A Restriction for Singularities on Collapsing Orbifolds

Abstract

Every point 𝑝 in an orbifold 𝑋 has a neighborhood that is homeomorphic to 𝐺_𝑝⧵𝐵_𝑟(0), where 𝐺_𝑝 is a finite group acting on 𝐵_𝑟(0)⊂ℝ^𝑛, so that 𝐺_𝑝(0)=0. Assume 𝑋 is a Riemannian orbifold with isolated singularities that is collapsing, that is, 𝑋 admits a sequence of metrics 𝑔_𝑖 with uniformly bounded curvature, so that, for any 𝑥∈𝑋, the volume of 𝐵₁(𝑥), with respect to the metric 𝑔_𝑖, goes to 0 as 𝑖→∞. For such 𝑋, we prove that |𝐺_𝑝|≤(2𝜋/0.47)^{𝑛(𝑛−1)} for all singularities 𝑝∈𝑋.

1. Introduction

An 𝑛-dimensional Riemannian orbifold, 𝑋, is a metric space so that the following is true: for any 𝑥∈𝑋, there exists 𝑟=𝑟(𝑥)>0 and a Riemannian metric 𝑔̃_𝑥 on 𝐵_2𝑟(0)⊂ℝ^𝑛, a finite group 𝐺_𝑥 (the isotropy group) acting on (𝐵_𝑟(0),𝑔̃_𝑥) by isometries, so that 𝐺_𝑥(0)=0, and there is an isometry 𝜄_𝑥∶𝐵_𝑟(𝑥)→𝐺_𝑥⧵𝐵_𝑟(0) with 𝜄_𝑥(𝑥)=0 (see [1]). We call 𝑥∈𝑋 a regular point if |𝐺_𝑥|=1; otherwise, 𝑥 is a singular point. We say the curvature of 𝑋 satisfies||𝐾_𝑋||≤𝜅²,(1.1) if the sectional curvature 𝐾 of every (𝐵_𝑟(0),𝑔̃_𝑥) above satisfies |𝐾|≤𝜅². We say 𝑋 is collapsing, if 𝑋 admits a sequence of metrics, 𝑔_𝑖, with uniformly bounded curvature, so that, for any 𝑥∈𝑋,lim_𝑖→∞Vol_𝑔𝑖𝐵₁(𝑥)=0.(1.2)

As an example, consider the standard ℤ_𝑚=ℤ/𝑚ℤ action on the sphere 𝑆².

The quotient orbifold 𝑋_𝑚=ℤ_𝑚⧵𝑆² will be arbitrarily collapsed when 𝑚→∞ (see Figure 1). However, for any fixed 𝑚, 𝑋_𝑚 can be collapsed only to a certain degree; it does not support a sequence of collapsing metrics. In fact, for each one of the two singularities on 𝑋_𝑚, there is a neighborhood that is isometric to ℤ_𝑚⧵ℝ², where ℝ² is equipped with some ℤ_𝑚 invariant metric. Therefore if 𝑔_𝑖 is a collapsing sequence of metrics on 𝑋_𝑚, we get a corresponding sequence 𝑔̃_𝑖 of pullback metrics on 𝑆²; every 𝑔̃_𝑖 is smooth. Observe Vol(𝑆²,𝑔̃_𝑖)=𝑚Vol(𝑋_𝑚,𝑔_𝑖), where 𝑚 is fixed and lim_𝑖→∞Vol(𝑋_𝑚,𝑔_𝑖)=0, thus lim_𝑖→∞Vol(𝑆²,𝑔̃_𝑖)=0. If the diameter of (𝑋_𝑚,𝑔_𝑖) stays bounded, we immediately get a contradiction to the Gauss-Bonnet theorem; in general, we can use the result in [2] to conclude that 𝑆² admits an F-structure, in particular the Euler characteristic 𝜒(𝑆²) vanishes—this is a contradiction since clearly 𝜒(𝑆²)=2.

Figure 1 

On the other hand, consider the double of a 2-dimensional rectangle. Clearly it admits a flat metric, thus we obtain a sequence of collapsing metrics by rescale. Notice, in this example, for each of the four singularities, the isotropy group 𝐺_𝑥 has order 2, a quite small number.

Intuitively, these examples suggest that when an orbifold is collapsing, a conelike singularity cannot be too “sharp,” that is, there should be some bound in |𝐺_𝑥|. The main result of this paper is as follows.

Theorem 1.1. Assume 𝑋 is a compact, collapsing orbifold, 𝑝∈𝑋 is an isolated singularity. Then |𝐺_𝑝|≤(2𝜋/0.47)^{𝑛(𝑛−1)}.

If 𝑋 has an isolated singularity 𝑝, then the dimension of 𝑋 must be even, and 𝐺_𝑝⊂𝑆𝑂(𝑛). Theorem 1.1 fails if we drop the requirement that 𝑥 is an isolated singularity; for example, we can take any orbifold 𝑋^ and let 𝑋=𝑋×𝑆¹; by shrinking the 𝑆¹ factor, we see 𝑋 is collapsing while there is no restriction on singularities of 𝑋^. The bound |𝐺_𝑝|≤(2𝜋/0.47)^{𝑛(𝑛−1)} has its root in the Bieberbach theorem of crystallographic groups and Gromov's almost flat manifold theorem.

Clearly, Theorem 1.1 is a corollary of the following.

Theorem 1.2. For any 𝐿>0, there is 𝜖=𝜖(𝑛,𝐿) so that if 𝑋 is an orbifold with all singularities 𝑞∈𝑋 satisfying |𝐺_𝑞|<𝐿, Vol(𝐵₁(𝑞))<𝜖, then |G_𝑝|≤(2𝜋/0.47)^{𝑛(𝑛−1)} for any isolated singularity 𝑝∈𝑋.

Remark 1.3. The bound |𝐺_𝑝|≤(2𝜋/0.47)^{𝑛(𝑛−1)} in Theorem 1.1 is not sharp. When 𝑛=2, it is not hard to see that either |𝐺_𝑝|=2 or 𝑋 is a flat orbifold. Therefore by Polya and Niggli's classification of crystallographic groups on ℝ² [3, page 105] or [4, page 228], we actually have |𝐺_𝑝|≤6 for collapsing 2 orbifolds.

A nilmanifold, Γ⧵𝑁, is the quotient of the (left) action of a discrete, uniform subgroup Γ⊂𝑁, on a simply connected nilpotent Lie group 𝑁. Left invariant vector fields (LIVFs) can be defined on Γ⧵𝑁. An affine diffeomorphism of Γ⧵𝑁 is a diffeomorphism that maps any local LIVF to some local LIVF. In general, a right invariant vector field (RIVF) cannot be defined globally in Γ⧵𝑁, unless this vector field is in the center of the Lie algebra of 𝑁. However, the right invariant vector fields, not the left invariant ones, are Killing fields of left invariant metrics on Γ⧵𝑁. An infranil orbifold is the quotient of a nilmanifold by the action of a finite group 𝐻 of affine diffeomorphisms. If the action 𝐻 is free, we get an infranil manifold.

In our previous work, [5], we generalized the Cheeger-Fukaya-Gromov nilpotent Killing structure [6] and the Cheeger-Gromov F-structure, [2, 7], to collapsing orbifolds. In particular, sufficiently collapsed 𝑋 can be decomposed into a union of orbits. Each orbit 𝒪_𝑝 is the orbit of the action of a sheaf 𝔫 of nilpotent Lie algebras, which comes from local RIVFs on a nilmanifold fibration in the frame bundle 𝐹𝑋. Therefore every 𝒪_𝑝 is an infranil orbifold. The proof of Theorem 1.2 is based on the relation between singularities on 𝑋 and singularities within an orbit 𝒪_𝑝 in 𝑋, as well as the nilmanifold fibration on 𝐹𝑋.

𝑋 is called almost flat, ifsup||𝐾_𝑋||^1/2⋅Diam𝑋≤𝛿_𝑛,(1.3) where Diam𝑋 is the diameter of 𝑋, 𝛿_𝑛 is a small constant that depends only on 𝑛. In [8], Gromov proved that an almost flat manifold 𝑀 has a finite, normal covering space 𝑀=Γ⧵𝑁 that is a nilmanifold. Subsequently, Ruh [9] proved that 𝑀 is diffeomorphic to Λ⧵𝑁, where Λ⊃Γ is a discrete subgroup in the affine transformation group of 𝑁. In [10], Ghanaat generalized this to an almost flat orbifold 𝑋, under the assumption that 𝑋 is good in the sense of Thurston [1], that is, 𝑋 is the global quotient of a simply connected manifold 𝑀. There are examples of orbifolds that are not good, see [1]. In fact, without much effort, one can remove the assumption that 𝑋 is good.

Proposition 1.4. If 𝑋 is an almost flat orbifold, then 𝑋 is an infranil orbifold.
Precisely, there is a nilmanifold 𝑋=Γ⧵𝑁, a finite group 𝐻 acting on 𝑋 by affine diffeomorphism, so that 𝑋 is diffeomorphic to 𝐻⧵𝑋. The order of 𝐻 is bounded by 𝑐_𝑛≤(2𝜋/0.47)^{𝑛(𝑛−1)/2}. Moreover, there is a sequence of metrics 𝑔_𝑗 so that Diam(𝑋,𝑔_𝑗)→0.

The proof is almost the same as [11, 12]; the only difference is one must replace the exponential map by the develop map (see [5, 13]) and modify the definition of Gromov product in [11] accordingly.

The proof of Theorem 1.2 does not depend on Proposition 1.4. On the other hand, Proposition 1.4 implies Theorem 1.2 for almost flat orbifolds immediately, even without the assumption that the singularities are isolated.

Remark 1.5. If 𝑝∈𝑋 is an isolated singularity, then, near 𝑝, 𝑋 is homeomorphic to (and in the metric sense, close to) a metric cone over a space form of dimension 𝑛−1. When 𝑛=4, the 4−1=3-dimensional space forms were first classified by Threlfall and Seifert, they used the fact that 𝑆𝑂(4) is locally isomorphic to 𝑆𝑂(3)×𝑆𝑂(3); [3, chapter 7] or [4] for details.

Remark 1.6. By the work of Anderson, Gao, Nakajima, Tian, Yang, and others, orbifolds with discrete singularities appear naturally as Gromov-Hausdorff limits of noncollapsing Einstein metrics with a uniform 𝐿^𝑛/2 curvature bound; see [14] for a recent survey. In particular, for Kähler-Einstein metrics, there is a complex structure on the limit 𝑋.

2. Proof of Theorem 1.2

If 𝑋 is an infranil orbifold, then it is easy to obtain the bound in Theorem 1.1. Since the proof contains some ideas for the general case, we give full details.

Lemma 2.1. Assume 𝑋 is an infranil orbifold. Then |𝐺_𝑥|≤(2𝜋/0.47)^{𝑛(𝑛−1)/2}.

Proof. Assume 𝑋=Λ⧵𝑁, where 𝑁 is a simply connected nilpotent Lie group, Λ is a discrete group of affine diffeomorphisms on 𝑁 so that 𝑋=Λ⧵𝑁 is compact. If 𝑁 is abelian, then 𝑋 is a flat orbifold, Λ is a discrete group of isometries on 𝑁=ℝ^𝑛 that acts properly discontinuously. So the conclusion follows from (the proof of) Bieberbach's theorem on crystallographic groups. In fact, it is well known that the maximal rotational angle of any 𝜆∈Λ is either 0 or at least 1/2. Thus the bound comes from a standard packing argument; notice 𝑛(𝑛−1)/2=dim𝑆𝑂(𝑛) and the bi-invariant metric on 𝑆𝑂(𝑛) has positive curvature.
We prove the general case by induction on dimension of 𝑋. Remember that Λ contains a normal subgroup Γ of finite index, so that Γ is a uniform, discrete subgroup of 𝑁 and 𝑋 is the quotient of the Λ/Γ action on the nilmanifold 𝑋=Γ⧵𝑁. Clearly 𝐺_𝑥 embeds in Λ/Γ, that is, 𝐺_𝑥={𝜆∈Λ/Γ|𝜆𝑥̃=𝑥̃}; here we choose a point 𝑥̃ in 𝑋=Γ⧵𝑁 that projects to 𝑥∈Λ⧵𝑁.
Let 𝐶 be the center of 𝑁, then 𝐶 is connected, of positive dimension. Since any 𝜆∈Λ is affine diffeomorphism, 𝜆 moves a 𝐶-coset in 𝑁 to a 𝐶-coset. Therefore Λ/Γ acts on the nilmanifold 𝑋^∗=(Γ/(Γ∩𝐶))⧵(𝑁/𝐶), the quotient 𝑋^∗ is an infranil orbifold of lower dimension. Let 𝜋∶𝑋→𝑋^∗ be the projection, and assume 𝜋(𝑥̃)=𝑥̃^∗. Thus we have a homomorphism ℎ∶𝐺_𝑥⟶𝐺_𝑥∗.(2.1)𝑋 is a torus bundle over 𝑋^∗, the fiber is 𝑇=(Γ∩𝐶)⧵𝐶. Assume 𝜆∈Λ/Γ is in Kerℎ, the kernel of ℎ, then 𝜆 fixes every 𝑇 fiber in 𝑋. If, in addition, 𝜆 fixes every point in the 𝑇 fiber passing through 𝑥̃, we claim 𝜆 must be identity. In fact, on 𝑁 we have 𝜆(𝑧)=𝑎⋅𝐴(𝑧), where 𝑎∈𝑁 and 𝐴 is a Lie group automorphism of 𝑁; if 𝜆 fixes every point in one 𝑇 fiber, then 𝐴 is identity on the center 𝐶⊂𝑁. This implies that 𝜆 is a translation on every 𝑇 fiber. Since 𝜆 is of finite order and fixes every point in one 𝑇-fiber, 𝜆 must be identity. Therefore any element 𝜆∈Kerℎ is decided by its restriction on the 𝑇 fiber passing through 𝑥; so Kerℎ is isomorphic to a finite group of affine diffeomorphisms on 𝑇 that fixes 𝑥̃∈𝑇, thus |Kerℎ| can be bounded by Bieberbach's theorem. Since ||𝐺_𝑥||≤||𝐺_𝑥∗||⋅||Kerℎ||,(2.2) the conclusion follows by induction.

In [5], the existence of nilpotent Killing structure of Cheeger-Fukaya-Gromov [6] is generalized to sufficiently collapsed orbifolds. We briefly review this construction.

As in the manifold case, one can define the frame bundle 𝐹𝑋 of an orbifold 𝑋. If 𝐵_𝑟(𝑥)⊂𝑋 is isometric to 𝐺_𝑥⧵𝐵_𝑟(0), where 𝐺_𝑥 is a finite group acting on 𝐵_𝑟(0)⊂ℝ^𝑛, then locally 𝐹𝑋 is 𝐺_𝑥⧵𝐹𝐵(0,𝑟), where 𝐹𝐵(0,𝑟) is the orthonormal frame bundle over 𝐵_𝑟(0), and 𝐺_𝑥 acts on 𝐹𝐵(0,𝑟) by differential, that is, 𝜏∈𝐺_𝑥 moves a frame 𝑢 to 𝜏_∗𝑢. Therefore 𝐹𝑋 is a manifold; strictly speaking, 𝐹𝑋 is not a fiber bundle. Let 𝜋∶𝐹𝑋→𝑋 be the projection.

Moreover, there is a natural 𝑆𝑂(𝑛) action on 𝐹𝑋; on the frames over regular points, this 𝑆𝑂(𝑛) action is the same one as in the manifold case; however, at (the frames over) singular points, this action is not free. As in the work of Fukaya [15], see also [5], any Gromov-Hausdorff limit 𝑌 of a collapsing sequence 𝐹𝑋_𝑖 is a manifold. Following [6], for sufficiently collapsing orbifolds, locally we have an 𝑆𝑂(𝑛)-equivariant fibration𝑍⟶𝐹𝑋⟶^𝑓𝑌,(2.3) where the fiber 𝑍 is a nilmanifold, 𝑌 is a smooth manifold with controlled geometry.

As in [6], we can put a canonical affine structure on the 𝑍 fibers, that is, a canonical way to construct a diffeomorphism from a fiber 𝑍 to the nilmanifold Γ⧵𝑁. In particular, there is a sheaf 𝔫, of a nilpotent Lie algebra of vector fields on 𝐹𝑋. Sections of 𝔫 are local right invariant vector fields on the nilmanifold fibers 𝑍. By integrating 𝔫, we get a local action of a simply connected nilpotent Lie group, 𝔑, on 𝐹𝑋. Therefore we also call a 𝑍 fiber an orbit, and we can write 𝑍=𝒪.

The fibration 𝑓∶𝐹𝑋→𝑌 is 𝑆𝑂(𝑛)-equivariant, so any 𝑄∈𝑆𝑂(𝑛) moves a 𝑍 fiber to a (perhaps another) 𝑍 fiber by affine diffeomorphism. Moreover, the 𝑆𝑂(𝑛) action on 𝔫 is locally trivial, that is, if 𝐴∈𝔰𝔬(𝑛) is sufficiently small, then 𝑒^𝐴∈𝑆𝑂(𝑛) moves a section, 𝔫(𝑈), of 𝔫 on any open set 𝑈⊂𝐹𝑋, to itself (over 𝑈∩𝑈𝑒^𝐴) ( [6, Proposition 4.3]). In particular, the sheaf 𝔫 induces a sheaf, which we also denote by 𝔫, on the orbifold 𝑋 away from the singular points. An orbit 𝒪_̃𝑞=𝑍 on 𝑋 projects down to an orbit 𝒪_𝑞 on 𝑋.

Assume 𝑞̃∈𝐹𝑋 is any frame over 𝑞∈𝑋. Let𝐼(𝑞)=𝑄∈𝑆𝑂(𝑛)∣𝑍_̃𝑞𝑄=𝑍_̃𝑞(2.4) be the the isotropy group of an orbit 𝑍_̃𝑞=𝒪_̃𝑞⊂𝐹𝑋. We will simply write 𝐼(𝑞) by 𝐼. Let 𝐼₀ be the identity component of 𝐼. It can be shown that, restricted on 𝑍=𝒪_̃𝑞, the action of 𝐼₀ is identical to the action of a torus, and the Lie algebra of this torus, 𝐼₀, is in the center of 𝔫 (see [5, 6] for more details). Consider the nilmanifold𝒪_̃𝑞=𝒪^̃𝑞𝐼₀.(2.5) Therefore 𝑍_̃𝑞=𝒪_̃𝑞 is a torus bundle over 𝒪_̃𝑞. Notice, on 𝑍=𝒪_̃𝑞, 𝐼 moves 𝐼₀ fibers to 𝐼₀ fibers, thus the orbit 𝒪_𝑞 is the quotient of 𝒪_̃𝑞 by the action of the finite group 𝐼/𝐼₀. Therefore 𝒪_𝑞 is an infranil orbifold. In particular, the singularities within 𝒪_𝑞 satisfy the bound in Lemma 2.1.

It is important to remark that the above structure is not trivial.

Lemma 2.2. Let 𝐿 be any integer. Then there is 𝜖=𝜖(𝑛,𝐿), so that if 𝑋 is an orbifold with |𝐺_𝑥|≤𝐿, Vol(𝐵₁(𝑥))≤𝜖 for all 𝑥∈𝑋, then every 𝔫-orbit 𝒪 on 𝑋 is of positive dimension.

Proof (sketch). For any unit vector 𝐴∈𝔰𝔬(𝑛), the bound in |𝐺_𝑥| implies that 𝑒^𝑡𝐴 does not have fixed point in 𝒪_̃𝑞 unless 𝑡=0 or |𝑡|>𝑐𝐿⁻¹. However, for sufficiently collapsed orbifolds, there is a vector 𝐵 in the center of 𝔫 so that 𝐵 generates a closed loop in 𝒪_̃𝑞 that is shorter than 𝑐𝐿⁻¹, therefore 𝐵 cannot be in the Lie algebra of 𝐼₀, which is in both 𝔰𝔬(𝑛) and the center of 𝔫. Thus the orbit 𝒪_̃𝑞 is not contained in a single 𝑆𝑂(𝑛) orbit in 𝐹𝑋, so 𝒪_𝑞 is of positive dimension in 𝑋=𝐹𝑋/𝑆𝑂(𝑛) (see [5] for more details).

Proof of Theorem 1.2. Assume 𝑝∈𝑋 is an isolated singular point, 𝑝̃∈𝜋⁻¹(𝑝) is in 𝐹𝑋. 𝑍_̃𝑝=𝒪_̃𝑝 is the fiber that projects to 𝒪_𝑝. Let 𝐼(𝑝), 𝐼₀, 𝒪_̃𝑞 be as above. Let 𝐾_𝑝={𝑄∈𝑆𝑂(𝑛)∣𝑝𝑄̃=𝑝}̃.(2.6) Thus 𝐾_𝑝 is a subgroup of 𝐼, and |𝐾_𝑝|=|𝐺_𝑝|. Let 𝐾^−𝑝=𝑄∈𝐾_𝑝∣𝑝^̃𝐼₀𝑄=𝑝^̃𝐼₀∀𝑝^̃∈𝑍_̃𝑝.(2.7) Thus 𝐾^−𝑝 is a normal subgroup of 𝐾_𝑝.
Lemma 2.3. If 𝑄∈𝐾_𝑝 fixes every point in 𝑍_̃𝑝, then 𝑄 is the identity in 𝑆𝑂(𝑛).
Proof. Potentially 𝑄 may fix every point in 𝑍_̃𝑝 while moving some points of 𝐹𝑋 that are outside 𝑍_̃𝑝. We will rule out this possibility.
By assumption, 𝑝 is an isolated singularity. For any 𝑄 that is not identity, the connected component of the fixed point set of 𝑄 that passes through 𝑝̃ must project to 𝑝 under 𝜋∶𝐹𝑋→𝑋, because away from 𝜋⁻¹(𝑝) the 𝑆𝑂(𝑛) action is free. Therefore 𝜋(𝑍_̃𝑝)=𝑝 is a single point in 𝑋, and this contradicts the fact that the 𝔫-orbits on 𝑋 are of positive dimension; see Lemma 2.2.
In particular, we have a faithful representation of 𝐾_𝑝 in the affine group of 𝑍_̃𝑝, that is, we can identify 𝐾_𝑝 with the restricted action of the group 𝐾_𝑝 on 𝑍_̃𝑝.Take any 𝑄∈𝐾^−𝑝 that is not identity in 𝑆𝑂(𝑛). By definition 𝑄 fixes 𝑝̃. If 𝑄 fixes every point in 𝑝𝐼̃₀, as in Lemma 2.1, 𝑄 is a translation on every 𝐼₀ fiber; because 𝑄 is of finite order and 𝑄 moves every 𝐼₀ fiber to itself, 𝑄 necessarily fixes every point in 𝑍; thus 𝑄 is identity. So 𝑄 rotates the tangent plane of 𝑝𝐼̃₀ at 𝑝̃. Therefore 𝐾^−𝑝 is isomorphic to a finite group of affine diffeomorphisms on the torus 𝑝𝐼̃₀. By the Bieberbach theorem, ||𝐾^−𝑝||≤2𝜋0.47^{𝑘(𝑘−1)/2},𝑘=dim𝐼₀.(2.8) Recall that Bieberbach's theorem implies that all finite subgroups of 𝑆𝐿(𝑛,ℤ) have a uniform upper bound in order.We have 𝒪_𝑝=𝐻𝑍_̃𝑝/𝐼₀,(2.9) where 𝐻=𝐼/𝐼₀ is a finite group. Let 𝐻_𝑝 be the subgroup of 𝐻 that fixes 𝑝. Now we get an embedding𝐾_𝑝𝐾^−𝑝⊂𝐻_𝑝.(2.10) By Lemma 2.1, ||𝐻_𝑝||≤2𝜋0.47^{𝑖(𝑖−1)/2},𝑖=dim𝒪_𝑝.(2.11) Thus ||𝐺_𝑝||=||𝐾_𝑝||=||||𝐾^𝑝𝐾^−𝑝||||⋅||𝐾_−𝑝||≤2𝜋0.47^{𝑛(𝑛−1)}.(2.12)

Acknowledgment

The author is grateful to Professor Tian for very helpful suggestions.