DIFFERENTIABLE RIGIDITY UNDER RICCI CURVATURE LOWER BOUND

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DIFFERENTIABLE RIGIDITY UNDER RICCI CURVATURE LOWER BOUND

Laurent Bessières, Gérard Besson, Gilles Courtois, Sylvestre Gallot

To cite this version:

Laurent Bessières, Gérard Besson, Gilles Courtois, Sylvestre Gallot. DIFFERENTIABLE RIGIDITY

UNDER RICCI CURVATURE LOWER BOUND. Duke Mathematical Journal, Duke University Press,

2012, 161 (1), pp.29-67. �10.1215/00127094-1507272�. �hal-00281855v3�

CURVATURE LOWER BOUND

L. BESSIÈRES, G. BESSON, G. COURTOIS, and S. GALLOT

Abstract

In this article we prove a differentiable rigidity result. Let .Y; g/ and .X; g 0 / be two closed n-dimensional Riemannian manifolds (n 3), and let f W Y ! X be a continuous map of degree 1. We furthermore assume that the metric g 0 is real hyper- bolic and denote by d the diameter of .X; g ₀ /. We show that there exists a number

" WD ".n; d / > 0 such that if the Ricci curvature of the metric g is bounded below by .n 1/g and its volume satisfies vol g .Y / .1 C"/ vol g

.X /, then the manifolds are diffeomorphic. The proof relies on Cheeger–Colding’s theory of limits of Riemannian manifolds under lower Ricci curvature bound.

1. Introduction

Let Y and X be two closed manifolds. The manifold Y is said to dominate X if there is a continuous map f W Y ! X of degree one. An n-dimensional hyperbolic man- ifold X has the smallest volume among the set of all Riemannian manifolds .Y; g/

such that Y dominates X and the metric g has Ricci curvature Ric g .n 1/g. In dimension n D 2 this is a consequence of the Gauss–Bonnet formula, and in dimen- sion n 3 this follows from the following.

THEOREM 1.1 ([3, p. 734])

Let .X; g 0 / be an n-dimensional closed hyperbolic manifold, and let Y be a closed manifold that dominates X. Then, for any metric g on Y such that Ric g .n 1/g, one has vol g .Y / vol g

.X /, and equality happens if and only if .Y; g/ and .X; g 0 / are isometric.

The minimal volume of a closed manifold Y is defined as minvol.Y / D inf ®

vol g .Y /I jK g j 1 ¯

DUKE MATHEMATICAL JOURNAL

Vol. 161, No. 1, © 2012 DOI 10.1215/00127094-1507272 Received 17 February 2010. Revision received 17 June 2011.

2010 Mathematics Subject Classification. Primary 53C20; Secondary 53C35.

Authors’ work partially supported by Agence Nationale de la Recherche grant ANR-07-BLAN-0251.

29

where K g is the sectional curvature of the Riemannian metric g. An n-dimensional hyperbolic manifold X is characterized by its minimal volume among the set of all Riemannian manifolds Y such that Y is homotopy equivalent to X . Namely, we have the following.

THEOREM 1.2 ([2, Theorem 1.1])

Let X be an n-dimensional closed hyperbolic manifold, and let Y be a closed man- ifold that dominates X. Then, minvol.Y / D minvol.X / if and only if X and Y are diffeomorphic.

The aim of this paper is to show the following gap result. It improves the above Theorem 1.2 since we now require a lower bound on the Ricci curvature instead of a pinching of the sectional curvature; moreover, under the hypothesis, we prove that if the volume of Y is close to the volume of X , then these two manifolds are diffeomor- phic. More precisely, we have the following.

THEOREM 1.3

Given any integer n 3 and d > 0, there exists ".n; d / > 0 such that the follow- ing holds. Suppose that .X; g 0 / is an n-dimensional closed hyperbolic manifold with diameter d and that Y is a closed manifold that dominates X; that is, there exists a degree-one map f W Y ! X. Then Y has a metric g such that

Ric g .n 1/g; (1)

vol g .Y / .1 C "/vol g

.X / (2) if and only if f is homotopic to a diffeomorphism.

In [10] the authors prove the existence of closed n-dimensional manifolds Y that are homeomorphic to a closed n-dimensional hyperbolic manifold .X; g 0 / but not diffeomorphic to it. An immediate corollary of the above theorem is the following.

COROLLARY 1.4

With the above notation, there exists " > 0 depending on n and on the diameter of X with the property that for any such Y and any Riemannian metric g on Y whose Ricci curvature is bounded below by .n 1/ one has

vol.Y; g/ > .1 C "/ vol.X; g 0 /:

To be more precise, in [10] the manifold Y is obtained as follows:

Y D X ]†;

where † is an exotic sphere. Not every closed hyperbolic manifold X gives rise to such a Y that is (obviously) homeomorphic but not diffeomorphic to X. Indeed, we may have to take a finite cover of X. But when we get one construction that works, it does on any finite cover X of X as well. The authors also prove that by taking covers of arbitrary large degree we can put on Y a metric whose sectional curvature is arbitrarily pinched around, say, 1. The stronger the pinching, the larger the degree.

Now assume that " could be taken independent of the diameter of X; applying the results of [3] one could show that the volumes of the two manifolds are very close when the pinching on Y is very sharp (close to 1). The volume of Y endowed with this pinched metric could then be taken smaller than .1 C "/ vol.X; g 0 /, by choosing a covering of large degree; the manifolds, though, are not diffeomorphic. This gives a contradiction and shows that “size” of X has to be involved in the statement of the theorem, for example, its diameter.

1.1. Sketch of the proof

We argue by contradiction. Suppose that there is a sequence .X k / k2N of closed hyper- bolic manifolds with diameter d and a sequence of closed manifolds Y k , of degree- one continuous maps f k W Y k ! X k and metrics g k on Y k satisfying hypotheses (3) and (4) for some " _k going to zero. Since f _k is of degree one and X _k is hyperbolic, it is equivalent to say (thanks to Mostow’s rigidity theorem) that f _k is homotopic to a diffeomorphism or simply that X k and Y k are diffeomorphic. We thus assume that Y k

and X k are not diffeomorphic. One then shows that up to a subsequence, for large k, Y k is diffeomorphic to a closed manifold Y; X k is diffeomorphic to a closed manifold X, and X and Y are diffeomorphic. One argues as follows: by the classical finiteness results we get the subconvergence of the sequence ¹X _k º. Indeed, the curvature is 1, the diameter is bounded by hypothesis, and there is a universal lower bound for the volume of any closed hyperbolic manifold of a given dimension, thanks to Margulis’s lemma (see [6, Theorem 37.1.1]). Cheeger’s finiteness theorem then applies. More- over, on a closed manifold of dimension 3, there is at most one hyperbolic metric, up to isometry. We can therefore suppose that X _k D X is a fixed hyperbolic manifold.

The inequality proved in Theorem 1.1 provides a lower bound for the volume of Y _k as it is explained below. We have no a priori bounds on the diameter of .Y _k ; g _k /, but we can use Cheeger–Colding’s theory to obtain subconvergence in the pointed Gromov–

Hausdorff topology to a complete metric space .Z; d / with small singular set. To

obtain more geometric control, the idea is to use the natural maps between Y k and

X (see [3]). One can show that they subconverge to a limit map between Z and X,

which is an isometry. Then X is an n-dimensional smooth closed Riemannian mani-

fold, which is the Gromov–Hausdorff limit of the sequence .Y k ; g k / of a Riemannian

manifold of dimension n satisfying the lower bound (3) on Ricci curvature; therefore X and Y _k are diffeomorphic for large k by a theorem of J. Cheeger and T. Colding.

The paper is organized as follows. The construction and the properties of the natural maps are given in Section 3. In Section 4, we construct the limit space Z and the limit map F W Z ! X. In Section 5, we prove that F is an isometry and conclude.

1.2. Maps of arbitrary degree and scalar curvature

For two closed manifolds Y and X we said above that Y dominates X if there exists a map of degree one from Y onto X. We could have required that there exist a map f W Y ! X of nonzero degree. The main theorem of [3] was stated and proved in this setup. More precisely, the following statement holds.

THEOREM 1.5 (see [3])

Let .X; g 0 / be an n-dimensional closed hyperbolic manifold, and let Y be a closed manifold such that there exists a map f W Y ! X with nonzero degree denoted deg.f /. Then, for any metric g on Y such that Ric g .n 1/g, one has vol g .Y / j deg.f /j vol g

.X /, and equality happens if and only if f is homotopic to a Rieman- nian covering (i.e., locally isometric) of degree j deg.f /j from .Y; g/ onto .X; g 0 /.

With the technique developed in this article, the following result can be proved.

THEOREM 1.6

Given any integer n 3 and d > 0, there exists ".n; d / > 0 such that the follow- ing holds. Suppose that .X; g 0 / is an n-dimensional closed hyperbolic manifold with diameter d and that Y is a closed manifold such that there exists a map f W Y ! X with nonzero degree. Then Y has a metric g such that

Ric g .n 1/g; (3)

vol g .Y / .1 C "/j deg.f /j vol g

.X / (4) if and only if f is homotopic to a covering of degree j deg.f /j.

The proof is essentially the one described above; it uses the technique described in the remainder of this text and the treatment of an arbitrary degree given in [2]. The fact that the degree can be, in absolute value, greater than one yields extra technicali- ties. For the sake of clarity we shall omit this proof in the present article and leave it to the reader. A corollary is the following.

COROLLARY 1.7

Let .X; g 0 / be a closed n-dimensional hyperbolic manifold. Then there exists " >

0, such that, for any metric g on the connected sum X ]X satisfying that its Ricci curvature of g is not smaller than .n 1/,

vol.X ]X; g/ 2.1 C "/ vol.X; g 0 /:

Indeed, X ]X does not carry a hyperbolic metric and hence is not a double cover of X. This can be proven, for example, by saying that the .n 1/-sphere on which the connected sum is made does not bound a topological n-ball, whereas in a hyperbolic manifold every such sphere bounds a ball. We may now ask whether such a result could be true with a lower bound on the scalar curvature instead of a lower bound on the Ricci curvature. The situation in dimension 3, completely clarified by Perelman’s work, shows that the answer to this question is negative. More precisely, if .X; g 0 / is a 3-dimensional closed hyperbolic manifold, a consequence of [1, inequality 2.10] is that

inf ®

vol.X ]X; g/I Scal.g/ 6 ¯

D 2 vol.X; g 0 /:

In dimensions greater than or equal to 4, it follows from [15] and the solution to the Yamabe problem that

inf ®

vol.X ]X; g/I Scal.g/ 6 ¯

2 vol.X; g 0 /:

2. Some a priori control on .Y; g/

Some a priori control on the metric g will be needed in Sections 2 and 3. We give here the necessary results.

Let .X; g 0 / be a hyperbolic manifold, and let Y be a manifold satisfying the assumptions of Theorem 1.3. For any Riemannian metric g on Y satisfying the cur- vature assumption (3), one has the following inequality:

vol g .Y / vol g

.X /: (5)

It is a consequence of Besson–Courtois–Gallot’s inequality (see [3])

h.g/ ⁿ vol g .Y / h.g 0 / ⁿ vol g

.X /; (6) where h.g/ is the volume entropy, or the critical exponent, of the metric g, that is,

h.g/ D lim

R!C1

1 R ln

vol g

.B g

.x; R//

;

where g Q is the lifted metric on Y Q . Indeed, any metric g on Y which satisfies (3) verifies, by Bishop’s theorem,

h.g/ h.g ₀ / D n 1: (7)

One can obtain a lower bound of the volume of some balls by Gromov’s isolation theorem (see [13, Theorem 0.5]). It shows that if the simplicial volume kY k—a topo- logical invariant also called Gromov’s norm—of Y is nonzero, then for any Rieman- nian metric g on Y satisfying the curvature assumption (3), there exists at least one point y g 2 Y such that

vol g

B.y g ; 1/

v n > 0: (8)

Here B.y g ; 1/ is the geodesic ball of radius 1 for the metric g, and v n is a universal constant. This theorem applies in our situation since, by an elementary property of the simplicial volume, kY k kXk if there is a degree-one map from Y to X (see [13]). On the other hand, X has a hyperbolic metric, and hence kXk > 0 by Gromov–

Thurston’s theorem (see [13]).

Given this universal lower bound for the volume of a unit ball B.y g ; 1/, the vol- ume of any ball B.y; r/ is bounded from below in terms of r and d.y g ; y/. Indeed, recall that under the curvature assumption (3), Bishop–Gromov’s theorem shows that for any 0 < r R, one has

vol g .B.y; r//

vol g .B.y; R// vol

.B

.r//

vol

.B

R// ; (9)

where B

.r/ is a ball of radius r in the hyperbolic space H ⁿ . As B.y g ; 1/ B.y; 1 C d.y g ; y/ C r/, one deduces from (9) that

vol g

B.y; r/

vol g

B.y; 1 C d.y g ; y/ C r/ vol

.B

.r//

vol

.B

.1 C d.y g ; y/ C r// (10) v n

vol

.B

.r//

vol

.B

.1 C d.y g ; y/ C r// : (11) The curvature assumption (3) and the volume estimates (9) or (11) are those required to use the noncollapsing part of Cheeger–Colding’s theory, as we shall see in Section 3.

3. The natural maps

In Sections 3.1 and 3.2 we recall the construction and the main properties of the natural maps defined in [3] (see also [4]).

3.1. Construction of the natural maps

Suppose that .Y; g/ and .X; g 0 / are closed Riemannian manifolds and that

f W Y ! X

is a continuous map of degree one. For the sake of simplicity, we assume that g 0 is hyperbolic. (The construction holds in a much more general situation.) Then, for any c > h.g/ there exists a C ¹ -map

F c W Y ! X;

homotopic to f , such that for all y 2 Y ,

j Jac F c .y/j c h.g 0 /

n

; (12)

with equality for some y 2 Y if and only if d _y F _c is a homothety of ratio _h.g ^c

0

/ . Inequality (6) is then easily obtained by integration of (12) and by taking a limit when c goes to h.g/. In general, to obtain global rigidity properties, one has to study carefully the behavior of F c as c goes to h.g/.

The construction of the maps is divided in four steps. Let Y Q and X Q be the univer- sal coverings of Y and X , respectively, and let f Q W Q Y ! Q X be a lift of f .

Step 1. For each y 2 Q Y and c > h.g/, let _y ^c be the finite measure on Y Q defined by d _y ^c .z/ D e

dv

.z/

where z 2 Q Y , g Q is the lifted metric on Y Q and .:; :/ is the distance function of . Y ; Q g/. Q Step 2. Pushing forward this measure gives a finite measure f Q ^c _y on X. Let us Q recall that it is defined by

f Q ^c _y .U / D _y ^c f Q

.U / :

Step 3. One defines a finite measure ^c _y on @ X Q by convolution of f Q _y ^c with all visual probability measures P x of X. Recall that the visual probability measure Q P x at x 2 Q X is defined as follows: the unit tangent sphere at x noted U x X Q projects onto the geometric boundary @ X Q by the map

v 2 U x X Q ! ^E

v .1/ 2 @ X ; Q

where v .t / D exp _x .t v/. The measure P x is then the pushforward by E x of the canonical probability measure on U x X; that is, for a Borel set Q A 2 @ X, Q P x .A/ is the measure of the set of vectors v 2 U x X Q such that v .C1/ 2 A.

Then

^c _y .A/ D Z

X

P x .A/ d f Q _y ^c .x/ D Z

Y

P f .z/

.A/ d _y ^c .z/:

One can identify @ X Q with the unit sphere in R ⁿ , by choosing an origin o 2 Q X and

using E ₀ . The density of this measure is given by (see [3])

d ^c _y . / D Z

Y

e

^{/B. f .z/; /}

e

dv

.z/

d;

where 2 @ X, Q d is the canonical probability measure on S ⁿ¹ , and B.:; / is a Busemann function on X Q normalized to vanish at x D o. We will use the notation

p.x; / D e

^{/B.x; /} : Step 4. The map

F c W Q Y ! Q X

associates to any y 2 Q Y the unique x 2 Q X , which minimizes on X Q the function x ! B.x/ D

Z

@ X

B.x; / d ^c _y . / (see [3, Appendix A]).

The maps F c are shown to be C ¹ and equivariant with respect to the actions of the fundamental groups of Y and X on their respective universal covers. The quotient maps, which are also denoted by F c W Y ! X, are homotopic to f . Note that F c

depends heavily on the metric g.

3.2. Some technical lemmas Let us give some definitions.

Definition 3.1

For y 2 Q Y , let _y ^c be the probability measure on @ X Q defined by _y ^c D ^c _y

^c _y .@ X / Q : Let us remark that we have

k ^c _y k D ^c _y .@ X / Q D Z

Y

e

dv

.z/ D k _y ^c k:

We consider two positive definite bilinear forms of trace equal to one and the corre- sponding symmetric endomorphisms.

Definition 3.2

For any y 2 Q Y , u; v 2 T F

.y/ X, Q h ^c _y .u; v/ D

Z

@ X

dB _.F

_{.y/; /} .u/ dB _.F

_{.y/; /} .v/ d _y ^c . / D g 0

H _y ^c .u/; v

:

And, for any y 2 Q Y , u; v 2 T y Y Q , h

_y .u; v/ D 1

^c _y .@ X / Q Z

Y

d .y;z/ .u/d .y;z/ .v/ d _y ^c .z/ D g

H _y

.u/; v :

LEMMA 3.3

For any y 2 Q Y , u 2 T y Y Q , v 2 T F .y/ X, one has Q ˇ ˇg ₀

.I H _y ^c /d y F c .u/; vˇˇ c

g 0 .H _y ^c .v/; v/ 1=2

g.H _y

.u/; u/ 1=2

: (13)

Proof

Since F c .y/ is an extremum of the function B, one has d F

.y/ B.v/ D

Z

@ X

dB .F

.y/; / .v/ d ^c _y . / D 0 (14) for each v 2 T F

.y/ X. By differentiating this equation in a direction Q u 2 T y Y Q , one obtains

Z

@ X

DdB .F

.y/; /

d y F c .u/; v d ^c _y . /

C Z

@ X

dB .F

.y/; / .v/ Z

Y

p f .z/; Q

c d .y;z/ .u/

d _y ^c .z/

d D 0:

Using Cauchy–Schwarz inequality in the second term, one gets ˇ ˇ

ˇ Z

@ X

DdB .F

.y/; /

d y F c .u/; v d ^c _y . /

ˇ ˇ ˇ

Z

@ X

jdB .F

.y/; / .v/j Z

Y

p f .z/; Q

d _y ^c .z/ 1=2

Z

Y

p f .z/; Q

jcd _.y;z/ .u/j ² d _y ^c .z/ 1=2

d;

which is, using Cauchy–Schwarz inequality again, c Z

@ X

j dB _.F

_{.y/; /} .v/j ² Z

Y

p f .z/; Q

d _y ^c .z/ d 1=2

Z

@ X

Z

Y

p f .z/; Q

jd _.y;z/ .u/j ² d _y ^c .z/ d 1=2

D c Z

@ X

j dB _.F

_{.y/; /} .v/j ² d ^c _y . / 1=2 Z

Y

jd _.y;z/ .u/j ² d _y ^c .z/ 1=2

D c ^c _y .@ X / Q

g ₀ .H _y ^c .v/; v/ 1=2

g.H _y

.u/; u/ 1=2

:

It is shown in [3, Chapter 5] that DdB D g 0 dB ˝ dB for a hyperbolic metric.

The left term of the inequality is thus ^c _y .@ X /g Q 0 ..I H _y ^c /d y F c .u/; v/. This proves the lemma.

Definition 3.4

Let 0 < ^c ₁ .y/ ^c _n .y/ < 1 be the eigenvalues of H _y ^c .

PROPOSITION 3.5

There exists a constant A WD A.n/ > 0 such that, for any y 2 Y , j JacF c .y/j c

h.g 0 / n

1 A X n iD1

^c _i .y/ 1 n

2

: (15)

Proof

The proof is based on the two following lemmas.

LEMMA 3.6 At each y 2 Q Y ,

j Jac F c .y/j c p n

n det.H y c / ¹⁼² det.I H y c

/ : Proof

Let ¹v i º be an orthonormal basis of T _F

_.y/ X Q which diagonalizes H y c

. We can assume that d _y F _c is invertible since otherwise the above inequality is obvious. Let u

_i D Œ.I H y c

/ ı d y F c

.v i /. The Schmidt orthonormalization process applied to .u

_i / gives an orthonormal basis .u i / at T y Y Q . The matrix of .I H y c

/ ı d y F c in the basis .u i / and .v i / is upper triangular; then

det.I H y c / Jac F c .y/ D Y n iD1

g 0

.I H y c / ı d y F c .u i /; v i

;

which gives, with (13),

det.I H y c /j Jac F c .y/j c ⁿ Y ⁿ

iD1

g 0

H y c .v i /; v i ¹⁼² Y ⁿ

iD1

g

H _y

.u i /; u i ¹⁼²

c ⁿ det.H _y ^c / ¹⁼² h 1 n

X n iD1

g

H _y

.u _i /; u _i i ⁿ⁼² :

This proves the desired inequality since trace.H _y

/ D 1.

LEMMA 3.7

Let H be a symmetric positive definite (n n)-matrix whose trace is equal to one.

Then, if n 3,

det.H / ¹⁼²

det.I H / n h.g 0 / ²

n=2 1 A

X n iD1

i 1 n

2

for some positive constant A.n/.

Proof

The proof is given in [3, Appendix B5]. This is the point where the rigidity of the natural maps fails in dimension 2.

This completes the proof of Proposition 3.5.

3.3. Some nice properties

We now show that when the volumes of .Y; g/ and .X; g 0 / are close, the natural maps F c have nice properties. More precisely, we will show that when vol.Y; g/

.1 C "/ vol.X; g 0 / and 0 c h.g/ ı with " and ı small enough, then dF c is almost isometric on a set of large relative volume (see Lemma 3.11). We then prove that F c is uniformly Lipschitz on balls of radius R when " ".R/ and ı ı.R/

are small enough (see Lemma 3.12). We finally end Section 3 by showing that F c is

“quasi-contracting” on balls of radius R (see Lemma 3.13).

In this section, we shall consider F c as a map from .Y; g/ to .X; g 0 /. We suppose that the metric g satisfies the curvature assumption (3) and the assumption on its volume (4) for some " > 0. Let us introduce some terminology.

Definition 3.8

Let 0 < ˛ < 1. We say that a property holds ˛-a.e. (˛-almost everywhere) on a set A if the set A

of points of A where the property holds has relative volume bigger than or equal to 1 ˛, that is,

vol.A

/

vol.A/ 1 ˛:

We show that dF _c is ˛-close to being isometric ˛-a.e. on Y for some positive

˛."; c/. Moreover, ˛."; c/ ! 0 as " ! 0 and c ! h.g/. On the other hand, given

any radius R > 0, one shows that kdF c k is uniformly bounded on balls B.y g ; R/,

provided c is close enough to h.g/. Recall that we have a lower bound for the volume

of .Y; g/ but we do not have an upper bound for its diameter. The key point is to show

that H _y ^c is ˛-close to ¹ _n Id on a set of large volume, and is bounded on a ball of fixed radius, with respect to the parameters "; c.

To estimate from above c h.g/ we introduce a parameter ı > 0. We suppose that the volume entropy of g satisfies the inequalities

h.g/ < c h.g/ C ı: (16)

Observe that (7), (15), and (16) imply that j Jac F c .y/j h.g/ C ı

h.g ₀ / n

1 C ı

n 1 n

; (17)

for all y 2 Y . The map F c is thus almost volume decreasing. On the other hand, as vol g .Y / is close to vol g

.X /, the set in Y where F c decreases the volume a lot must have a small measure. Equivalently, j Jac F c j must be close to 1 in the L ¹ -norm. We now give a precise statement.

LEMMA 3.9

If ı is small enough, there exists ˛ 1 D ˛ 1 ."; ı/ > 0 such that for ˛ 1 -a.e. on Y one has,

1 ˛ 1 j Jac F c .y/j; (18)

and for all y 2 Y one has

j Jac F c .y/j 1 C ˛ 1 : (19)

Moreover, ˛ 1 ."; ı/ ! 0 as " and ı ! 0.

Proof Let

˛ D max r 1 C ı

n 1 n

1; p

"

:

Thus

1 C ı

n 1 n1

1 C ˛ ²

and " ˛ ² . In particular, j JacF c .y/j 1 C ˛ ² 1 C ˛ for all y 2 Y , if ı is small enough so that ˛ is less than 1. (We also assume that " is small.)

As F c has degree one, we have vol g

.X / D

Z

Y

F _c

.dv

/ D Z

Y

Jac F c .y/ dv

.y/:

Denote by Y ˛

the set of points y 2 Y such that j Jac F c .y/j 1 ˛:

We have

vol g

.X / Z

Y

j Jac F c .y/j dv

.y/ (20) D

Z

Y

j JacF c .y/j dv

.y/ C Z

Y

j JacF c .y/j dv

.y/ (21) .1 C ˛ ² / vol g .Y ˛

/ C .1 ˛/ vol g .Y n Y ˛

/ (22) D vol g .Y / C ˛ ² vol g .Y ˛

/ ˛ vol g .Y n Y ˛

/: (23) Then, using assumption (4) and the inequality (5) on the volume, we get

vol g .Y n Y ˛

/ vol g .Y / vol g

.X /

˛ C ˛ vol g .Y ˛

/ (24)

"

˛ C ˛

vol g .Y / (25)

2˛ vol g .Y /: (26)

Clearly, 1 2˛ j Jac F c .y/j on Y ˛

and j Jac F c .y/j 1 C 2˛ on Y , which proves the lemma with ˛ 1 ."; ı/ D 2˛.

From this lemma, we deduce that F c is almost injective. Indeed, let x 2 X; one defines N.F c ; x/ 2 N [ ¹1º to be the number of preimages of x by F c . As F c

has degree one, one has N.F c ; x/ 1 for all x 2 X. We then define X 1 WD ¹x 2 X; N.F c ; x/ D 1º. Observe that N.F c ; x/ 2 on X n X 1 .

LEMMA 3.10

There exists ˛ 2 D ˛ 2 ."; ı/ > 0 such that

vol g

.X 1 / .1 ˛ 2 / vol g

.X / (27) and

Z

X

N.F _c ; x/ dv

.x/ ˛ ₂ ."; ı/ vol g

.X /: (28) Moreover, ˛ 2 ."; ı/ ! 0 as " and ı ! 0.

In particular, there exists ˛

> 0 such that N.F c ; x/ D 1 ˛

-a.e. on X .

Proof One defines

˛ 2 ."; ı/ D 2 1 C ı

n 1 n

.1 C "/ 1

:

From (15) and the area formula (see [16, Section 3.7]), we have c

h.g 0 / n

vol g .Y / Z

Y

j Jac F c .y/j dv

.y/ (29) D

Z

X

N.F c ; x/ dv

.x/ (30)

D Z

X

N.F c ; x/ dv

.x/

C Z

X

N.F c ; x/ 1 C 1

dv

.x/ (31) D vol g

.X / C

Z

XnX

N.F c ; x/ 1

dv

.x/ (32) and

vol g

.X n X 1 / Z

XnX

N.F c ; x/ 1

dv

.x/ (33)

c h.g 0 /

n

vol g .Y / vol g

.X / (34) c

h.g ₀ / n

.1 C "/ 1

vol g

.X / (35) ˛ 2 ."; ı/

2 vol g

.X /: (36)

Thus, since N.F c ; x/ 2.N.F c ; x/ 1/ on X n X 1 , we get vol g

.X n X 1 /

Z

XnX

N.F c ; x/ dv

.x/ ˛ 2 ."; ı/ vol g

.X /;

and this proves the lemma.

The following lemma says that dF c .y/ is almost isometric at points y where Jac F c .y/ is almost equal to 1.

LEMMA 3.11

There exists ˛ 3 D ˛ 3 ."; ı/ > 0 such that the following holds. Let Y ˛

be the set of

points where (18) holds, that is, 1 ˛ 1 ."; ı/ j Jac F c .y/j. Let y be a point in Y ˛

, and let u 2 T y Y ; then

.1 ˛ 3 /kuk g kd y F c .u/k g

.1 C ˛ 3 /kuk g : (37) Moreover, ˛ 3 ."; ı/ ! 0 as ", ı ! 0.

Proof

The inequality (15) implies that, for all y 2 Y ,

H _y ^c 1 n Id

² 1

A

1 j Jac F c .y/j 1 C _n1 ^ı n

:

Let us define

ˇ 1 D ˇ 1 ."; ı/ D 1 A ¹⁼²

1 1 ˛ 1 ."; ı/

1 C _n1 ^ı n

1=2

; (38)

where ˛ 1 ."; ı/ is the constant from Lemma 3.9. Clearly, ˇ 1 ."; ı/ ! 0 as " and ı ! 0.

Let Y ˛

be the set of points where (18) holds. On Y ˛

, one has

H _y ^c Id n ² ˇ 1 2

: (39)

Let ¹u i º iD1;:::;n be an orthonormal basis of T y Y , and let v i D d y F .u i /. Writing Id H _y ^c D ⁿ¹ _n Id C _n ¹ Id H _y ^c , one gets

ˇ ˇg ₀

.Id H _y ^c /d y F c .u i /; d y F c .u i /ˇˇ

ˇ ˇ ˇ ˇ g 0

n 1 n Id

d y F c .u i /; d y F c .u i / ˇˇ ˇ ˇ

ˇ ˇ ˇ ˇ g 0

1

n Id H _y ^c

d y F c .u i /; d y F c .u i / ˇˇ ˇ ˇ (40) n 1

n kd y F c .u i /k ² _g

0

1

n Id H _y ^c :kd y F c .u i /k ² _g

0

(41)

n 1 n ˇ ₁

kd _y F _c .u _i /k ² _g

0

: (42)

Writing H _y ^c D _n ¹ Id C H _y ^{c 1} _n Id, one has g 0

H _y ^c d y F c .u i /; d y F c .u i / 1=2

g 0

1 n Id

d y F c .u i /; d y F c .u i /

1=2

C ˇ ˇ ˇ ˇ g ₀

H _y ^c 1 n Id

d _y F _c .u _i /; d _y F _c .u _i / ˇˇ ˇ ˇ

1=2

(43) 1

p n C ˇ ¹⁼² ₁

kd _y F _c .u _i /k _g

: (44) Taking the trace of the right-hand side of (13) and using the Cauchy–Schwarz inequality, one has

X n iD1

g 0

H _y ^c d y F c .u i /; d y F c .u i / 1=2

g

H _y

.u i /; u i 1=2

1

p n C ˇ ₁ ¹⁼² X ⁿ

iD1

kd y F c .u i /k ² _g

0

1=2 X ⁿ

iD1

g

H _y

.u i /; u i ¹⁼² (45)

D 1

p n C ˇ ₁ ¹⁼² X ⁿ

iD1

kd y F c .u i /k ² _g

1=2

: (46)

By (13), the trace of (42) is not greater than the right-hand side of (46) multiplied by c, and hence

n 1 n ˇ 1

X ⁿ

iD1

kd y F c .u i /k ² _g

0

c 1

p n C ˇ ₁ ¹⁼² X ⁿ

iD1

kd y F c .u i /k ² _g

0

1=2

;

and X ⁿ

iD1

kd y F c .u i /k ² _g

0

1=2

c

p

1

n C ˇ ₁ ¹⁼²

n1 n ˇ 1

p n

1 C ı n 1

1 C p nˇ ¹⁼² ₁ 1 _n1 ⁿ ˇ 1

:

Let us define

ˇ 2 WD ˇ 2 ."; ı/ D 1 C ı

n 1 2

1 C p nˇ ₁ ¹⁼² 1 _n1 ⁿ ˇ 1

2

1:

Clearly, ˇ ₂ ."; ı/ ! 0 as " and ı ! 0. One has X n

iD1

kd y F c .u i /k ² _g

n.1 C ˇ 2 /:

Let L be the endomorphism of T _y Y defined by L D .d _y F _c /

ı d _y F _c . We have trace.L/ D

X n iD1

g

L.u _i /; u _i D

X n iD1

g

d _y F _c .u _i /; d _y F _c .u _i /

n.1 C ˇ ₂ /: (47)

On the other hand,

j1 ˛j ² j Jac F c .y/j ² D det.L/ trace.L/

n n

.1 C ˇ 2 / ⁿ ;

which shows that there is almost equality in the arithmetic-geometric inequality. We then get that there exists some ˛ 3 ."; ı/ > 0, with ˛ 3 ."; ı/ ! 0 as "; ı ! 0, such that

kL Idk ˛ 3 ."; ı/:

Thus for any y 2 Y ˛

and u 2 T y Y ,

.1 ˛ 3 /kuk kd y F c .u/k g

.1 C ˛ 3 /kuk; (48) and d y F c is almost isometric.

We now prove that given a fixed radius R > 0, the natural maps F c have uni- formly bounded differential dF _c on B.y _g ; R/ if the parameters ", ı are sufficiently small. Recall that the point y g has been chosen such that (8) holds; namely, vol g .B.y g ; 1// v n .

LEMMA 3.12

Let R > 0. Then there exist ".R/ > 0 and ı.R/ > 0 such that for any 0 < " < ".R/

and 0 < ı < ı.R/, and for any y 2 B.y _g ; R/, kd y F c k 2 p

n: (49)

Proof

We first prove that for all y 2 Y , kd y F c k is bounded from above by ^c _n .y/, the maxi- mal eigenvalue of H _y ^c (see Definition 3.4). Recall that all eigenvalues of H _y

are less than one and that 0 < ^c _n < 1. Let u be a unit vector in T y Y Q , and let v D d y F c .u/.

Equation (13) gives 1 ^c _n .y/ˇˇg 0

d y F c .u/; d y F c .u/ˇˇ c ^c _n .y/ ¹⁼² g 0

d y F c .u/; d y F c .u/ 1=2

: (50) Hence

kd y F c .u/k g

c p ^c _n .y/

1 ^c _n .y/ : (51)

We thus have to show that ^c _n .y/ is not close to 1. More precisely, let ˇ > 0 such that

1

n C ˇ < 1; one then defines

.ı; ˇ/ WD n 1 C ı n 1 nˇ

p 1 C nˇ 1 > 0:

Clearly, .ˇ; ı/ ! 0 as ı; ˇ ! 0. One can check that if ^c _n .y/ ¹ _n C ˇ, then kd y F c .u/k g

p

n.1 C /. For our purpose, we may suppose that 1. Now let ı _n > 0 and ˇ _n > 0 be such that if 0 < ı 10ı _n and 0 < ˇ 10ˇ _n , then .ı; ˇ/ 1.

Moreover, we define " n > 0 such that if 0 < " < " n and 0 < ı 10ı n , then with the notation of (38) and Lemma 3.9, ˇ 1 ."; ı/ ˇ n . In what follows, we suppose " and ı to be sufficiently small.

By (39) we have that j ^c _n .y/ ¹ _n j ˇ 1 ."; ı/ on Y ˛

. Recall that Y ˛

has a large relative volume in Y . The idea is first to estimate ^c _n on a neighborhood of Y ˛

and then to show that this neighborhood contains B.y _g ; R/ if the parameters " and ı are sufficiently small relative to R.

For this purpose we need to estimate the variation of ^c _n . Recall that H _y ^c is defined by

g ₀

H _y ^c .u/; v D

Z

@ X

dB _.F

_{.y/; /} .u/ dB _.F

_{.y/; /} .v/ d _y ^c . /:

Let U , V be parallel vector fields near F c .y/ extending unit tangent vectors at F c .y/, u, and v. We compute the derivative of g 0 .H _y ^c .U /; V / in a direction w 2 T y Y :

w g ₀

H _y ^c .U /; V D

Z

@ X

DdB _.F

_{.y/; /}

d _y F .w/; U

dB _.F

_{.y/; /} .V / d _y ^c . /

C Z

@ X

dB _.F

_{.y/; /} .U /DdB _.F

_{.y/; /}

d _y F .w/; V d _y ^c . /

C Z

@ X

dB .F

.y/; / .U / dB .F

.y/; / .V /w d _y ^c . /:

The Buseman functions of the hyperbolic space satisfies kDdB k 1 and kdBk 1, and thus

ˇ ˇw g 0

H _y ^c .U /; V ˇˇ 2kd y F c .w/k g

C ˇ ˇ ˇ Z

@ X

w d _y ^c . / ˇ ˇ ˇ : Recall that

d _y ^c . / D d ^c _y . / ^c _y .@ X / Q D

R

Y

p. f .z/; /e Q

dv

.z/

R

Y

e

dv

.z/ d:

Differentiating this formula yields w d _y ^c . / D

R

Y

p. f .z/; /.c Q d _.y;z/ .w//e

dv

.z/

^c _y .@ X / Q d (52)

d ^c _y . / ^c _y .@ X / Q ²

Z

Y

c d _.y;z/ .w/

e

dv

.z/: (53)

Since jd .y;z/ .w/j kwk g , we have ˇ ˇ

ˇ Z

@ X

w d _y ^c . / ˇ ˇ ˇ Z

@ X

2ckwk g d _y ^c . / D 2ckwk g ; (54) which gives jw g 0 .H _y ^c .U /; V /j 2kd y F c .w/k g

C 2ckwk g . If w is a unit vector, then (51) yields

ˇ ˇw g 0

H _y ^c .U /; V ˇˇ 2c

p ^c _n .y/

1 ^c _n .y/ C 1

: (55)

Let us now consider small constants > ˇ > 0 and define

r.ı; ˇ; / WD ˇ

2.n 1 C ı/

nC

1

nC

C 1 > 0:

Our goal is to prove that inf

°

d.y 0 ; y 1 / ˇ ˇ ˇ y 0 ; y 1 2 Y; ^c _n .y 0 / 1

n C ˇ; ^c _n .y 1 / 1 n C ±

r.ı; ˇ; /:

Let y 0 2 Y so that ^c _n .y 0 / ¹ _n C ˇ. Assume that there exists y 2 Y such that ^c _n .y/

1

n C . One defines

r WD inf

°

d.y 0 ; y/ ˇ ˇ ˇ y 2 Y; ^c _n .y/ 1 n C ±

:

By continuity, there exists y 1 2 Y such that ^c _n .y 1 / D _n ¹ C and d.y 0 ; y 1 / D r . Let W Œ0; r ! Y be a minimizing geodesic from y 0 to y 1 . We easily see that ^c _n ..t // < _n ¹ C for any 0 t < r. Let U.t / be a parallel vector field in X along F c . / such that U.r/ is a unit eigenvector of H _y ^c

1

. Then, using (55) with P

:g 0 .H _{.t /} ^c U.t /; U.t // D _dt ^d g 0 .H _{.t /} ^c U.t /; U.t //, one has j ^c _n .y 1 / ^c _n .y 0 /j ˇ

ˇg 0

H _.r/ ^c U.r/; U.r/

g 0

H _.0/ ^c U.0/; U.0/ˇˇ (56) D

ˇ ˇ ˇ

Z r 0

d dt g 0

H _{.t /} ^c U.t /; U.t / dt

ˇ ˇ

ˇ (57) 2c

Z r 0

p ^c _n ..t //

1 ^c _n ..t // C 1

dt (58)

2cr q

1 n C 1 ₁

n C C 1

: (59)

As a consequence,

r ˇ

2.n 1 C ı/

nC

1

nC

C 1

D r.ı; ˇ; /:

We now set D 2ˇ n so that .ı; / 1 for any ı ı n . One then defines r n WD r.ı n ; ˇ n ; 2ˇ n /. Let us recall that for " " n and ı ı n , we have ˇ 1 ."; ı/ ˇ n . On Y ˛

, one has ^c _n .y/ ¹ _n C ˇ 1 ."; ı/ ¹ _n C ˇ n . Hence, if ^c _n .y 1 / ¹ _n C 2ˇ n , one has

d.y 1 ; Y ˛

/ r

ı; ˇ 1 ."; ı/; 2ˇ n

r.ı n ; ˇ n ; 2ˇ n / D r n :

We thus have proved that in the r n -neighborhood of Y ˛

, one has ^c _n .y/ ¹ _n C 2ˇ n . This implies that

kd y F c k

1 C .ı; 2ˇ n / p n 2 p

n:

Let us denote by V _r

.Y _˛

/ the r _n -neighborhood of Y _˛

. It remains to show that B.y g ; R/ V r

.Y ˛

/, if " ".R/ and ı ı.R/. Let us recall that

vol g .Y ˛

/

vol g .Y / 1 ˛ ₁ ; and hence

vol g .Y n Y _˛

/ ˛ ₁ vol g .Y / ˛ ₁ .1 C "/ vol g

.X / WD v."; ı/:

Clearly, v."; ı/ ! 0 when ", ı ! 0. On the other hand, by (11) for any y 2 B.y g ; R/, we have

vol g

B g .y; r 0 / v n

vol

.B

.r 0 //

vol

.B

.1 C R C r 0 // WD v 0 .R/ > 0: (60) If v 0 .R/ > v."; ı/, then for any y 2 B.y g ; R/ one has B g .y; r n / 6 Y n Y ˛

, which means that B _g .y; r _n / intersects Y _˛

. This shows that d.y; Y _˛

/ < r _n and y 2 V r

.Y ˛

/.

The lemma is proved if we define " D ".R/ > 0 and ı D ı.R/ > 0 to be suffi- ciently small constants such that v."; ı/ < v 0 .R/.

We now prove that F c is almost 1-Lipschitz.

LEMMA 3.13

For any fixed R > 0, there exists " 2 .R/ > 0 and ı 2 .R/ > 0 such that for every 0 < " <

" 2 .R/ and 0 < ı < ı 2 .R/, there exists D ."; ı; R/ > 0 such that on B g .y g ; R/, d _g

F _c .y ₁ /; F _c .y ₂ /

.1 C /d _g .y ₁ ; y ₂ / C : (61)

Moreover, ."; ı; R/ ! 0 as ", ı ! 0.

Proof

The idea goes as follows. We have proved that d y F c is almost isometric on Y ˛

. On the other hand, kd y F c k is uniformly bounded in B.y g ; R/ if the parameters "

and ı are chosen sufficiently small. To prove the lemma one computes the lengths of F c . / where is a minimizing geodesic in B.y g ; R/ whose intersection with Y ˛

is large. Existence of such geodesics follows from an integral geometry lemma due to T. Colding.

Fix some R > 0. We define the following constants.

If d > 0,

c 1 .n; d / WD sup

0<s=2<r<s<d

vol

.@B

.s//

vol

.@B

.r// : If > 0, R > 0,

c 2 .n; ; R/ WD c 1 .n; 2R/

2 vol

.B

. //

: If " > 0, ı > 0,

."; ı/ WD 2˛ ₃ ² ."; ı/vol g

.X / C 2.4n C 1/˛ 1 ."; ı/ vol g

.X /:

Clearly, ."; ı/ ! 0 as "; ı ! 0.

Let ."; ı; R/ > 0 be the function implicitly defined by

vol

. / WD ."; ı/ 2c 1 .n; 2R/ vol

.1 C R C 1/ ²

v _n ² :

Again, we easily see that, for fixed R, ."; ı; R/ ! 0 as ", ı ! 0. We also choose

" ₂ .R/ > 0 and ı ₂ .R/ > 0 such that " ₂ .R/ ".2R/, ı ₂ .R/ < ı.2R/ and such that, if 0 < " " 2 .R/ and 0 < ı < ı 2 .R/, then ."; ı; R/ 1.

Finally, one defines ."; ı; R/ WD max.2 p n p

; 8 p

/. From the remarks above we can choose " 2 .R/ and ı 2 .R/ so that ."; ı; R/ < 1=R (for 0 < " " 2 .R/, 0 < ı <

ı 2 .R/ and R big).

There are two cases.

Case (i). Let y 1 , y 2 in B g .y g ; R/ be such that d.y 1 ; y 2 / p

. Using (49), if 0 < " < ".2R/, 0 < ı < ı.2R/ one has

d

F c .y 1 /; F c .y 2 / 2 p

n p

: (62)

Case (ii). Let y 1 , y 2 in B g .y g ; R/ be such that d.y 1 ; y 2 / p

. We will use the

following theorem, due to J. Cheeger and T. Colding (see [7, Theorem 2.11]), which

we describe now in a particular case. We keep the notations of [7].

Let us define A 1 D B g .y 1 ; /, A 2 D B g .y 2 ; / and W D B g .y g ; 2R/ where y 1

and y 2 are points as above sitting on a complete Riemannian manifold .Y; g/ with Ric g .n 1/g. For any z ₁ 2 A ₁ and any unit vector v ₁ 2 T _z

Y , the set I.z ₁ ; v ₁ / defined by

I.z 1 ; v 1 / D ®

t ˇ ˇ .t / 2 A 2 ;

is minimal;

.0/ D v 1

¯

has a measure jI.z 1 ; v 1 /j bounded above by 2 . Thus D.A ₁ ; A ₂ / WD sup

z

;v

jI.z ₁ ; v ₁ /j 2;

and, similarly, D.A 2 ; A 1 / 2 . For any z 1 2 A 1 and z 2 2 A 2 , let z

z

be a min- imizing geodesic from z 1 to z 2 . Clearly, B.y g ; 2R/. Then, by [7, Theorem 2.11], we have for any nonnegative integrable function e defined on Y ,

Z

A

Z d.z

;z

/ 0

e. z

;z

/.s/ ds c 1 .n; 2R/

D.A 1 ; A 2 / vol.A 1 / C D.A 2 ; A 1 / vol.A 2 /

Z

W

e.y/ dv

.y/: (63)

By Bishop’s theorem, for i D 1, 2 we have vol g .A i / vol

B

. /

; and thus

c ₁ .n; 2R/

D.A ₁ ; A ₂ / vol.A ₁ / C D.A ₂ ; A ₁ /vol.A ₂ /

c ₂ .n; ; R/:

Therefore, applying (63) to the function e.y/ D sup

u2U

Y

kd y F c .u/k kuk 2

and using (37) on W \ Y ˛

and (49) on W n Y ˛

, we get Z

A

Z d.z

;z

/ 0

e. _z

_;z

/.s/ ds

c ₂ .n; ; R/ Z

W

e.y/ dv

.y/ C Z

W

e.y/ dv

.y/

c 2 .n; ; R/

˛ ² ₃ : vol g .Y / C .4n C 1/ vol g .Y n Y ˛

/

c ₂ .n; ; R/."; ı/: (64)

Now, if we denote by WD z

z

, then we have j`.F c ı / `. /j D ˇ ˇ ˇ

Z d.z

;z

/ 0

kd _.s/ F c . /k k P P kds ˇ ˇ ˇ

Z d.z

;z

/ 0

sup

u2T

Y

ˇ ˇkd _.s/ F _c .u/k kuk ˇ ˇ ds:

Using Cauchy–Schwarz inequality we have j`.F c ı / `. /j ²

d.z 1 ; z 2 / . R d.z

;z

/

0 sup _u jkd _.s/ F c .u/k kukj ds/ ² d.z 1 ; z 2 /

Z d.z

;z

/ 0

e .s/

ds:

Integrating on A 1 A 2 , we deduce from (64) that Z

A

j`.F c ı z

z

/ `. z

z

/j ²

d.z 1 ; z 2 / dv

.z 1 / dv

.z 2 / c 2 .n; ; R/."; ı/: (65) By (11), for i D 1, 2 one has

vol g .A i / v n

vol

.B

. //

vol

.B

.1 C R C // WD v 0 .; R/ > 0:

From the obvious inequality c 2 .n; ; R/."; ı/ 1

v 0 .; R/ ² Z

A

c 2 .n; ; R/."; ı/ dv

.z 1 / dv

.z 2 /;

we get Z

A

j`.F c ı z

z

/ `. z

z

/j ² d.z 1 ; z 2 /

Z

A

c 2 .n; ; R/."; ı/

v 0 .; R/ ² : (66) As a consequence there exist z 1 2 A 1 and z 2 2 A 2 such that

j`.F c ı z

z

/ `. z

z

/j ² d.z 1 ; z 2 / c 2 .n; ; R/."; ı/

v 0 .; R/ ² : On the other hand, one can check that by definition of ,

c 2 .n; ; R/."; ı/

v 0 .; R/ ² D ."; ı/ 2c 1 .n; 2R/ vol

.1 C R C 1/ ² v ² _n vol

. / D ² : This yields

j`.F _c ı _z

_z

/ `. _z

_z

/j ² d.z ₁ ; z ₂ / ² ;

and

d

F c .z 1 /; F c .z 2 /

`.F c ı z

z

/ d.z 1 ; z 2 / C p

d.z 1 ; z 2 /:

Since d.y i ; z i / < and d.y 1 ; y 2 / p

, we have

d.z 1 ; z 2 / d.y 1 ; y 2 / C 2 d.y 1 ; y 2 /.1 C 2 p /:

With our choice of very small compared to 1, we also have d.z 1 ; z 2 / d.y 1 ; y 2 / 2

p 2 : We then have

d

F c .y 1 /; F c .y 2 / d

F c .y 1 /; F c .z 1 / C d

F c .z 1 /; F c .z 2 / C d

F c .z 2 /; F c .y 2 / (67) 2 p

n C d.z 1 ; z 2 / C

d.z 1 ; z 2 / 1=2

C 2 p

n (68)

4 p

n C d.y 1 ; y 2 / d.z 1 ; z 2 / d.y 1 ; y 2 /

1 C .d.z 1 ; z 2 //

(69) 4 p

n C d.y 1 ; y 2 /.1 C 2 p

/.1 C p

2 ³⁼⁴ / (70)

4 p

n C d.y 1 ; y 2 /.1 C 8 p

/: (71)

We finally get

d

F c .y 1 /; F c .y 2 /

C .1 C /d.y 1 ; y 2 /; (72) in case (ii).

4. A limit map on the limit space

In this section, we consider a sequence .Y k ; g k / k2N of closed Riemannian n-mani- folds satisfying the curvature bound (3) and the assumption that there exist an closed hyperbolic n-manifold .X; g 0 /, degree-one maps f k W Y k ! X, and a sequence " k ! 0 such that

vol g

.Y _k / ! vol g

.X /; (73) as k goes to C1. From (8), for every k 2 N, there exists y g

2 Y k satisfying the local volume estimate; that is, vol.B g

.y g

; 1// v n > 0. For the sake of simplicity we shall use the notation y _k instead of y g

.

Below, we prove that .Y _k ; g _k ; y _k / subconverges in the pointed Gromov–Haudorff

topology to a limit metric space .Y

; d

; z

/. Moreover, there exists a sequence of

natural maps F c

W .Y k ; g k / ! .X; g 0 /, with suitably chosen parameters c k , which subconverges to a “natural map” F W Y

! X .

Let us recall the definition of the Gromov–Hausdorff topology. For two subsets A; B of a metric space Z the Hausdorff distance between A and B is

d ^Z

.A; B/ WD inf ®

" > 0 ˇ ˇ B V " .A/ and A V " .B/ ¯

2 R [ ¹1º:

It is a distance on compact subsets of Z (see [11]).

Definition 4.1 (see [14])

Let X 1 , X 2 be two metric spaces. Then the Gromov–Hausdorff distance d

.X 1 ; X 2 / 2 R [ 1 is the infimum of the numbers

d ^Z

f 1 .X 1 /; f 2 .X 2 /

for all metric spaces Z and all isometric embeddings f i W X i ! Z.

It is a distance on the space of isometry classes of compact metric spaces. One says that a sequence .X i / i2N of metric spaces converges in the Gromov–Hausdorff topology to a metric space X

if d

.X _i ; X

/ ! 0 as i ! 1. Let x _i 2 X _i and x

2 X

. One says that the sequence .X i ; x i / i2N converges to .X

; x

/ in the pointed Gromov–Hausdorff topology if for any R > 0, d

.B X

.x i ; R/; B X

.x

; R// ! 0 as i ! C1. (In fact, this definition holds only for length spaces, which will be sufficient in our situation.)

To deal with the Gromov–Hausdorff distance between X 1 and X 2 , it is convenient to avoid the third space Z by using "-approximations between X ₁ and X ₂ .

Definition 4.2

Given two metric spaces X 1 ,X 2 and " > 0, an "-approximation (or "-isometry) from X 1 to X 2 is a map f W X 1 ! X 2 such that

(1) for any x; x

2 X 1 , jd X

.f .x/; f .x

// d X

.x; x

/j < ";

(2) the "-neighborhood of f .X 1 / is equal to X 2 .

Then one can show (see [5, Corollary 7.3.28]) that d

.X 1 ; X 2 / < " if there exists a 2"-approximation from X 1 to X 2 and similarly an "-approximation exists if d

.X 1 ; X 2 / < 2". Let us insist on the fact that these approximations may be neither continuous nor even measurable.

Our goal is to prove the following.

PROPOSITION 4.3

Up to extraction and renumbering, the sequence .Y _k ; g _k ; y _k / satisfies the following.

(1) There exists a complete pointed length space .Y

; d

; y

/ such that .Y k ; g k ; y _k / converges in the pointed Gromov–Hausdorff topology to a metric space .Y

; d

; y

/. Moreover, .Y

; d

/ has Hausdorff dimension equal to n.

(2) There exist sequences of positive numbers " k ! 0, ı k ! 0, c k 2 h.g k /;

h.g k / C ı k Œ, R k ! C1, such that " k ".R k / and ı k ı.R k /, where "./

and ı./ are given by Lemma 3.12, and such that the following holds. Let F c

W .Y k ; g k / ! .X; g 0 /

be the natural map as defined in Section 2. Then F c

ı _k converges uniformly on compact sets to a map

F W Y

! X;

which is 1-Lipschitz.

The proof is divided in two steps described in the following sections.

Existence of the limit and its properties

Under the curvature bound (3) and the local volume estimate (11), the sequence .Y k ; g k / is “noncollapsing” and part (1) of Proposition 4.3 is a straightforward appli- cation of the Gromov and Cheeger–Colding compactness theorem (see [8, Theo- rem 1.6]). Before proving point (2) of Proposition 4.3, let us describe some features of the convergence and of the limit space which will be used later.

The continuity of the volume under the (pointed) Gromov–Hausdorff conver- gence is crucial for our purposes. For ` > 0, note H <sup>`</sup> the `-dimensional Hausdorff measure of a metric space (see [5, Definition 1.7.7]).

THEOREM 4.4 ([8, Theorem 5.9])

Let p i 2 Y i , let p

2 Y

be their limit, and let R > 0. Then

i!C1 lim vol g

B.p i ; R/

D H ⁿ

B.p

; R/

: (74)

In particular, Y

satisfies the Bishop–Gromov inequalities (9) and the Bishop inequality. By definition, a tangent cone at p 2 Y

is a complete pointed Gromov–

Hausdorff limit, ¹Y

; d

; p

º of a sequence of rescaled space, ¹.Y

; r _i

d; p/º,

where ¹r i º is a positive sequence such that r i ! 0. Indeed, by [12, Proposition 5.2],

every such sequence has a convergent subsequence, but the limit might depend on the

choice of the subsequence. Notice that this notion is different from the one described

in [5, Chapter 8] where the authors require that the limit be unique (does not depend

on the subsequence).

Definition 4.5

The regular set R consists of those points, p 2 Y

, such that every tangent cone at p is isometric to R ⁿ . The complementary S D Y

n R is the singular set.

Let B ₀ ⁿ .1/ R ⁿ be the unit ball.

Definition 4.6

The "-regular set R " consists of those points, p 2 Y

, such that every tangent cone, .Y

; p

/, satisfies d _GH .B.p

; 1/; B ₀ ⁿ .1// < ". A point in Y

n R _" D S _" is called

"-singular.

THEOREM 4.7 ([8, Theorem 5.14]) There exists " n > 0 such that for " " n ,

ı

R " has a natural smooth manifold structure.

Moreover, for this parameterization, the metric on

ı

R " is bi-Hölder equivalent to a smooth Riemannian metric. The exponent ˛."/ in this bi-Hölder equivalence satisfies

˛."/ ! 1 as " ! 0.

THEOREM 4.8 ([8, Theorem 6.1]) We have

H ⁿ² .S/ D 0: (75) Remark 4.9

Clearly, R D T

">0 R " . The sets R " , R are not necessarily open. However, for any

" > 0, there is some ı 2 .0; "/ such that R ı

ı

R " (see [8, Appendix A.1.5]). In [9, Section 3], it is also proved that

ı

R " is path connected. This important fact will be used in the last part of this text.

We now study the density of the Hausdorff measure. A consequence of Bishop’s inequality is that

lim sup

r!0

H ⁿ .B.p; r//

vol

.r/ 1:

Definition 4.10

The density at p of Y

is

.p/ WD lim inf

r!0

H ⁿ .B.p; r//

vol

.r/ : (76)