Dificultades en la enseñanza

For an integern≥2, a nondecreasing functionκ: (0,∞)→[0,∞) and constantsr0, v0 >0,

we denote byOn,κ,r0,v0 the space of isometry classes of pointed complete connected smooth

1

Riemanniann-orbifolds (O, p), such that|sec| ≤κ◦d(p,·), i.e. such that for everyr >0 the sectional curvature onBr(p) is bounded by |sec| ≤κ(r), and such that vol(Br0(p))≥v0.

The local lower curvature bounds imply via Bishop-Gromov volume comparison (Prop. 3.1.4) that the thickness (volume non-collapse) in the base point propagates across the orbifold and can be bounded below in terms of the distance from the base point, i.e. there are uniform estimates vol(Br(x)) ≥ v(n, κ, r0, v0, d(p, x), r), cf. Lemma 4.0.1. Moreover,

the size of ǫ-nets in metric balls can be bounded above, and hence On,κ,r0,v0 ⊂ LN for a

suitable counting function N(r, ǫ) = Nn,κ,r0,v0(r, ǫ). In particular, the space Od,κ,r0,v0 is

Gromov-Hausdorffprecompact.

We are interested in the Gromov-Hausdorff closure ofOn,κ,r0,v0, more precisely, in the

local structure of limit spaces in the Gromov-Hausdorff boundary ∂GH_O

n,κ,r0,v0, and in

compactness properties ofOn,κ,r0,v0 and related spaces with respect to the finer topologies

of smooth convergence. Since by construction lower Alexandrov curvature bounds pass to Gromov-Hausdorff limits, we already know that the limit spaces in ∂GH_O

n,κ,r0,v0 have

again local lower curvature bounds in the Alexandrov sense.

If we impose more regularity on a space of orbifolds, we can also expect the limit spaces to be more regular and to obtain stronger compactness properties. For an integer n ≥2, a function D :N0×(0,∞)→ [0,∞) and constants r0, v0 >0, we denote by On,D,r0,v0 the

space of isometry classes of pointed complete connected smooth Riemannian n-orbifolds (O, p) for which vol(Br0(p)) ≥v0 and k∇

l_{Rk ≤ D(l, r) on B}

r(p) for all l ≥ 0 and r > 0.

Clearly, Od,D,r0,v0 ⊂ Od,κ,r0,v0 for suitable κ(r) = κD(0,·)(r) and On,D,r0,v0 is also Gromov-

1

Since we are only interested in curvature at the moment, it would actually suffice to require only regularity of classC4

4.2 Compactness of thick orbifolds 47

Hausdorff precompact.

We consider a sequence of pointed Riemannian orbifolds (Ok, pk) ∈ On,κ,r0,v0. Due to

the Gromov-Hausdorff precompactness of On,κ,r0,v0, we may assume after passing to a

subsequence that the (Ok, pk) converge to a limit space,

(Ok, pk) GH

−→(X_∞, p_∞).

When investigating the local structure ofX_∞ near some pointx_∞, we fix a constantR >> d(p_∞, x_∞) and a realization of the Gromov-Hausdorff convergence of balls (BOk

R (pk), pk)GH→

(BX∞_R (p_∞), p_∞) in some auxiliary ambient metric spaceZ.

We now apply our uniform lower injectivity radius bounds obtained in section 4.1 to the orbifoldsOk.

Proposition 4.2.4. Any point in X_∞ is contained in an open metric ball which is a Gromov-Hausdorff limit of standard balls in theOk.

Proof. Given a pointx_∞ ∈X_∞, Corollary 4.1.6 implies that there exists a point x′

∞ ∈X∞

(arbitrarily close to x_∞) and a sequence of points x′

k ∈Ok such thatx′k →x′∞ (inZ) and

lim infk→∞inj(x′k)> d(x∞, x′∞). Then for d(x∞, x′∞)< r <lim infk→∞inj(x′k) we have

(BOk

r (x′k), x′k) GH

−→(B_rX∞(x′_∞), x′_∞). (4.2.5) and the BOk

r (x′k) are standard balls for large k.

Since there are uniform curvature bounds on the ballsBOk

r (x′k) and since their volumes

are bounded below uniformly ink, we have that lim infk→∞vol(Σx′

kOk)>0. Therefore the

links Σx′

kOk fall into finitely many isometry types, compare Remark 3.2.4, and after passing

to a subsequence we may assume that they are isometric to each other. Then there exist a finite subgroup Γ ⊂ O(n) and isometric identifications ik : Rn/Γ

∼ = → T_x′ kOk (preserving origins). Let us denote byek :BR n_/Γ r (0) ∼ = → BOk

r (x′k) the diffeomorphisms obtained from restrict-

ing exp_x′

k◦ik, by πΓ :

Rn _{→ R}n_{/Γ the quotient projection, and ˆe}

k := ek ◦πΓ : BR

n

r (0) →

BOk

r (x′k). Pulling back the orbifold Riemannian metrics gk from Ok via ˆek yields smooth

Riemannian metrics ˆhk := ˆe∗kgk on the ballBr(0)⊂Rn. Ifris chosen sufficiently small, say

r < π/2pκ(R), the Rauch estimates imply that the ˆhk are uniformly bilipschitz equivalent

to the euclidean metric.

Depending on the additional regularity assumptions which one imposes on the Ok, one

can deduce regularity for the limit space X_∞. We assume now that (Ok, pk) ∈ On,D,r0,v0.

The function κ(r) can be chosen so that On,D,r0,v0 ⊂ On,κ,r0,v0. Then there are uniform

bounds for the curvature tensors of the metrics ˆhk and all their covariant derivatives (for

48 4. Convergence of thick orbifolds Lemma 4.2.6 (cf. [Ba07, Lem. 3.2.1]). For r < ∞, l, n ∈ N _{and constants 0 <}

D0, . . . , Dl <∞ there are constants0< D0′, . . . , Dl′ <∞ such that the following holds: Let

(M, g) be an n-dimensional smooth Riemannian manifold with x ∈ M and suppose that

the exponential map is defined on Br(0)⊂TxM. If k∇kRk< Dk on M for all 0≤k≤l,

then kdl_exp∗_{gk< D}′

l on Br(0).

Based on the lemma, we may apply Arzel`a-Ascoli and pass to a subsequence so that the ˆhk converge to a Γ-invariant smooth Riemannian limit metric ˆh∞ on Br(0),

ˆ

hk C ∞

→ ˆh_∞.

The smooth convergence implies Gromov-Hausdorff convergence with respect to the asso- ciated intrinsic path metrics, (Br(0), dˆ_hk)GH→ (Br(0), dˆ_h∞), and equally for the Γ-quotients,

(Br(0), dˆ_hk)/ΓGH→ (Br(0), dˆ_h∞)/Γ.

Note that the balls BOk

r (x′k) may not be convex, but their intrinsic path metrics are

at leastlocallyisometric to their extrinsic metrics obtained by restricting the path metrics

dgk on the Ok. More precisely, for a point y

′

k ∈ BrOk(x′k), both metrics coincide on the

ball around y′

k with radius 12(r−dgk(x

′

k, yk′)). Hence the ball quotients (Br(0), dˆ_hk)/Γ

are in a uniform way locally isometric to the balls BOk

r (x′k) equipped with their extrinsic

metrics. We conclude with (4.2.5) that BX∞

r (x′∞) is locally isometric to the Riemannian

ball quotient (Br(0), dˆ_h∞)/Γ. It follows that the limit space X∞ is a smooth Riemannian

n-orbifold O_∞, and that O_∞ inherits from the Ok the same rate of local non-collapsedness

and the same bounds for the curvature tensor along with all its derivatives. We therefore have completed the proof of

Theorem 4.2.7 (Gromov-Hausdorff compactness). On,D,r0,v0 is Gromov-Hausdorff

compact.

Using an isometric identificationi_∞ :Rd_/Γ→∼= _T x′

∞O∞, we convert theik into isometries

ιk =ik◦i−_∞1 :Tx′ ∞O∞

∼

=

→Tx′

kOk (preserving origins). For the corresponding diffeomorphisms

of standard balls φk := expO_x′k k ◦ιk◦(exp O∞ x′ ∞) −1 _:BO∞ r (x′∞)−→BrOk(x′k) (4.2.8) then holds φ∗_kgk C ∞ −→g_∞|_BO∞ r (x′∞) (4.2.9)

where we denote byg_∞the Riemannian metric onO_∞. The Gromov-Hausdorff convergence (BOk

r (x′k), x′k) GH

−→(B_rO∞(x′_∞), x′_∞), (4.2.10) compare (4.2.5), and the smooth convergence (4.2.9) can be coordinated so that

φk → id_BO∞

4.2 Compactness of thick orbifolds 49

uniformly. A priori one has only that for large k the φk are close to isometries ψk of the

limit ball BO∞

r (x′∞), but (4.2.11) can be achieved by replacing theφk with φk ◦ψ_k−1.

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