Caracterización del profesor Richard

Throughout this section, we closely follow the well-known proof of the Soul Theorem for manifolds. Let us consider a complete non-compact non-negatively curved orbifold O

without boundary. Then for every point p∈ O there is a ray ρ (an isometric embedding

ρ: [0,∞)→ O) emanating from p. For any such ray we define the associated Busemann function as βρ(x) := limt→∞(d(ρ(t), x)−t) for all y ∈ O. (Because (d(ρ(t), x)−t) is

3.5 The orbifold Soul Theorem 37

nonincreasing in t and bounded below by −d(p, x), βρ is well defined on all of O and

1-Lipschitz.)

By Proposition 3.4.4 the function βρ is concave along (not necessarily minimizing)

geodesics: Suppose there is a geodesic segment c : [0,1] → O and s ∈ (0,1) such that

βρ(c(s))< s·βρ(c(1)) + (1−s)·βρ(c(0)). For sufficiently larget > 0, we choose minimizing

segments from c(0), c(s) and c(t) to ρ(t). For these segments we still have d(c(s), ρ(t)) < s·d(c(1), ρ(t)) + (1−s)·d(c(0), ρ(t)). On the other hand, we compute for the Euclidean comparison angle thatd0(¯c(s),ρ¯(t))≤s·d0(¯c(1),ρ¯(t))+(1−s)·d(¯c(0),ρ¯(t)). It follows that

we have comparison angles (with respect to the comparison curvature valuek = 0) satisfy- ing ˜∠c(s)(c(0), ρ(t)) + ˜∠c(s)(c(1), ρ(t)) > π, and hence∠c(s)(c(0), ρ(t)) +∠c(s)(c(1), ρ(t))> π

which is impossible. Thus, the function β := minρβρ (where the minimum runs over all

raysρin Owithρ(0) =p; this is a compact family of rays) is also well-defined, 1-Lipschitz and concave along geodesics. For every point x ∈ O there is a ray ρx emanating from x

such thatβ(ρx(s)) =β(x)−s.

It follows that for all t ≥ 0 the subsets Ct := {β ≥ t} ⊂ O are totally convex in

the sense that for any two points x1, x2 ∈ Ct all geodesic segments between x1 and x2

are entirely contained in Ct. Moreover, C0 is closed and compact (or it would contain

another ray emanating from p∈C0), and hence so are all Ct for t≥0. Hence in any local

uniformization πU : ˜U → U near a point x ∈ Ct, the lift π−1(Ct ∩U) ⊂ U˜ is a totally

convex subset of ˜U. But locally closed totally convex subsets of Riemannian manifolds are embedded submanifolds with (possibly nonsmooth) boundary locally supported by a cone in the tangential space (see [CE75, 8.6 – 8.8]). Hence the setsCt are contained in the

closure of embedded suborbifolds in O. We set C(1) _{:= C}

tmax where tmax is the maximal

value of t such that Ct is nonempty. If N(1) is the maximal embedded suborbifold in O

such thatC(1)_=N(1)_{, we set ∂C}(1)_{= C}(1)_−N(1)_{. We have dimN}(1)_<dimO.

Suppose that∂C(1)_{6= ∅. Using Rauch comparison, we can proceed as in the manifold}

case (see [CE75, 8.9 – 8.10]) to show that the sets Ca :={x∈C(1) | d(x, ∂C(1))≥a} are

also totally convex closed compact subsets of O. We set C(2) _{= C}

amax where again amax

is the maximal value of a such that Ca is nonempty. If N(2) is the maximal embedded

suborbifold inO such that C(2) _=N(2)_{, we have dimN}(2)_<dimN(1)_{. After repeating this}

process a finite number of times if necessary, we arrive at a totally convex, totally geodesic compact suborbifold S without boundary such that dimS < dimO, a so-called soul of

O. Moreover, the distance function d(S,·) has no critical points on O−S: Every point

x∈O−S is contained in ∂C for some totally convex compact subset ofO (either one of the Ct or one of the Ca), and therefore a lift of x to a local uniformization is contained

in the boundary of a totally convex subset of a Riemannian manifold. Again from the study of such sets, we can find nearx a vector field making angle> π

2 with all minimizing

38 3. Geometric properties of Riemannian orbifolds

The normal bundleν(S) of the soulSis diffeomorphic to a small tubular neighbourhood of Bǫ(S) ⊂ O. On the other hand, we can construct a smooth gradient-like vector field

X on O −Bǫ/2(S) such that d(S,·) decreases along the integral curves of X at least

linearly (i.e. has uniform negative directional derivative): By the last observation in the last paragraph, such vector fields exist locally and we can average them using a partition of unity (cf. [Ka89, 5.2.2]). We have now completed an orbifold version of the proof of the Soul Theorem which was originally given by Cheeger and Gromoll [CG72] for non-compact non-negatively curved manifolds:

Proposition 3.5.1 (Soul Theorem for orbifolds). A complete non-compact Rieman- nian orbifold with sectional curvature ≥ 0 contains a totally convex and totally geodesic closed suborbifold, a so-called soul, and is diffeomorphic to the normal bundle of the soul.

3.5.1

Classification of non-compact 3-orbifolds with sec

≥

0

To derive the classification of all non-compact complete 3-orbifolds with sec≥0 from the Soul Theorem, we will also use the splitting theorem for general Alexandrov spaces (see [BBI01, Sec. 10.5]) which states that if an Alexandrov space of curvature≥0 contains a line (an isomtric embedding ofR), it splits off a line metrically. Thus, if a complete Riemannian orbifold O with sectional curvature ≥ 0 contains a line, it splits as the metrical product

R_×O′ _{with O}′ _{a complete Riemannian orbifold with dimO}′_{= dimO−1.}

Proposition 3.5.2. Every connected complete non-compact Riemannian 3-orbifold with sectional curvature ≥ 0 is either diffeomorphic to a discal (dimS = 0) or a solid toric (dimS = 1) 3-orbifold, or isometric to Σ2_{×R or (Σ}2 _×R)/Z

2 where Σ2 is a closed 2- orbifold with sectional curvature≥0, andZ2 operates isometrically onΣ2×R(dimS = 1). Proof. The proposition is clear for dimS = 0. For dimS = 2 it is sufficient to observe that after passing to a double covering if necessary, S separates ν(S) and hence O. Thus, O

contains a line and the splitting theorem applies.

If dimS = 1 andS is a circle, we obtainO by starting withD2_{(p)×[−1,1] orV}2_(p)×

[−1,1] and identifying the boundary components. Thus, O is diffeomorphic to a bundle over S1 _{with fiber R}2_{/Γ for some finite subgroup Γ ⊂ O(2). If S is diffeomorphic to}

the mirrored interval, O can be obtained from D2_{(p)×[−1,1], p ≥ 1, by gluing each of}

the boundary components D2_{(p)× {±1} either to itself via a half-rotation or reflection}

or by making it a reflector boundary component (there are six such orbifolds), or from

V2_{(p)×[−1,1], p≥ 1, by gluing each of the boundary components V}2_{(p)× {±1} either}

to itself via the reflection at its bisector or by making it a reflector boundary component (there are three such orbifolds). In both cases the orbifold O is diffeomorphic to a solid

3.5 The orbifold Soul Theorem 39

toric 3-orbifold.

Remark 3.5.3. We also note an alternative construction for the case where the soul is a mirrored interval: In this case, O can also be obtained by starting with two quotients of the 3-ball, each with boundary RP2_{(p), S}2_{(2,2, p) or D}2_{(p) for some fixed p≥ 1, and}

glueing them together along a closed pointed disc D2_{(p) contained in both boundaries.}

The (interior of the) resulting 3-orbifolds are then diffeomorphic to the ones obtained from

D2_{(p)×[−1,1] discussed in the proof.}

Similarly, we could start with two quotients of the 3-ball, each with boundaryD2_{(; 2,2, p)}

orD2_{(2;p) for some fixedp≥1, and then identify two sectorsV}2_{(p). In this way, we obtain}

4. Convergence of thick orbifolds

LetOn≥2 _{be a complete connected Riemannian orbifold with bounded sectional curvature}

|sec| ≤κ. We will assume a certain thickness(non-collapsedness) in the volume sense in a base point p∈O, namely that

vol(Br(p))≥v

for certain constants r, v > 0. Since we are assuming a lower curvature bound, thickness in the base point propagates over the entire orbifold.

Lemma 4.0.1 (Thickness propagates). ForR, r′ _{>0there is v}′ _=v′_{(n, κ, r, v, R, r}′_)>

0such that the following holds: If for p, p′ _{∈O holds vol(B}

r(p))≥v andd(p′, p)≤R, then

vol(Br′(p′))≥v′.

Proof. Since vol(Br+R(p′)) ≥ v by our assumption, Bishop-Gromov volume comparison

(3.1.4) yields a lower bound for vol(Br′(p′)) ifr′ ≤r+R.

If one imposes an upper bound on the order of local isotropy groups, equivalently, a lower bound on the volume of links, then small local volume implies small injectivity radius (again by Bishop-Gromov). The converse is not true, since the injectivity radius tends to zero as one approaches a singular stratum ofS. The main result of the next section (Prop. 4.1.4) is that this is the only way for the injectivity radius to become small in a thick (i.e. volume non-collapsed) region, or in other words that close to a point with small injectivity radius there is a more singular point.