Complete flat Lorentzian three-manifolds Introduction Proper group - - PowerPoint PPT Presentation

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

Complete flat Lorentzian three-manifolds

Jeffrey Danciger

with F . Gu´ eritaud and F . Kassel

University of Texas – Austin jdanciger@math.utexas.edu XXIInd Rolf Nevanlinna Colloquium August 8, 2013

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

Complete flat Lorentzian manifolds

Question

• 1. Classify the topology of manifolds M admitting a flat Lorentzian metric,

which is geodesically complete.

• 2. Understand the geometry of those metrics.

These manifolds are quotients of Minkowski space Rn,1: M = Γ\Rn,1, where Γ ⊂ Isom(Rn,1) = O(n, 1) ⋉ Rn+1, a group of isometries.

Question

• 3. Which groups Γ can act properly discontinuosly and freely by isometries
• n Minowski space?
• 4. Given such Γ, classify proper isometric group actions Γ → Isom(Rn,1).

Study dimension n = 3.

• 3. long known due to Fried–Goldman
• 4. nearly complete (Fried–Goldman, Goldman–Labourie–Margulis,

D–Gu´ eritaud–Kassel) Today: give a topological classification (Γ finitely generated); in particular all complete flat Lorentian three-manifolds are tame.

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

Which groups act properly discontinuously?

Let Γ ⊂ O(2, 1) ⋉ R3 be a group of isometries. Then, T ∈ Γ is of the form T(x) = Ax + b A ∈ O(2, 1) is the linear part and b ∈ R3 is the translational part. Denote the homomorphism L : T → A. Easy example: Γ = Z × Z × Z, generated by three independent translations. Here, the linear part L(Γ) is trivial and M = Γ\R3 ∼ = T3. Fried–Goldman ’83: If Γ acts properly and freely, then either

◮ Γ is solvable, or ◮ L : Γ → O(2, 1) is injective and discrete.

So Γ ∼ = L(Γ) is a surface group, necessarily non-compact (Mess). = ⇒ Γ is a free group. In fact, dichotomy holds more generally for Γ ⊂ Aff(R3).

Conjecture (Auslander ’64)

A group of affine transformations acting properly discontinuously and cocompactly on affine space is virtually solvable.

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

Free groups acting properly discontinuously

Fried–Goldman ’83: If Γ acts properly, freely and Γ is not solvable then the linear part L(Γ) ∼ = Γ is a non-compact surface group. = ⇒ Γ a free group.

◮ Γ is called affine deformation of L(Γ). ◮ Margulis ’83: first examples. ◮ M called a Margulis spacetime.

So, in n = 3, it remains to:

◮ Understand which affine deformations of a surface group give proper

actions.

◮ Determine the topology of the quotients.

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

Topology: tameness

Let the free group Γ ⊂ O(2, 1) ⋉ R2,1 act properly discontinuously on R2,1. Let M = Γ\R2,1 be the quotient manifold.

◮ Expect M to be a handle-body. But is M even tame? ◮ A manifold is tame if it is the interior of a compact manifold with boundary. ◮ Exist wild manifolds which are homotopically trivial (Whitehead). ◮ Tameness of hyperbolic three-manifolds: conjectured by Marden in ’74. only

recently proven by Agol and Calegari–Gabai in ’06.

◮ Drumm (’90) constructed examples Γ by building fundamental domains

(crooked planes) ( = ⇒ tame).

◮ Drumm–Goldman conjectured that all Margulis spacetimes can be built by

Drumm’s technique. therefore

Conjecture (Drumm-Goldman ’90s)

All Margulis spacetimes are tame

◮ Partial results due to Charette–Drumm–Goldman, in case rank(Γ) = 2.

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

Topology: tameness

Theorem (D–Gu´ eritaud–Kassel, Choi–Goldman)

Let Γ ⊂ O(2, 1) ⋉ R3 be a free group which acts properly on R2,1. Assume the linear part L(Γ) is convex cocompact. Then the quotient M = Γ\R2,1 is tame, diffeomorphic to a handle-body.

◮ In progress: general case allowing for linear part L(Γ) to have parabolics.

will classify the topology of all complete affine three-manifolds.

◮ We prove that M is fibered in (time-like) geodesics over a hyperbolic

surface S = L(Γ)\H2, where L(Γ) is the linear part. = ⇒ M ∼ = S × R.

◮ point of view: Margulis spacetimes (flat) behave like infinitesimal versions

• f AdS manifolds (curved).

◮ AdS = anti de Sitter geometry: constant negative curvature model space. ◮ can make the intuition precise:

(paraphrased) Theorem (D-Gu´ eritaud-Kassel)

The complete flat Lorentzian metric on M is realized as a limit of complete metrics of constant curvature.

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

Hyperbolic surfaces

H2 = upper hemisphere in CP1. Isom0 H2 = PSL(2, R) =: G.

◮ Γ0 ⊂ PSL(2, R) discrete =

⇒ Γ0 acts properly discontinuously.

◮ b/c point stabilizer is compact. ◮ If Γ0 is f.g. and acts freely then S = Γ0\H2 is a surface of finite type

(tameness in two-dimensions).

◮ Γ0 called convex cocompact if all elements are hyperbolic (translation

length > 0).

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

Deforming representations

G = PSL(2, R). g = sl(2, R). Consider a path ρt : Γ0 → G with ρ0 = id : Γ0 → Γ0.

◮ Derivative ρ′ = d

dt

• t=0

is a representation Γ0 → TG ∼ = G ⋉ g. = ⇒ ρ′(γ) = (γ, u(γ)) where u : Γ0 → g is u(γ) = d dt

• t=0

ρt(γ) γ−1 “infinitesimal deformation” of Γ0 in G. cocycle condition: u(γ1γ2) = u(γ1) + Adγ1 u(γ2).

◮ γ → (γ, u(γ)) is a representation Γ0 → G ⋉ g.

Notation

Γu := {(γ, u(γ))} ⊂ G ⋉ g.

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

The associated affine action

Notation

Γu := {(γ, u(γ))} ⊂ G ⋉ g. G ⋉ g acts by affine transformations on g: (g, u) ∈ G ⋉ g acts by (g, u) · v := Adg v + u

◮ preserves indefinite metric on g coming from Killing form. ◮ For G = PSL(2, R), signature = (2, 1) ◮

= ⇒ identify R2,1 = g.

◮ therefore G ⋉ g ⊂ Isom R2,1. ◮ In fact, G ⋉ g = Isom0 R2,1. ◮ by Fried–Goldman, any proper isometric action on R2,1 must have the

form Γu

0 (up to f.i.).

Which Γu

0 act properly?

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

Margulis’s opposite sign lemma

linear translation exponential growth exponential decay Action of (γ, u(γ)):

Margulis invariant:

Observation

signed translation length of (γ, u(γ)) given by dλu(γ).

Notation

◮ λ(γ) = hyperbolic translation length of γ ∈ PSL(2, R). ◮ dλu(γ) = rate of change of λ(γ) under the infinitesimal deformation u(γ).

Lemma (Margulis)

If there are γ1, γ2 ∈ Γ0 such that dλu(γ1) ≥ 0 and dλu(γ2) ≤ 0 then Γu

0 does

not act properly discontinuously on R2,1.

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

The GLM criterion

Notation reminder

◮ Γ = Γu 0 =

• (γ, u(γ)) : γ ∈ Γ0

◮ λ(γ) = hyperbolic translation length of γ ∈ PSL(2, R). ◮ dλu(γ) = rate of change of λ(γ) under the infinitesimal deformation u(γ).

Theorem (Goldman-Labourie-Margulis ’06)

Suppose Γ0 ⊂ PSL(2, R) is convex cocompact and u is a Γ0-cocycle. Then Γu acts properly discontinuously on R2,1 if and only if: sup

γ∈Γ0

dλu(γ) λ(γ) < 0. up to switching u with −u.

◮ The lengths of closed geodesics on S decrease in the direction u (at a

rate proportional to length).

◮ “up to switching u with −u”: cocycles which expand lengths also proper.

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

Example

γ1 = 3 1/3

• γ2 =

3/ √ 5 2/ √ 5 2/ √ 5 3/ √ 5

• u(γ1) =

−1 1

• u(γ2) =

−1 −1

• γ1

γ2

◮ Γ0 = γ1, γ2 is a one-holed torus group. ◮ The generators have orthogonal axes. ◮ Deforming in u direction preserves orthogonality but shrinks the lengths of

γ1 and γ2.

◮ Therefore (exercise): all lengths of closed geodesics decrease.

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

Macroscopic version

G = PSL(2, R) G × G acts on G by (g, h) · k := hkg−1. preserves bi-invariant Killing metric: Lorentzian, constant negative curvature. “anti de Sitter geometry” AdS3 = G with Isom0 AdS3 = G × G.

Notation

Let ρ : Γ0 → G be a representation. Γρ

0 = {(γ, ρ(γ)) : γ ∈ Γ0}

⊂ G × G When does Γρ

0 act properly on G = AdS3?

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

Kassel’s properness criterion

Notation reminder

◮ Γ0 ⊂ G = PSL(2, R) a surface group. ◮ Γρ 0 = {(γ, ρ(γ)) : γ ∈ Γ0} ⊂ G × G.

Theorem (Kassel ’10, Gu´ eritaud-Kassel ’13)

Γρ

0 acts properly discontinuously on PSL(2, R) if and only if either of the

following equivalent conditions holds:

◮ Lipschitz contraction: There exists a ρ-equivariant Lipschitz map

f : H2 → H2 with Lip(f) < 1.

◮ Length contraction:

sup

γ∈Γ0

λ(ρ(γ)) λ(γ) < 1. *where if ρ is discrete faithful, then conditions must also be tested with the roles of Γ0 and ρ(Γ0) switched.

◮ think: f is a map Γ0\H2 → ρ(Γ0)\H2 contracting all distances on the

• surface. (only if ρ discrete faithful)

◮ *uniformly longer representations ρ also give proper actions.

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

Lipschitz maps and fibrations

Notation reminder

◮ Γ0 ⊂ G = PSL(2, R) a surface group. ◮ Γρ 0 = {(γ, ρ(γ)) : γ ∈ Γ0} ⊂ G × G.

Suppose Lipschitz contraction: There exists a ρ-equivariant Lipschitz map f : H2 → H2 with Lip(f) < 1.

◮ For each g ∈ G, consider the fixed point problem

g−1f(p) = p. Unique solution p ∈ H2 because Lip(g−1f) = Lip(f) < 1.

◮ The map g → p defines a fibration.

S1

G

Π

• H2

The fiber above p: a coset of SO(2): Lp = {g : g(p) = f(p)}.

◮ Π takes action of Γρ

0 to action of Γ0.

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

Lipschitz maps and fibrations

Notation reminder

◮ Γ0 ⊂ G = PSL(2, R) a surface group. ◮ Γρ 0 = {(γ, ρ(γ)) : γ ∈ Γ0} ⊂ G × G.

Theorem (D-Gu´ eritaud-Kassel)

Suppose Γρ

0 acts properly discontinuously on G. Then the quotient manifold

M = Γρ

0\G) is a principal S1-bundle over S = Γ0\H2:

S1

Γρ

0\G Π

• Γ0\H2

◮ Want a similar theorem in the R2,1 = sl(2, R) case. ◮ Need an infinitesimal version of Lipschitz contraction.

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

Infinitesimal Lipschitz contraction

Let ρt : Γ0 → G be a path of representations such that ρ0 = Id : Γ0 → Γ0 d dt

• t=0

ρt(γ)γ−1 = u(γ) Suppose ft : H2 → H2 are ρt-equivariant maps with f0 = IdH2. Then

d dt ft

• t=0 = X a vector field on H2.

d dt

• ft(γp) = ρt(γ)f(p)
• −

→ X(γp) = γ∗X(p) + u(γ)(γp) where u(γ) is interpreted as a Killing vector field. X is called u-equivariant.

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

Infinitesimal Lipschitz contraction

X = d

dt ft

• t=0 a vector field.

Notation

d′(X(p), X(q)) = d dt

• t=0

d(ft(p), ft(q)). p q X(p) X(q)

Definition

X is called k-lipschitz if for all p = q, d′(X(p), X(q)) ≤ k d(p, q) and lip(X) = best possible k.

◮ “ lip(X) = d

dt Lip(ft) ”

◮ if lip(X), then X is contracting.

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

A new properness criterion

Notation reminder

◮ Γ0 ⊂ G = PSL(2, R) a surface group. ◮ Γu 0 = {(γ, u(γ)) : γ ∈ Γ0} ⊂ G ⋉ g.

Theorem (D–Gu´ eritaud–Kassel)

Assume Γ0 is convex cocompact. Then the group Γu

0 acts properly

discontinuously on sl(2, R) = R2,1 if and only if either of the following holds:

◮ Lipschitz contraction: There exists a u-equivariant lipschitz vector field X

with lip(X) < 0.

◮ Length contraction:(as in GLM):

sup

γ∈Γ0

dλu(γ) λ(γ) < 0 * up to switching u and −u;

◮ * X expanding also implies properness.

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

Fibrations

Suppose infinitesimal lipschitz contraction: There exists a u-equivariant lipschitz vector field X with lip(X) < 0.

◮ For each Killing field V ∈ g, consider the equation

(X − V)(p) = 0. Unique solution p ∈ H2 because lip(X − V) = lip(X) < 0 so X − V points inward on a sufficiently large ball.

◮ The map V → p defines a fibration.

R

g

π

• H2

Fiber above p: a coset of so(2): ℓp = {V : V(p) = X(p)}.

◮ π takes Γu

0 action to Γ0 action.

Theorem (D-G-K)

Let M = Γu

0\R2,1 is naturally a principle R bundle over the surface Γ0\H2.

In particular, M is tame.

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

Illustration

zero vector field contracting

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

References

• 1. V. CHARETTE, T. DRUMM, W. M. GOLDMAN, Proper affine deformations of

two-generator Fuchsian groups, in preparation.

• 2. S. CHOI, W. M. GOLDMAN, Topological tameness of Margulis spacetimes, in

preparation.

• 3. J. DANCIGER, F. GU´

ERITAUD, F. KASSEL, Geometry and topology of complete

Lorentz spacetimes of constant curvature, arXiv.1306.2240

• 4. T. DRUMM, Fundamental polyhedra for Margulis space-times, Topology 31 (1992),
• p. 677–683.
• 5. D. FRIED, W. M. GOLDMAN, Three-dimensional affine crystallographic groups, Adv.
• Math. 47 (1983), p. 1–49.
• 6. W. M. GOLDMAN, F. LABOURIE, G. A. MARGULIS, Proper affine actions and

geodesic flows of hyperbolic surfaces, Ann. of Math. 170 (2009), p. 1051–1083.

• 7. F. GU´

ERITAUD, F. KASSEL, Maximally stretched laminations on geometrically finite

hyperbolic manifolds, arXiv.1307.0250

• 8. G. A. MARGULIS, Free completely discontinuous groups of affine transformations (in

Russian), Dokl. Akad. Nauk SSSR 272 (1983), p. 785–788.

• 9. G. A. MARGULIS, Complete affine locally flat manifolds with a free fundamental

group, J. Soviet Math. 1934 (1987), p. 129–139, translated from Zap. Naucha. Sem.

• Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 134 (1984), p. 190–205.
• 10. G. MESS, Lorentz spacetimes of constant curvature (1990), Geom. Dedicata 126

(2007), p. 3–45.

• 11. W. P. THURSTON, Minimal stretch maps between hyperbolic surfaces, preprint

(1986), arXiv:9801039.

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

Proof sketch

Notation reminder

◮ Γ0 ⊂ G = PSL(2, R) a surface group. ◮ Γu 0 = {(γ, u(γ)) : γ ∈ Γ0} ⊂ G ⋉ g.

Theorem

Assume Γ0 convex cocompact. Then Γu

0 acts properly discontinuously on

sl(2, R) = R2,1 if and only if either of the following holds:

◮ Lipschitz contraction: There exists a u-equivariant lipschitz vector field X

with lip(X) < 0.

◮ Length contraction:(as in GLM): supγ∈Γ0

dλu(γ) λ(γ)

< 0. (up to switching u and −u) Follow strategy due to Gu´ eritaud-Kassel in the AdS3 case. Based on idea from Thurston’s Lipschitz metric on Teichmuller space.

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

Proof sketch

Assume dλu(α) < 0 for some α ∈ Γ0 (by switching u ↔ −u). Theorem almost follows from:

Proposition

Let k be the infimum over lipschitz constants of u-equivariant vector fields and suppose k ≥ 0. Then for any u-equivariant k-lipschitz vector field X, there exists a geodesic lamination Λ in the convex core of Γ0 which is k-stretched: If p = q lie in the same leaf of Λ, then d′(X(p), X(q)) = k d(p, q).

◮ If k < 0, then Γu

0 acts properly by fibration argument. (easy direction).

◮ Let γn ∈ Γ0 be a sequence with translation axes approximating Λ. ◮ If k = 0, show directly: (γn, u(γn)) fails to take a compact neighborhood

• ff of itself.

◮ If k > 0, some γn has dλu(γn) > 0. Apply Margulis opposite sign lemma

to γn and α.

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

Proof sketch

Problem: A vector field with optimal lipschitz constant might not exist. For let Xn be u-equivariant vector fields with lip(Xn) → k.

◮ The Xn satisfy an a priori bound (in the convex core.) ◮ Extract a Hausdorff limit X∞: a closed subset of TH2, non-empty (at least

• ver points of the convex core).

◮ X ∩ TpH2 is a closed, convex set: X is a convex field.

slope − 1

n

Xn X∞

Must work with convex fields instead of vector fields.

◮ Prove that a u-equivariant convex field X with minimal lipschitz constant

k ≥ 0 must k-stretch a lamination. (in general false when k < 0)

◮ Prove that there are u-equivariant vector fields close to X (with close

lipschitz constants).

◮ technical tools: extension theory for lipschitz convex fields (mimicing

extension theory for Lipschitz maps a la Kirzbraun and Valentine).

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

Margulis spacetimes are limits of AdS spacetimes

Notation reminder

◮ Γ0 ⊂ G = PSL(2, R) convex cocompact. ◮ Γu 0 = {(γ, u(γ)) : γ ∈ Γ0} ⊂ G ⋉ g.

Theorem

Let M = Γu

0\R2,1 be a Margulis spacetime. Then M is a limit of AdS manifolds

in the following sense: There exists a path of complete AdS structures At on S × S1, defined for t > 0 such that

• 1. The At collapse as t → 0.
• 2. There are real projective structures Pt on S × S1, conjugate to the At,

such that P0 = limt→0 Pt exists.

• 3. M is the restriction of P0 to S × {π}.

◮ Use geodesic fibrations (and appropriate sections) to build developing

maps directly.

◮ Control the geometry of the collapsing At: the geodesic fibrations of At

converge to a fibration of M.

Complete flat Lorentzian three-manifolds

• J. Danciger

Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition

Margulis spacetimes are limits of AdS spacetimes

Notation reminder

◮ S = Γ0\H2 is convex cocompact. ◮ M = Γu 0\R2,1 a Margulis spacetime.

Corollary

There exist complete AdS metrics ̺t on S × S1 such that when restricted to S × (−π, π), the metrics t−2̺t converge uniformly on compact sets to a complete flat Lorentzian metric ̺ that makes S × (−π, π) isometric to M.