EXAMPLES OF FOUR-DIMENSIONAL GEOMETRIC ...seppi.perso.math.cnrs.fr/slides_nice.pdfestablished how to...

EXAMPLES OF FOUR-DIMENSIONAL GEOMETRIC TRANSITION

Joint with S. Riolo

Nice, 17th January 2019

WHAT IS GEOMETRIC TRANSITION?

Roughly speaking, a geometric transition is a deformation of geometric structures on a manifold, which at some point “transitions” from one geometry to another.

That is, one has a path Pt of (G,X)-structures on a manifold M, for t in (-ε,ε), where G and X suddenly change for some t0.

A classical example of geometric transition is obtained as a deformation from hyperbolic structures to spherical structures, “going through” Euclidean structures:

X = ℍn, G = Isom(ℍn) for t < 0X = 𝔼n, G = Isom(𝔼n) for t = 0X = 𝕊n, G = Isom(𝕊n) for t > 0

A simple example of this transition is obtained by considering a hyperbolic sphere with three cone points, i.e. the double of a (regular, say) hyperbolic triangle.

AN EXAMPLE

HYPERBOLIC-EUCLIDEAN-SPHERICAL TRANSITION

One can then “zoom" around this limit point, and obtain a transition of the geometric structure to a Euclidean sphere with cone points.

2 CHAPTER 1. INTRODUCTION

of a geometric transition. The focus of this thesis is the study of geometric transitions in the

context of a di↵erent degeneration behavior, that of three-dimensional structures collapsing

down onto a hyperbolic plane.

Figure 1.2: Fundamental domains for hyperbolic cone manifolds collapse onto a hyperbolicplane.

To begin the discussion, we focus our attention on hyperbolic structures on a compact

three-manifold M with boundary. Let Dt : fM ! H3 be a family of developing maps de-

fined for t > 0 and suppose Dt converges to D0 a local submersion onto a two dimensional

hyperbolic plane P. Then D0 defines a co-dimension one transversely hyperbolic foliation.

The problem of regenerating hyperbolic structures from this data was examined by Hodg-

son [Hod86], and later in a specific case by Porti [Por10]. However, it had not yet been

established how to construct a geometric transition in this context. Our point of view,

based on projective geometry, is that such a degeneration naturally suggests a transition

to AdS geometry. As motivation, note that the isometry groups Isom(H3) = SO(3, 1) and

Isom(AdS3) = SO(2, 2) are naturally subgroups of the projective group PGL(4,R). The

intersection SO(3, 1) \ SO(2, 2) is, with suitable choice of coordinates, exactly the copy of

O(2, 1) that preserves the hyperbolic plane P.

Roughly, the transition from hyperbolic to AdS geometry is constructed similarly to

the transition from hyperbolic to spherical, alluded to above. As the hyperbolic structures

degenerate, the collapsing direction is rescaled so that the structures converge to a robust

three-dimensional structure called a half-pipe (HP) structure. Interpreting the hyperbolic

structures as projective structures, this rescaling is really a projective change of coordinates.

This construction is given in Chapter 3.

The key for constructing transitions is that HP structures contain precisely the informa-

tion needed to regenerate to both hyperbolic and AdS structures. In fact, if a regeneration

can be constructed on the level of representations, then an HP structure can be “expo-

nentiated” to produce a regeneration to robust geometric structures. We simply note here

that HP geometry plays the same central role in the transition between hyperbolic and AdS

In this talk, we will be interested in hyperbolic structures which “collapse" to a co-dimension one totally geodesic hyperplane.

An example (from Cooper-Hodgson-Kerckhoff's book):

By glueing opposite faces, and then doubling, one constructs a hyperbolic structure on T3, singular along a link, which collapses to a hyperbolic torus with one cone point.

We want to study "rescaled limits” for this type of degenerations.





























In dimension 3, Danciger in 2011 introduced the “right” limit geometry (half-pipe) and understood a fairly general condition which ensures the existence of transition on a compact three-manifold M, with a very simple (conical) singularities along a knot K.

He proves that collapsing hyperbolic structures on M, with cone singularities along a knot K, admits a geometric transition to half-pipe and Anti-de Sitter structures under the following condition on the character variety:

for a representation

HALF -PIPE GEOMETRY

WHY DO WE CARE?

H1Adρ(π1(M∖K), 𝔰𝔬(2,1)) ≅ ℝ .

ρ : π1(M∖K ) → SO(2,1) .

In general, it is a difficult problem to determine which manifolds admit a transition of geometric structures.

In this talk, we are interested in higher-dimensional examples.

It is already pretty hard to produce higher-dimensional examples of hyperbolic manifolds (or other geometric structures) and study their deformations.

We will produce here a class of examples of geometric transition from a hyperbolic structure to half-pipe and Anti-de Sitter structures on a closed finite-volume four-manifold with a simple not too complicated singular locus.

HALF -PIPE GEOMETRY

WHY DO WE CARE?

In 2010, Kerckhoff and Storm described an interesting deformation of hyperbolic polytopes in dimension 4, depending on a real parameter.

Starting from the 24-cell and removing two walls, they construct a polytope which can now be deformed to a family Pt, collapsing to a three-dimensional polytope (cuboctahedron) when t=0.

This family of polytopes has been used by several authors (Martelli-Riolo, Saratchandran, Kolpakov-Riolo, Riolo-Slavich).

In their paper, Kerckhoff and Storm mention that when t=0 the family of polytopes Pt is expected to have interesting geometric limits. It turns out that half-pipe geometry is well-suited to describe this infinitesimal behaviour.

THE KERCKHOFF-STORM POLYTOPES

WHY DO WE CARE?

A REMINDER

REAL PROJECTIVE STRUCTURES

A real projective structure on a manifold M is given by an atlas with values in n-dimensional projective space, where transition functions are the restrictions of projective transformations.

Ui Uj

'i(Ui)⊆X 'j(Uj)⊆X

'i :Ui!X 'j :Uj!X'j◦'−1

i 2G

This is the definition of (G,X)-structure, for:

X = ℝPn

G = PGL(n + 1,ℝ)

SPECIAL CASES

REAL PROJECTIVE STRUCTURES

Here we are interested in three special types of real projective structures:

• Hyperbolic structures

• Anti-de Sitter structures

• Half-pipe structures

X1 = ℍn = P {−x20 + x2

1 + … + x2n−1 + x2

n < 0}, G1 = PO(n,1)

X−1 = 𝔸d𝕊n = P {−x20 + x2

1 + … + x2n−1 − x2

n < 0}, G−1 = PO(n − 1,2)

X0 = ℍℙn = P {−x20 + x2

1 + … + x2n−1+0 ⋅ x2

n < 0},

G0 = {[ A 0vT ±1] : A ∈ O(n − 1,1), v ∈ ℝn} < PGL(n + 1,ℝ)

1. LIMIT OF HYP&ADS GEOMETRY

MOTIVATION FOR HALF-PIPE GEOMETRY

Consider the projective transformations

Then define

Namely, Xt is defined by the inequality

Then Xt converges to:

rt = [diag (1,…,1,1t )] .

{Xt = rt(ℍn) t > 0Xt = r|t|(𝔸d𝕊n) t < 0

Xt = P {−x20 + x2

1 + … + x2n−1 + t | t |x2

n < 0} .

X0 = ℍℙn = P {−x20 + x2

1 + … + x2n−1 < 0} .

1. LIMIT OF HYP&ADS GEOMETRY


Moreover, Gt converges to:

which i s what we def ine as the “isometry" group of half-pipe space.

Half-pipe space X0 is topologically the product

However, its geometry is substantially different. For instance, the second f a c t o r re p re s e n t s a d e g e n e r a t e direction.

G0 = {[ A 0vT ±1] : A ∈ O(n − 1,1), v ∈ ℝn}

X0 ≅ ℍn−1 × ℝ .

2. DUALITY WITH MINKOWSKI SPACE


There is another reason why the "isometry group" G0 of half-pipe geometry is naturally defined as before.

In fact, n-dimensional half-pipe space naturally parameterizes spacelike hyperplanes in Minkowski space

Given (x0,…,xn), we associate the hyperplane defined by

This is a spacelike hyperplane as a consequence of the condition

and, by homogeneity, the correspondence is well-defined on

ℝn−1,1 .

{(y0, …yn−1) ∈ ℝn−1,1 : − x0y0 + x1y1 + … + xn−1yn−1 = xn} .

−x20 + x2

1 + … + x2n−1 < 0 ,

ℍℙn .

2. DUALITY WITH MINKOWSKI SPACE


The "isometry group" G0 of half-pipe space is then the group induced by the action of

In fact, it is an exercise to check that the action on spacelike hyperplanes defines an isomorphism

Isom(ℝn−1,1) ≅ O(n − 1,1) ⋉ ℝn−1,1 .

Isom(ℝn−1,1) ≅ G0 = {[ A 0vT ±1] : A ∈ O(n − 1,1), v ∈ ℝn} .

There is more additional structure in half-pipe geometry determined by the structure of the group G0. In fact, it makes sense to talk about a (non-degenerate) totally geodesic hyperplane, which corresponds to the hyperplanes in Minkowski space passing through a given point.

HOW TO PRODUCE EXAMPLES OF TRANSITION

Here’s my favourite recipe:

• Take a continuous family of finite-volume hyperbolic polytopes Pt, for t in [0,ε), such that P0 collapses to a polytope in a co-dimension one totally geodesic subspace.

• Glue a finite number of copies of Pt, so as to obtain a family of hyperbolic structures on a fixed manifold (with some singularities!)

• Apply the “rescaling” transformations rt=[diag(1,…,1,1/t)].

• Show that the rescaled polytopes glue to a half-pipe structure when t=0, and even continue towards Anti-de Sitter structures for t in (-ε,0].

EXAMPLES OF TRANSITIONA FIRST EXAMPLE IN 3D





























Let us now go back to our first 3d example of geometric transition.

For simplicity, we send some of the vertices at infinity.

Here is an affine picture of the hyperbo l i c po lyhedron we obtain (in the projective model).


We can now deform this hyperbolic (ideal) polyhedron, by “flattening” towards a hyperbolic quadrilateral.

By glueing faces/doubling, one obtains a deformation of geometric structures (with cone singularities!).


Let us now apply the projective rescaling rt=[diag(1,…,1,1/t)]. One shows that the polytope limits to a polytope in half-pipe space, and continues to Anti-de Sitter space!

THE STRUCTURE OF THE SINGULAR LOCUS

When we glue opposite faces, and double, we get finite-volume “cusped” structures, with singularities along a two-component link.

• When t>0, this is a cone singularity, for which the cone angle approaches 2π.

• When t<0, this is a tachyon singularity.

• When t=0, the singularity is the infinitesimal version of cone angles and tachyons.


Why does there exist such an element of G0, the analogue of a rotation along a line, in half-pipe geometry?

Recalling that two totally geodesic hyperplanes P1 and P2 in half-pipe space correspond to two points q1 and q2 in Minkowski space, P1 and P2

intersect if and only if q1 and q2 are separated by a spacelike segment.

In this case, the element of G0 that we are looking for corresponds to the translation sending q1 to q2.

P2

P1

q1

q2

EXAMPLES OF TRANSITIONA SECOND EXAMPLE IN 3D

We shall now see another simple example which will be useful later on.

All the angles of this polytopes are π/2, except at the three top edges.

Let us now send these three dihedral angles to π.

EXAMPLES OF TRANSITIONA SECOND EXAMPLE IN 3D

Again, by applying the projective rescaling rt=[diag(1,…,1,1/t)], one obtains a limit polytope in half-pipe space. As before, the three lateral faces are degenerate.

We c a n a g a i n p r o d u c e a t rans i t ion o f f i n i te -vo lume structures on a manifold, by doubling along lateral faces, and then along top and bottom faces.

The s ingular locus is more complicated here (a θ-graph).

FOUR-DIMENSIONAL EXAMPLES

DEFINITION: A cuboctahedral manifold is a hyperbolic finite-volume 3-manifold obtained by glueing a finite number of copies of an ideal right-angled cuboctahedron. It is good if the glueings preserve a checkerboard coloring of the triangular faces.

THEOREM (Riolo-S.):

Let M be a good cuboctahedral mani fo ld. Then MxS1 admits a g e o m e t r i c t r a n s i t i o n f r o m a hyperbolic structure to an Anti-de Sitter structure, singular along a 2-dimensional simplicial complex.

FOUR-DIMENSIONAL EXAMPLES

The starting point to construct this transition is the family of 4-dimensional polytopes Pt introduced by Kerckhoff and Storm, by starting from the hyperbolic 24-cells and removing two walls.

For all t, the polytope Pt intersects a totally geodesic plane in a cuboctahedron. When t→0, Pt collapses to the cuboctahedron itself.

We show that, after rescaling, one can continue the polytopes rt(Pt) in projective space to a path of polytopes.

For t=0, one obtains a half-pipe polytope, for which some of the walls are degenerate, and this continues to Anti-de Sitter space.

Moreover, all isometries of the cuboctahedron extend to isometries of Pt. This permits to “glue" copies and obtain geometric structures.

SOME WORDS ABOUT THE COMBINATORICS

To get a flavour of how this looks like, here is the list of hyperplanes defining the polytope Pt.FOUR-DIMENSIONAL GEOMETRIC TRANSITION 15

0+ =h�p2 : 1 : 1 : 1 : 1

�|t|i, 0� =

h�p2 : 1 : 1 : 1 : �t

i,

1+ =h�p2 : 1 : �1 : 1 : �1

�|t|i, 1� =

h�p2 : 1 : �1 : 1 : t

i,

2+ =h�p2 : 1 : �1 : �1 : 1

�|t|i, 2� =

h�p2 : 1 : �1 : �1 : �t

i,

3+ =h�p2 : 1 : 1 : �1 : �1

�|t|i, 3� =

h�p2 : 1 : 1 : �1 : t

i,

4+ =h�p2 : �1 : 1 : �1 : 1

�|t|i, 4� =

h�p2 : �1 : 1 : �1 : �t

i,

5+ =h�p2 : �1 : 1 : 1 : �1

�|t|i, 5� =

h�p2 : �1 : 1 : 1 : t

i,

6+ =h�p2 : �1 : �1 : 1 : 1

�|t|i, 6� =

h�p2 : �1 : �1 : 1 : �t

i,

7+ =h�p2 : �1 : �1 : �1 : �1

�|t|i, 7� =

h�p2 : �1 : �1 : �1 : t

i,

A =h�1 :

p2 : 0 : 0 : 0

i, B =

h�1 : 0 :

p2 : 0 : 0

i,

C =h�1 : 0 : 0 :

p2 : 0

i, D =

h�1 : 0 : 0 : �

p2 : 0

i,

E =h�1 : 0 : �

p2 : 0 : 0

i, F =

h�1 : �

p2 : 0 : 0 : 0

i.

Table 1. The half-spaces that define Pt are given by these elements of S4,⇤

and denoted by the same symbols.

It can also be checked that the faces of P (t) are pairwise orthogonal, except the pairs(1+

,3+), (1+,5+) and (3+

,5+) which form an angle ✓(t) (for t > 0) such that:

cos(✓(t)) = �1 +3

2t2.

Hence at time t =p

2/3, the planes bounding the half-spaces 1+, 3+ and 5+ meet in theboundary at infinity of H3.

4. A four-dimensional transition

Recall the notation introduced in Section ?? for half-spaces of the projective sphere Sn.We define

Pt ⇢ S4

to be the intersection of the half-spaces listed in Table 1, depending on the time parameter

t 2✓�1,

q1�3

�.

When t � 0, each element [v] in the list of Table 1 satisfies hv, vi1 > 0, and thus definesa half-space of H

4. For t > 0, the set Pt ⇢ H4 is indeed a hyperbolic 4-polytope, first

introduced in [?] and then studied in [?]. As t ! 0 the polytope collapses to a three-dimensional polyhedron in

H3 = H \H

4 = H \ AdS4 ⇢ S4,

where H ⇢ S4 is the hyperplane defined by [0 : 0 : 0 : 0 : 1]. PROMEMORIA: definire

nei preliminari carta a�ne


A technical point: we work in the double cover of real projective space. For instance, the vector A=[-1:√2:0:0] defines the half-space

The polytope Pt is actually contained in a single affine chart {x0>0}.

−x0 + 2x1 ≤ 0

FOUR-DIMENSIONAL GEOMETRIC TRANSITION 15

0+ =h�p2 : 1 : 1 : 1 : 1

�|t|i, 0� =

h�p2 : 1 : 1 : 1 : �t

i,

1+ =h�p2 : 1 : �1 : 1 : �1

�|t|i, 1� =

h�p2 : 1 : �1 : 1 : t

i,

2+ =h�p2 : 1 : �1 : �1 : 1

�|t|i, 2� =

h�p2 : 1 : �1 : �1 : �t

i,

3+ =h�p2 : 1 : 1 : �1 : �1

�|t|i, 3� =

h�p2 : 1 : 1 : �1 : t

i,

4+ =h�p2 : �1 : 1 : �1 : 1

�|t|i, 4� =

h�p2 : �1 : 1 : �1 : �t

i,

5+ =h�p2 : �1 : 1 : 1 : �1

�|t|i, 5� =

h�p2 : �1 : 1 : 1 : t

i,

6+ =h�p2 : �1 : �1 : 1 : 1

�|t|i, 6� =

h�p2 : �1 : �1 : 1 : �t

i,

7+ =h�p2 : �1 : �1 : �1 : �1

�|t|i, 7� =

h�p2 : �1 : �1 : �1 : t

i,

A =h�1 :

p2 : 0 : 0 : 0

i, B =

h�1 : 0 :

p2 : 0 : 0

i,

C =h�1 : 0 : 0 :

p2 : 0

i, D =

h�1 : 0 : 0 : �

p2 : 0

i,

E =h�1 : 0 : �

p2 : 0 : 0

i, F =

h�1 : �

p2 : 0 : 0 : 0

i.

Table 1. The half-spaces that define Pt are given by these elements of S4,⇤

and denoted by the same symbols.

It can also be checked that the faces of P (t) are pairwise orthogonal, except the pairs(1+

,3+), (1+,5+) and (3+

,5+) which form an angle ✓(t) (for t > 0) such that:

cos(✓(t)) = �1 +3

2t2.

Hence at time t =p

2/3, the planes bounding the half-spaces 1+, 3+ and 5+ meet in theboundary at infinity of H3.

4. A four-dimensional transition

Recall the notation introduced in Section ?? for half-spaces of the projective sphere Sn.We define

Pt ⇢ S4

to be the intersection of the half-spaces listed in Table 1, depending on the time parameter

t 2✓�1,

q1�3

�.

When t � 0, each element [v] in the list of Table 1 satisfies hv, vi1 > 0, and thus definesa half-space of H

4. For t > 0, the set Pt ⇢ H4 is indeed a hyperbolic 4-polytope, first

introduced in [?] and then studied in [?]. As t ! 0 the polytope collapses to a three-dimensional polyhedron in

H3 = H \H

4 = H \ AdS4 ⇢ S4,

where H ⇢ S4 is the hyperplane defined by [0 : 0 : 0 : 0 : 1]. PROMEMORIA: definire

nei preliminari carta a�ne

• When t→0, the “plus” walls converge to a totally geodesic plane.

• The “minus" & “letter" walls are timelike when t < 0 , a n d b e c o m e degenerate in the half-pipe limit.


We already know some of the walls!

The “letter” walls look like today's first 3d example…

…whereas the “minus" walls are copies of our second example.

We can compare them with the cuboctahedron, which is the “horizontal” slice of t h e f o u r - d i m e n s i o n a l polytope.


The singular locus of these geometric structures on MxS1 is a compact foam, namely a 2-complex locally modelled on the cone over the 1-skeleton of the tetrahedron.

To visualize the singularities, let us picture the link of a singular point. It is endowed with:

• a spherical structure for t>0,

• a HS-structure for t<0, namely, is locally modelled on the space of rays from a point in ,

• an intermediate geometric structure for t=0, which is locally modelled on the space of rays in the tangent space of a point in half-pipe space.

ℝ3,1


On the 2-strata of the singular locus, we have a conical singularity.

For t>0, the (spherical) link of a point looks like:

The analogue for t<0 is a HS-structure on the 3-sphere:

θ

θ

The (interior of) the red balls in S3 represent the timelike rays.

There is a transition of the geometric structure in the link of this singular point, from spherical to HS. In the transition, the two red spheres collapse to points, representing the degenerate direction in half-pipe geometry.


The other points on the singular locus have a more complicated structure:

At these points, the singular 2-complex is of the form:

F i n a l l y , t h e m o s t complicated singularity is of the form:

THE STRUCTURE OF THE CUSPS

Finally, let us visualize the structure of the cusps along the transition.

In a suitable coordinate system, we can write hyperbolic space (t>0), Anti-de Sitter space (t<0) and half-pipe space (t=0) as the upper half-plane with the bilinear form:

Hence the link of an ideal point is endowed with:

• a Euclidean structure for t>0,

• a Minkowski structure for t<0,

• an intermediate structure for t=0, which is the flat analogue of half-pipe geometry.

dx20 + dx2

1 + dx22 + t | t |dx2

3

x20

THE STRUCTURE OF THE CUSPSA section of the cusp for the hyperbolic structures when t>0 looks like (glue opposite faces by translations):

After rescaling, the geometric structure on a section of the cusp looks like:

t = 0

t < 0t = 0t > 0

THANKS FOR YOUR ATTENTION!

EXAMPLES OF FOUR-DIMENSIONAL GEOMETRIC ...seppi.perso.math.cnrs.fr/slides_nice.pdfestablished how to...

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