argument applies to the differentials. (cid:3) N ASYMPTOTICALLY FLAT MANIFOLDS WITH NON MAXIMAL VOLUME GROWTH. 35
Local fibration gluing.
Now, we need to adjust the local fibrations so as to makethem compatible. The technical device is essentially the same as in [CFG]. The followinglemma will be widely used in this process.
Lemma — The setting is that of lemma 3.12. Given twopoints x and x ′ in M \ K with αr ( x ) ≤ d ( x, x ′ ) ≤ βr ( x ) for some real numbers < α < β < .We assume that on B ( x, γr ( x )) and B ( x ′ , γr ( x ′ )) , some fibrations f x and f x ′ as in 3.11 aredefined, that B ( x, δr ( x )) and B ( x ′ , δr ( x ′ )) have nonempty intersection, with < δ < γ , andthat a map φ x ′ ,x as in 3.12 is defined. We can then build a fibration ˜ f x ′ on B ( x ′ , δr ( x ′ )) ,with the same properties as f x ′ , plus: ˜ f x ′ = φ x ′ ,x ◦ f x on B ( x, δr ( x )) ∩ B ( x ′ , δr ( x ′ )) . Moreover, this new fibration coincides with the old f x ′ on B ( x, γr ( x )) and wherever we already had f x ′ = φ x ′ ,x ◦ f x .Proof. We set ˜ f x ′ ( y ) = λ ( y ) φ x ′ ,x ( f x ( y )) + (1 − λ ( y )) f x ′ ( y )with λ ( y ) = θ (cid:18) f x ( y ) r ( x ) (cid:19) where θ : R −→ [0 ,
1] is a truncature function equal to 1 on the ball centered in 0 and withradius δ , equal to 0 outside the ball centered in 0 and with radius γ . Using the bounds on f x ,we find (cid:12)(cid:12) ∇ k λ (cid:12)(cid:12) ≤ C k r ( x ) − k , and the announced estimates can be obtained by differentiatingthe equation ˜ f x ′ ( y ) − f x ′ ( y ) = λ ( y ) (cid:0) φ x ′ ,x ◦ f x ( y )) − f x ′ ( y ) (cid:1) . (cid:3) Lemma — The setting is that of lemma 3.13. Giventhree points x , x ′ and x ′′ in M \ K with αr ( x ) ≤ d ( x, x ′ ) , d ( x ′ , x ′′ ) , d ( x, x ′′ ) ≤ βr ( x ) forsome real numbers < α < β < . We assume that on B ( x, γr ( x )) , B ( x ′ , γr ( x ′ )) and B ( x ′′ , γr ( x ′′ )) , some fibrations f x , f x ′ and f x ′′ as in 3.11 are defined, that the intersectionof B ( x, δr ( x )) , B ( x ′ , δr ( x ′ )) and B ( x ′′ , δr ( x ′′ )) is nonempty for some < δ < γ and thatmaps φ x ′ ,x , φ x,x ′′ and φ x ′ ,x ′′ as in 3.13 are defined. We can then build a new diffeomorphism ˜ φ x ′ ,x ′′ , with the same properties as φ x ′ ,x ′′ , plus: ˜ φ x ′ ,x ′′ = φ x ′ ,x ◦ φ x,x ′′ on f x ′′ ( B ( x, δr ( x )) ∩ B ( x ′ , δr ( x ′ )) ∩ B ( x ′′ , δr ( x ′′ )) . Moreover, this new diffeomorphism coin-cides with φ x ′ ,x ′′ on B ( x ′′ , γr ( x ′′ )) and wherever we already had φ x ′ ,x ′′ = φ x ′ ,x ◦ φ x,x ′′ .Proof. We simply set˜ φ x ′ ,x ′′ ( v ) = λ ( v ) φ x ′ ,x ◦ φ x,x ′′ ( v ) + (1 − λ ( v )) φ x ′ ,x ′′ ( v )with λ ( v ) = θ | v | r ( x ) ! where θ is the same function as in the previous proof. (cid:3) Theorem — Let ( M , g ) be a complete hyperk¨ahler manifold with Z M | Rm | rdvol < ∞ and ∀ x ∈ M, ∀ t ≥ , At ≤ vol B ( x, t ) ≤ Bt ( < A ≤ B ). Then there exists a compact set K in M such that M \ K is endowed witha smooth circle fibration π over a smooth open manifold X . Besides, there is a geometricpositive constant C such that fibers have length pinched between C − and C and secondfundamental form bounded by Cr − . Remark . The proof will show that for any point x in M \ K , there is a diffeomorphism ψ x between a neighborhood of π ( x ) in X and a ball in R such that ψ x ◦ π is a fibration satisfyingestimates as in proposition 3.11.Proof. We take a maximal set of points x i , i ∈ I , such that for all indices i = j , d ( x i , x j ) ≥ κr ( x i ) /
8. This provides a uniformly locally finite covering of M by the balls B ( x i , κr ( x i ) / i , we let f i be the local fibration given by 3.11. We will work with the minimalsaturated (for f i ) sets Ω i ( α ) containing the balls B ( x i , αr ( x i )), where α is a parameter inferiorto κ . As in [CFG], we divide I into packs S , ..., S N such that any two distinct points x i , x j whose indices are in the same pack are far from each other: ∃ a ∈ [1 , N ] , { i, j } ⊂ S a ⇒ d ( x i , x j ) ≥ κ min( r ( x i ) , r ( x j )) . In particular, Ω i ( α ) and Ω j ( α ) have empty intersection if i and j are in different packs;in this case, if the number of the pack of i is greater than for j , one denotes by φ i,j thediffeomorphism given by 3.12 and by φ j,i its inverse.In order to improve the approximations f i ≈ φ i,j f j into equalities f i = φ i,j f j , we set upan adjustment campaign in the following way. The idea consists in giving priority to packswith small number. To do so, given an area where several fibrations are defined, we willmodify them so that they all fit with the fibration with smallest number among them. Theorder of implementation is important. We will distinguish several stages, indexed by subsets A := { a < · · · < a k } of [1 , N ]. We implement these 2 N stages by increasing order of a ,then decreasing order of k , then increasing order of a , then increasing order of a , etc. Torephrase it, we have { a < · · · < a k } ≺ { b < · · · < b l } if one of these exclusive conditions is realized: • a < b ; • a = b and k > l ; • a i = b i for i ≤ i and k = l and a i < b i .We denote by m A the rank of A in this order and set α m := κ · (cid:18) (cid:19) m N . Along the campaign, the fibration domains Ω i ( α ) will be shrinked: α m A will be the domainsize at stage A .At stage A := { a < · · · < a k } , we consider all elements I = ( i , · · · , i k ) of S a × · · · × S a k : to each such element corresponds one step. At step I , we are interested in Ω I :=Ω i ( α m A +1 ) ∩ · · · ∩ Ω i k ( α m A +1 ). One should notice that our choice of packing ensures allthe intersections Ω i ( α m A ) ∩ · · · ∩ Ω i k ( α m A ) treated at the same stage are away from eachother, so that the following modifications are independent (during the stage). Essentially, the N ASYMPTOTICALLY FLAT MANIFOLDS WITH NON MAXIMAL VOLUME GROWTH. 37 fibration f i will overrule its neighbour on Ω I . Given 2 ≤ p ≤ k , we build ˜ f i p on Ω i p ( α m A +1 ),from f i and f i p , as in 3.14, so as to obtain • ˜ f i p = φ i p ,i f i sur Ω i p ( α m A +1 ) ∩ Ω i ( α m A +1 ), • ˜ f i p = f i p sur Ω i p ( α m A +1 ) \ Ω i ( α m A ).We also build, for 2 ≤ p < q ≤ k , ˜ φ i p ,i q on ˜ f i q (Ω i p ( α m A +1 ) ∩ Ω i q ( α m A +1 )) from φ i p ,i φ i ,i q and φ i p ,i q , as in 3.15, so that • ˜ φ i p ,i q = φ i p ,i φ i ,i q sur ˜ f i q (Ω i p ( α m A +1 ) ∩ Ω i q ( α m A +1 ) ∩ Ω i ( α m A )), • ˜ φ i p ,i q = φ i p ,i q sur ˜ f i q (Ω i p ( α m A +1 ) ∩ Ω i q ( α m A +1 ) \ Ω i ( α m A )).After this, we can add that wherever it makes sense, we have for every { p, q } ⊂ [2 , k ]:˜ φ i q ,i p ˜ f i p = φ i q ,i φ i ,i p φ i p ,i f i = φ i q ,i f i = ˜ f i q . Now forget the tildes. We have just ensured that on Ω I , for all relevant indices i, j , one has f i = φ i,j f j .We proceed, independently, for all possible I at this stage, then we go on with the nextstage, following the chosen order.At the moment we pass from a stage { a < · · · } to a stage { b < · · · } , with a = b , we cannotice the fibrations f i and the diffeomorphisms φ i,j are definitively fixed on the sets withnumber in the pack S a : indeed, the device of 3.14 and 3.15 does not modify the fibrationswhich are already consistent. Afterwards, on these areas, we have definitively ensured the equalities f i = φ i,j f j .For the same reason, at the moment we pass from a stage { a < · · · < a k } to a stage { a < · · · < b k − } , the fibrations f i and the diffeomorphisms φ i,j are definitively fixed onthe sets Ω I , where I is a k -tuple beginning with an element of S a . Therefore, on theseintersections of order k , we have definitively ensured the equalities f i = φ i,j f j and all thatis done afterwards on intersections of order k − f i on the sets Ω i := Ω i ( κ/
2) anddiffeomorphisms φ i,j such that φ i,j ◦ f j = f i on Ω i ∩ Ω j . The initial estimates still hold, withdifferent constants.Let us define an equivalence relation: x and y are considered equivalent if there is anindex i such that x and y belong to Ω i and f i ( x ) = f i ( y ). Denote by X the quotienttopological space and by π the corresponding projection. Maps f i induce homeomorphisms(from their domain to their image) ˇ f i , which endow X with a structure of smooth 3-manifold:for every (relevant) pair i, j , ˇ f i ˇ f j − = φ i,j is a diffeomorphism between open sets in R . Byconstruction, π is then a smooth fibration. (cid:3) The circle fibration geometry.
In this whole paragraph, the setting is a completehyperk¨ahler manifold ( M , g ) with Z M | Rm | rdvol < ∞ and ∀ x ∈ M, ∀ t ≥ , At ≤ vol B ( x, t ) ≤ Bt (0 < A ≤ B ). We have built a circle fibration π : M \ K −→ X . The vectors that are tangentto the fibers will be called “vertical” whereas vectors orthogonal to the fibers will be called“horizontal”. Let us average the metric g along the fibers of this fibration. Given a point x in M \ K , we can choose a unit vector field V , defined on a saturated neighborhood of x and vertical (there are two choices of sign). Let φ t be the flow of V . Denote by l x the length ofthe fiber π − ( π ( x )). We define a scalar product on T x M by the formula h x := 1 l x Z l x φ ∗ t g dt. This definition does not depend on the choice of V . We thus obtain a Riemannian metric h on M \ K and the flows φ t are isometries for h . To estimate the closeness of h to g , weproceed to a few estimations.First we show that a local unit vertical field V is almost parallel and almost Killing. Lemma — The covariant derivatives of V can be estimated by |∇ V | ≤ C r − and ∀ k ≥ , (cid:12)(cid:12)(cid:12) ∇ k V (cid:12)(cid:12)(cid:12) ≤ C k r − k . Proof.
Let f : Ω −→ R be one of the local fibrations. By construction, we have df ( V ) = 0.Differentiation yields:(31) ∇ f ( V, . ) = − df ( ∇ V ) . Since V has constant norm, one has(32) ( ∇ V, V ) = 0so, with (3.11): |∇ V | ≤ C (cid:12)(cid:12) ∇ f (cid:12)(cid:12) ≤ Cr − . We then make an inductive argument, assumingthe result up to order k −
1. Differentiating k − df ( ∇ k V ) = k − X i =1 ∇ k − i f ∗ ∇ i V + k − X i =0 ∇ k − i f ∗ ∇ i V, which enables us to bound the horizontal part of ∇ k V by (cid:12)(cid:12)(cid:12) ∇ k V ⊥ (cid:12)(cid:12)(cid:12) ≤ C k k − X i =1 (cid:12)(cid:12)(cid:12) ∇ k − i f (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) ∇ i V (cid:12)(cid:12) + C k k − X i =0 (cid:12)(cid:12)(cid:12) ∇ k − i f (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) ∇ i V (cid:12)(cid:12) . Induction assumption and (3.11) yield: (cid:12)(cid:12) ∇ k V ⊥ (cid:12)(cid:12) ≤ C k ( r − k + r − k ) ≤ C k r − k . Differentiating(32), we get (cid:12)(cid:12)(cid:12) ( ∇ k V, V ) (cid:12)(cid:12)(cid:12) ≤ C k k − X i =1 (cid:12)(cid:12)(cid:12) ∇ k − i V (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) ∇ i V (cid:12)(cid:12) , so that, by induction assumption: (cid:12)(cid:12) ( ∇ k V, V ) (cid:12)(cid:12) ≤ C k r − k . All in all: (cid:12)(cid:12) ∇ k V (cid:12)(cid:12) ≤ C k r − k . (cid:3) Lemma — The Lie derivative of g along V satisfies: | L V g | ≤ C r − , and ∀ k ≥ , (cid:12)(cid:12)(cid:12) ∇ k L V g (cid:12)(cid:12)(cid:12) ≤ C k r − − k . Proof.
The formula L V g ( X, Y ) = ( ∇ X V, Y )+( ∇ Y V, X ) ensures that for any natural number k , (cid:12)(cid:12) ∇ k L V g (cid:12)(cid:12) is estimated by (cid:12)(cid:12) ∇ k +1 V (cid:12)(cid:12) . So we can apply lemma 3.17. (cid:3) If φ t is the flow V , we are interested in the family of metrics g t := φ t ∗ g , with Levi-Civitaconnection ∇ t and curvature Rm t . First, a nice formula. Lemma — For every vector fields X and Y , ddt ∇ tX Y = Rm t ( X, V ) Y − ∇ t, X,Y V. N ASYMPTOTICALLY FLAT MANIFOLDS WITH NON MAXIMAL VOLUME GROWTH. 39
Proof.
The connection ∇ t is obtained by transporting ∇ thanks to the isometry φ t :(33) ∇ tX Y = φ t ∗ ∇ φ t ∗ X φ t ∗ Y. Thus: ddt φ t ∗ ∇ tX Y = ddt ∇ φ t ∗ X φ t ∗ Y, hence φ t ∗ [ V, ∇ tX Y ] + φ t ∗ ddt ∇ tX Y = ∇ [ V,φ t ∗ X ] φ t ∗ Y + ∇ φ t ∗ X [ V, φ t ∗ Y ] , which, thanks to (33) and the invariance of V under its flow, simplifies into ddt ∇ tX Y = ∇ t [ V,X ] Y + ∇ tX [ V, Y ] − [ V, ∇ tX Y ] . Develop and simplify: ddt ∇ tX Y = ∇ t [ V,X ] Y + ∇ tX ∇ tV Y − ∇ tX ∇ tY V − ∇ tV ∇ tX Y + ∇ t ∇ tX Y V = Rm t ( X, V ) Y − ∇ t, X,Y V. (cid:3) This formula gives a control on the covariant derivatives of g t (with respect to g ). Lemma — For every t , g t satisfies | g t − g | ≤ C r − and ∀ k ∈ N ∗ , (cid:12)(cid:12)(cid:12) ∇ k g t (cid:12)(cid:12)(cid:12) ≤ C k r − − k . Proof.
Let X be a vector field. The definition of the Lie derivative reads: ddt g t ( X, X ) = ( φ t ∗ L V g )( X, X ) . So, denoting by L the supremum of | L V g | on the fiber under consideration, we get −L g t ( X, X ) ≤ ddt g t ( X, X ) ≤ L g t ( X, X ) . After integration, we obtain g ( X, X ) e −L t ≤ g t ( X, X ) ≤ g ( X, X ) e L t . Lemma 3.18 bounds L : g ( X, X ) e − Cr − ≤ g t ( X, X ) ≤ g ( X, X ) e Cr − , hence the first estimate. Now, we consider three vector fields X , Y , Z . We have( ∇ tX g t )( Y, Z ) = 0 = X · g t ( Y, Z ) − g t ( ∇ tX Y, Z ) − g t ( Y, ∇ tX Z ) , and ( ∇ X g t )( Y, Z ) = X · g t ( Y, Z ) − g t ( ∇ X Y, Z ) − g t ( Y, ∇ X Z ) , so, if A t := ∇ t − ∇ , we arrive at( ∇ X g t )( Y, Z ) = g t ( A t ( X, Y ) , Z ) + g t ( Y, A t ( X, Z )) , which we write(34) ∇ g t = g t ∗ A t . Lemma 3.19 implies A t = Z t (Rm s ( ., V ) − ∇ s, V ) ds. Since the curvature is invariant under isometries, we find(35) Rm t = φ t ∗ Rm and, thanks to (33) and the invariance of V under the flow,(36) ∇ t, V = φ t ∗ ∇ V. We have bounds on g t , Rm and ∇ V : (33) leads to (cid:12)(cid:12) Rm t (cid:12)(cid:12) ≤ Cr − (and even r − ), (cid:12)(cid:12) ∇ t, V (cid:12)(cid:12) ≤ Cr − and (cid:12)(cid:12) A t (cid:12)(cid:12) ≤ Cr − .Now assume (by induction) that for some k ≥
1, for every 0 ≤ i ≤ k − t , (cid:12)(cid:12) ∇ i ( g t − g ) (cid:12)(cid:12) ≤ Cr − − i , (cid:12)(cid:12) ∇ i Rm t (cid:12)(cid:12) ≤ Cr − − i , (cid:12)(cid:12) ∇ i ∇ t, V (cid:12)(cid:12) ≤ Cr − − i . In particular, we get ∀ t, ∀ i ∈ [0 , k − , (cid:12)(cid:12) ∇ i A t (cid:12)(cid:12) ≤ Cr − − i . Fix t . Differentiating (34), we obtain the formula ∇ k g t = k − X i =0 ∇ k − − i g t ∗ ∇ i A t . Induction assumption yields (cid:12)(cid:12) ∇ k g t (cid:12)(cid:12) ≤ Cr − − k . To go on, we need to estimate (cid:12)(cid:12) ∇ t,i A t (cid:12)(cid:12) , i ≤ k −
1. To do this, we write ∇ t = ∇ + A t . In this way, we see that (cid:12)(cid:12) ∇ t,i A t (cid:12)(cid:12) can becontrolled by a sum of a bounded number of terms like i − Y α =0 (cid:12)(cid:12) ∇ α A t (cid:12)(cid:12) m α ! (cid:12)(cid:12)(cid:12) ∇ β A t (cid:12)(cid:12)(cid:12) with natural numbers m α , β satisfying i − X α =0 (1 + α ) m α + β = i. Induction assumption implies each of these terms is bounded by Cr − (2+ α ) m α − − β ≤ Cr − − i ,so (cid:12)(cid:12) ∇ t,i A t (cid:12)(cid:12) ≤ Cr − − i . Now, writing ∇ = ∇ t − A t , we estimate (cid:12)(cid:12) ∇ k Rm t (cid:12)(cid:12) by a sum of a bounded number of termslike k − Y α =0 (cid:12)(cid:12) ∇ t,α A t (cid:12)(cid:12) m α ! (cid:12)(cid:12)(cid:12) ∇ t,β Rm t (cid:12)(cid:12)(cid:12) with natural numbers m α , β satisfying k − X α =0 (1 + α ) m α + β = k. With (35) and (33), we bound (cid:12)(cid:12) ∇ t,β Rm t (cid:12)(cid:12) by (cid:12)(cid:12) ∇ β Rm (cid:12)(cid:12) and thus by r − − β . Eventually, wefind (cid:12)(cid:12) ∇ k Rm t (cid:12)(cid:12) ≤ Cr − − k . In the same way, we get (cid:12)(cid:12) ∇ k ∇ t, V (cid:12)(cid:12) ≤ Cr − − k and we conclude byinduction. (cid:3) Lemma — The length l of the fibers is controlled by: | dl | ≤ C r − and ∀ k ≥ , (cid:12)(cid:12)(cid:12) ∇ k l (cid:12)(cid:12)(cid:12) ≤ C k r − k . N ASYMPTOTICALLY FLAT MANIFOLDS WITH NON MAXIMAL VOLUME GROWTH. 41
Proof.
By construction, we have the identity φ l ( x ) ( x ) = x , at every point x in M \ K . Differen-tiation yields dl ⊗ V + T φ l = id . Taking the scalar product with V , we obtain dl = ( g − g l )( V, . ).Differentiating this leads to ∇ k l = k − X i =0 ∇ i ( g − g l ) ∗ ∇ k − − i V. Now we use (3.20), (3.17) and the bound on l : (cid:12)(cid:12) ∇ k l (cid:12)(cid:12) ≤ Cr − k . (cid:3) We can finally control the metric h , obtained by averaging g along the fibers. Proposition — The averaged metric h obeys the estimates: | h − g | ≤ C k r − and ∀ k ≥ , (cid:12)(cid:12)(cid:12) ∇ k h (cid:12)(cid:12)(cid:12) ≤ C k r − − k . Proof.
The definition of h can be written h − g = 1 l Z l ( g t − g ) dt The first estimate follows immediately from (3.20). Let us differentiate: ∇ h = dll ⊗ ( g l − h ) + 1 l Z l ∇ g t dt. An induction yields for every k ≥ ∇ k h = k X i =1 C ik ∇ i ll ⊗ ∇ k − i ( g l − h ) + 1 l Z l ∇ k g t dt. (3.20) and (3.21) then lead, by induction, to (cid:12)(cid:12) ∇ k h (cid:12)(cid:12) ≤ Cr − − k . (cid:3) Since g has cubic curvature decay, we deduce the Corollary — The curvature of h has cubic decay. Now, let us push h down into a Riemannian metric ˇ h on X : for every point y in X , forevery vector w in T y X , we choose a lift x of y ( π ( x ) = y ) and we set ˇ h y ( w, w ) = h x ( v, v )where v is the horizontal lift of w in T x M ; this definition makes sense because the flow φ t isisometric for h . Proposition — The manifold X is diffeomorphic to the complementary set of a ballin R , mod out by the action of a finite subgroup of O (3) . Moreover, ˇ h is an ALE metric oforder − , that is ˇ h = g R + O ( r − τ ) for every τ < . Proof.
Observe the volume of a ball of radius t in ( X , ˇ h ) is comparable to t . To estimatethe curvature on the base, we use O’Neill formula ([Bes]), which asserts that if Y and Z areorthogonal unit horizontal vector fields on M \ K , thenSect ˇ h ( π ∗ Y ∧ π ∗ Z ) = Sect h ( Y ∧ Z ) + 34 h ([ Y, Z ] , V ) . The first term decays at a cubic rate by 3.23. Moreover, h ([ Y, Z ] , V ) = − ( ∇ Y h )( Z, V ) − h ( Z, ∇ Y V ) + ( ∇ Z h )( Y, V ) + h ( Y, ∇ Z V ) . Lemma 3.17 and corollary 3.22 yield | h ([ Y, Z ] , V ) | ≤ Cr − . Hence: (cid:12)(cid:12)
Sect ˇ h ( π ∗ Y ∧ π ∗ Z ) (cid:12)(cid:12) ≤ Cr − . This cubic curvature decay, combined with Euclidean volume growth, enables us to applythe main theorem of [BKN]. (cid:3)
What have we proved ?
We have proved the following theorem.
Theorem — Let ( M , g ) be a complete hyperk¨ahler manifold satisfying Z M | Rm | r dvol < ∞ and ∀ x ∈ M, ∀ t ≥ , At ν ≤ vol B ( x, t ) ≤ Bt ν with < A ≤ B and ≤ ν < . Then there is compact set K in M , a ball B in R , a finitesubgroup G of O (3) and a circle fibration π : M \ K −→ ( R \ B ) /G . Moreover, the metric g obeys g = π ∗ ˜ g + η + O ( r − ) , where η measures the projection along fibers and ˜ g is an ALE metric of order − . Let us precise the topology at infinity, that is the topology of the connected space E = M \ K , which, according to theorem 3.25, is a circle bundle over X = R \ B/G . Thanks tothe projection p : ¯ X = R \ B −→ X = R \ B/G , we can pull back the fibration π into acircle fibration ¯ π : ¯ E −→ ¯ X . The space ¯ E is a finite covering of E , with order | G | :¯ E = (cid:8) (¯ x, e ) ∈ ¯ X × E, p (¯ x ) = π ( e ) (cid:9) and ¯ π is given by the projection onto the first factor ( pr ).¯ E pr −−−−→ E y ¯ π y π ¯ X p −−−−→ X Of course, ¯ X = R \ B has the homotopy type of S , so that we can classify its circlefibrations. Moreover, the homotopy groups of ¯ E can be computed thanks to the long exacthomotopy sequence associated to ¯ π . In this way, we obtain essentially two cases, which aredistinguished by the homotopy groups at infinity (those of M \ K ). • If the fundamental group at infinity is finite, then a finite covering of M \ K is R \ B and the circle fibration is the Hopf fibration, up to a finite group action. In this case,the π at infinity is trivial. This is typically the “Taub-NUT” situation . • If the fundamental group at infinity is infinite, then a finite covering of M \ K is R \ B × S and the circle fibration comes from the trivial one. The π at infinity isthen Z .It is easy to adapt the arguments above in order to obtain the following result. Theorem — Let ( M n , g ) be a complete manifold satisfying ∀ k ∈ N , (cid:12)(cid:12)(cid:12) ∇ k Rm (cid:12)(cid:12)(cid:12) = O ( r − − k ) and ∀ x ∈ M, ∀ t ≥ , At n − ≤ vol B ( x, t ) ≤ ω ( t ) t n for some positive number A and some function ω going to zero at infinity. Assume moreoverthere is c ≥ such that the holonomy H of any geodesic loop based at x and with length N ASYMPTOTICALLY FLAT MANIFOLDS WITH NON MAXIMAL VOLUME GROWTH. 43 L ≤ r ( x ) /c satisfies | H − id | ≤ cLr ( x ) . Then there is compact set K in M , a ball B in R n − , a finite subgroup G of O ( n − and acircle fibration π : M \ K −→ ( R n − \ B ) /G . Moreover, the metric g obeys g = π ∗ ˜ g + η + O ( r − ) , where η measures the projection along fibers and ˜ g is an ALE metric of order − ( if n ≥ . Remark . The required estimates on the curvature are satisfied on a Ricci flat manifoldwith cubic curvature decay. This allows one to englobe the Schwarzschild metrics ( [Min] forinstance) in this setting. Note a little topology ensures the fibration is trivial if n ≥ . Appendix A. Curvature decay.
The following result is proved in [Min]. Recall we always distinguish a point o in ourmanifolds. We will use the measure µ defined by dµ = r n vol B ( o,r ) dvol . Theorem
A.1 — Let ( M n , g ) be a complete Ricci flat manifold. Assume there are numbers ν > and C > such that ∀ t ≥ s > , vol B ( o, t )vol B ( o, s ) ≥ C (cid:18) ts (cid:19) ν . Then the integral bound Z M | Rm | n dµ < ∞ implies the pointwise bound | Rm | = O ( r − a ( n,ν ) ) with a ( n, ν ) = max (cid:18) , ( ν − n − n − (cid:19) . It should be stressed that the integral assumption R | Rm | n dµ < ∞ is weaker than | Rm | = O ( r − − ǫ ) for some positive ǫ . In this paper, we use the Corollary
A.2 — Let ( M n , g ) be a complete Ricci flat manifold, with n ≥ . Assumethere are positive numbers A and B such that ∀ t ≥ , At n − ≤ vol B ( o, t ) ≤ Bt n − . Then the integral bound Z M | Rm | n r dvol < ∞ implies the pointwise bound | Rm | = O ( r − ( n − ) . These estimates follow from a Moser iteration, which is self improved thanks to a globalweighted Sobolev inequality. In this appendix, we wish to obtain similar estimates on thecovariant derivatives of the curvature tensor. To do this, we need a technical inequality.
Lemma
A.3 (Moser iteration with source term) — Let ( M n , g ) be a complete noncompactRiemannian manifold with nonnegative Ricci curvature and let E −→ M be a smooth Eu-clidean vector bundle, endowed with a compatible connection ∇ . We denote by ∆ = ∇ ∗ ∇ theBochner Laplacian and suppose V is a continuous field of symmetric endomorphisms of E whose negative part satisfies | V − | = O ( r − ) . Given a locally bounded section φ and a locallyLipschitz section σ such that ( σ, ∆ σ + V σ ) ≤ ( σ, φ ) , the following estimate holds for large R : sup A ( R, R ) | σ | ≤ C vol B ( o, R ) k σ k L ( A ( R/ , R/ + CR k φ k L ∞ ( A ( R, R )) . Proof.
Set u := | σ | + F , with F := R k φ k L ∞ ( A ( R, R )) . The case φ = 0 is treated in [Min].Actually, in [Min], the estimation is written assuming a global weighted Sobolev inequality.But since we work at a fixed scale R , there is no need for such a global inequality: thelocal Sobolev inequality of L. Saloff-Coste [SC], with controlled constant, is sufficient for ourpurpose; and its validity only requires Ric ≥
0. Therefore we assume F = 0.To avoid troubles on the zero set of σ , let us consider the regularizations v ǫ := q | σ | + ǫ and u ǫ := v ǫ + F . Observing the inequalities v ǫ ∆ v ǫ ≤ ( σ, ∆ σ ) ≤ | σ | ( | V − | | σ | + | φ | | σ | ) ≤ v ǫ ( | V − | | σ | + | φ | | σ | ) , we deduce ∆ v ǫ ≤ | V − | v ǫ + | φ | and thus find∆ u ǫ ≤ | V − | u ǫ + | φ | ≤ (cid:18) | V − | + | φ | F (cid:19) u ǫ . Our choice of F enables us to use the estimate without source term in [Min]:sup A ( R, R ) u ǫ ≤ C vol B ( o, R ) m k u ǫ k L m ( A ( R/ , R/ Let ǫ go to zero, so as to obtainsup A ( R, R ) | σ | ≤ sup A ( R, R ) u ≤ C vol B ( o, R ) m k σ k L m ( A ( R/ , R/ + CF, which is what we want. (cid:3)
We will use this lemma on tensor bundles, with the induced Levi-Civita connection, inorder to prove that on a Ricci flat manifold, if the curvature decays at infinity, then thecovariant derivatives of the curvature also decay.
Proposition
A.4 — Let ( M n , g ) be a complete noncompact Ricci flat manifold. If a ≥ ,the estimate | Rm | = O ( r − a ) implies for positive integer i : (cid:12)(cid:12) ∇ i Rm (cid:12)(cid:12) = O ( r − a − i ) . Proof.
Since M is Ricci flat, its curvature tensor obeys an elliptic equation [BKN]∆ Rm = Rm ∗ Rm , which implies for every k in N [TV]:(37) ∆ ∇ k Rm = k X i =0 ∇ i Rm ∗∇ k − i Rm . Let us prove the result by induction on i . The case i = 0 is contained in the assumptions.Suppose that the result is established for i ≤ k , with k ≥
0. Formula (37) can be written(∆ − Rm ∗ ) ∇ k +1 Rm = k X i =1 ∇ i Rm ∗∇ k +1 − i Rm . N ASYMPTOTICALLY FLAT MANIFOLDS WITH NON MAXIMAL VOLUME GROWTH. 45
Since the right-hand side is bounded by C k +1 r − a − k − , lemma A.3 yields:(38) sup A ( R, R ) (cid:12)(cid:12)(cid:12) ∇ k +1 Rm (cid:12)(cid:12)(cid:12) ≤ C k +1 vol B ( o, R ) (cid:13)(cid:13)(cid:13) ∇ k +1 Rm (cid:13)(cid:13)(cid:13) L ( A ( R/ , R/ + C k +1 R − a − k . Let χ be a positive smooth function equal to 1 on A ( R/ , R/ A ( R/ , R ) c and withdifferential bounded by 10 /R . Then we can write Z A ( R/ , R/ (cid:12)(cid:12)(cid:12) ∇ k +1 Rm (cid:12)(cid:12)(cid:12) ≤ Z A ( R/ , R ) (cid:12)(cid:12)(cid:12) ∇ (cid:16) χ ∇ k Rm (cid:17)(cid:12)(cid:12)(cid:12) and, after integration by parts, we find Z A ( R/ , R/ (cid:12)(cid:12)(cid:12) ∇ k +1 Rm (cid:12)(cid:12)(cid:12) ≤ Z A ( R/ , R ) | dχ | (cid:12)(cid:12)(cid:12) ∇ k Rm (cid:12)(cid:12)(cid:12) + Z A ( R/ , R ) χ ( ∇ k Rm , ∆ ∇ k Rm) . With (37), we obtain the upper bound Z A ( R/ , R/ (cid:12)(cid:12)(cid:12) ∇ k +1 Rm (cid:12)(cid:12)(cid:12) ≤ R Z A ( R/ , R ) (cid:12)(cid:12)(cid:12) ∇ k Rm (cid:12)(cid:12)(cid:12) + C k +1 k X i =0 Z A ( R/ , R ) (cid:12)(cid:12)(cid:12) ∇ k Rm (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) ∇ i Rm (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∇ k − i Rm (cid:12)(cid:12)(cid:12) . Using a ≥
2, we estimate this by Z A ( R/ , R/ (cid:12)(cid:12)(cid:12) ∇ k +1 Rm (cid:12)(cid:12)(cid:12) ≤ C k +1 vol B ( o, R ) (cid:16) R − − a − k + R − a − k (cid:17) ≤ C k +1 vol B ( o, R ) R − − a − k . As a result, (38) impliessup A ( R/ , R/ (cid:12)(cid:12)(cid:12) ∇ k +1 Rm (cid:12)(cid:12)(cid:12) ≤ C k +1 (cid:16) R − − a − k + R − a − k (cid:17) ≤ C k +1 R − − a − k , hence (cid:12)(cid:12)(cid:12) ∇ k +1 Rm (cid:12)(cid:12)(cid:12) ≤ C k +1 r − a − ( k +1) . (cid:3) Corollary
A.5 — Let ( M n , g ) be a complete Ricci flat manifold. Assume there are numbers ν > and C > such that ∀ t ≥ s > , vol B ( o, t )vol B ( o, s ) ≥ C (cid:18) ts (cid:19) ν . Then the integral bound Z M | Rm | n dµ < ∞ implies for every k in N : (cid:12)(cid:12)(cid:12) ∇ k Rm (cid:12)(cid:12)(cid:12) = O ( r − a ( n,ν ) − k ) with a ( n, ν ) = max (cid:18) , ( ν − n − n − (cid:19) . Corollary
A.6 — Let ( M n , g ) be a complete Ricci flat manifold, with n ≥ . Assumethere are positive numbers A and B such that ∀ t ≥ , At n − ≤ vol B ( o, t ) ≤ Bt n − . Then the integral bound Z M | Rm | n r dvol < ∞ implies for every k in N : (cid:12)(cid:12)(cid:12) ∇ k Rm (cid:12)(cid:12)(cid:12) = O ( r − ( n − − k ) . Appendix B. Distance and curvature.
The following lemma sums up some comparison estimates on the distance function. Upto order two, it is quite classical. Higher order estimates do not seem to be proved in thelitterature, so we include a proof.
Lemma
B.1 — Consider a complete Riemannian manifold ( M, g ) , a point x in M and anumber a ≥ such that inj( x ) > ǫ ≥ and ∀ i ∈ [0 , k ] , (cid:12)(cid:12) ∇ i Rm (cid:12)(cid:12) ≤ cǫ − a − i on the ball B ( x, ǫ ) . Then there is a constant C such that on this ball, the function ρ = d ( x, . ) / satisfies: • | dρ | ≤ ǫ ; • (cid:12)(cid:12) ∇ ρ − g (cid:12)(cid:12) ≤ Cǫ − a ; • for ≤ i ≤ k , (cid:12)(cid:12) ∇ i ρ (cid:12)(cid:12) ≤ Cǫ − a − i .Proof. The first estimate is obvious and the second follows from [BK]. Let us turn to higherorder estimates. We consider the gradient N of r := d ( x, . ) and use the Riccati equation forthe second fundamental form ∇ N of geodesic spheres: ∇ N S = − S − Rm(
N, . ) N. Identifying quadratic forms to symmetric endomorphisms, we write the endomorphism E := ∇ ρ − Id as E = dr ⊗ N + rS − Id . Setting V = grad ρ = rN , we obtain the equation ∇ V E = − E − E − Rm(
V, . ) V. Since ∇ V = Id + E and ∇ V ∇ E = ∇∇ V E − ∇ ∇ V E + Rm( V, . ) E, it follows that ∇ V ∇ E = − ∇ E + E ∗ ∇ E + ∇ Rm ∗ V ∗ V + Rm ∗∇ V ∗ V + Rm ∗ V. Observing that for k ≥
2, we have ∇ k V = ∇ k − E , an induction yields: ∇ V ∇ k E = − ( k + 1) ∇ k E + X i + j = k ∇ i E ∗ ∇ j E + X i + j + l = k ∇ i Rm ∗∇ j V ∗ ∇ l V + X i + j = k − ∇ i Rm ∗∇ j V, for every natural number k . We set F k = r k +1 ∇ k E and G = E/r , so that ∇ N F k = G ∗ F k + H k N ASYMPTOTICALLY FLAT MANIFOLDS WITH NON MAXIMAL VOLUME GROWTH. 47 with H k = r − k − X i =1 F i ∗ F k − i + r k X i + j + l = k ∇ i Rm ∗∇ j +1 ρ ∗ ∇ l +1 ρ + X i + j = k − ∇ i Rm ∗∇ j +1 ρ Along a geodesic starting from x , we find ∂ r | F k | ≤ C k | F k | | G | + | H k | and since the order two estimate ensures r | G | is small, we can bound | F k | by r sup | H k | , upto a constant. We will prove by induction the estimate | F k | ≤ C k r k +1 ǫ − a − k that is (cid:12)(cid:12)(cid:12) ∇ k E (cid:12)(cid:12)(cid:12) ≤ C k ǫ − a − k , or (cid:12)(cid:12)(cid:12) ∇ k +2 ρ (cid:12)(cid:12)(cid:12) ≤ C k ǫ − a − ( k +2) . It will conclude the proof. Initialization ( k = 0) follows from the order two estimate on ρ .Assume the estimates up to order k −
1. It follows that H k , up to some constant, is boundedby: r − r k +2 ǫ − a − k + r k (cid:16) ǫ − a − k +4 − a − − a − + ǫ − a − k +1+4 − a − (cid:17) hence | H k | ≤ C k r k (cid:16) ǫ − a − k + ǫ − a − k +2 − a + ǫ − a − k (cid:17) . With a ≥
2, we find | H k | ≤ C k r k ǫ − a − k ≤ C k r k ǫ − a − k and therefore we get the promisedestimate | F k | ≤ C k r k +1 ǫ − a − k , hence the result. (cid:3) H. Kaul [Kau] proved a control on Christoffel coefficients in the exponential chart, givenbounds on Rm and ∇ Rm. We need the following
Proposition
B.2 — Consider a complete Riemannian manifold ( M, g ) , a point x in M and a number a ≥ such that | Rm | ≤ cǫ − a and |∇ Rm | ≤ cǫ − a − on the ball B ( x, ǫ ) , with ǫ ≥ . Then there is a constant C such that, on the ball ˆ B (0 , ǫ ) in T x M , the connection ∇ ˆ g of the metric ˆ g = exp ∗ x g and the flat connection ∇ are related by (cid:12)(cid:12)(cid:12) ∇ ˆ g − ∇ (cid:12)(cid:12)(cid:12) ≤ Cǫ − a . A better control on the distance function stems from this.
Lemma
B.3 — Consider a complete Riemannian manifold ( M, g ) , a point x in M and anumber a ≥ such that | Rm | ≤ cǫ − a and |∇ Rm | ≤ cǫ − a − on the ball B ( x, ǫ ) , with ǫ ≥ . Then there is a constant C such that if v and w belong to ˆ B (0 , C − ǫ ) , endowed with ˆ g , then | ( dρ v ) w − g x ( w − v, . ) | ≤ Cǫ − a . Proof.
First, choose a sufficiently large C to ensure the convexity of the ball under consid-eration. Observe the expression ( dρ v ) w = − ˆ g w (Exp − w v, . ) , where Exp is the exponential map of ˆ g . Comparison yields(39) | ˆ g w − g x | ≤ Cǫ − a ǫ = Cǫ − a . Suppose γ parameterizes the geodesic connecting w to v in unit time. The geodesic equation ∇ ˆ g ˙ γ ˙ γ = 0 can be written ¨ γ + (cid:16) ∇ ˆ g ˙ γ − ∇ (cid:17) ˙ γ = 0 . With (B.2), we obtain | ¨ γ | ≤ Cǫ − a ǫ = Cǫ − a . Taylor formula γ (1) − γ (0) − ˙ γ (0) = Z (1 − t )¨ γ ( t ) dt then yields (cid:12)(cid:12) v − w − Exp − w v (cid:12)(cid:12) ≤ Cǫ − a . To conclude, we write | ( dρ v ) w − g x ( w − v, . ) | = (cid:12)(cid:12) ˆ g w (Exp − w v, . ) − g x ( v − w, . ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) (ˆ g w − g x )(Exp − w v, . ) (cid:12)(cid:12) + (cid:12)(cid:12) g x (Exp − w v, . ) − g x ( v − w, . ) (cid:12)(cid:12) ≤ Cǫ − a ǫ + Cǫ − a ≤ Cǫ − a . (cid:3) References [A2] U. Abresch,
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