AFFINE DEFORMATIONS OF QUASI-DIVISIBLE CONVEX CONES Joint work with - - PowerPoint PPT Presentation

AFFINE DEFORMATIONS OF QUASI-DIVISIBLE CONVEX CONES

Teichmüller theory: Classical, Higher, Super and Quantum

Luminy, 9/10/2020 Joint work with Xin Nie

HIGHER TEICHMÜLLER THEORY

HIGHER TEICHMÜLLER THEORY

Let be a closed orientable surface with [with punctures].

Σ χ(Σ) < 0

HIGHER TEICHMÜLLER THEORY

Let be a closed orientable surface with [with punctures].

Σ χ(Σ) < 0

Given a Lie group that contains , higher Teichmüller theory deals with the study of deformations of Fuchsian representations into and their associated geometric structures. For instance:

G SO0(2,1) ρ : π1(Σ) → SO0(2,1) G

HIGHER TEICHMÜLLER THEORY

Let be a closed orientable surface with [with punctures].

Σ χ(Σ) < 0

Given a Lie group that contains , higher Teichmüller theory deals with the study of deformations of Fuchsian representations into and their associated geometric structures. For instance:

G SO0(2,1) ρ : π1(Σ) → SO0(2,1) G

• 1. For

, these deformations give rise to maximal globally hyperbolic Minkowski manifolds, introduced by Mess (1990).

G = SO0(2,1) ⋉ ℝ3

HIGHER TEICHMÜLLER THEORY

Let be a closed orientable surface with [with punctures].

Σ χ(Σ) < 0

Given a Lie group that contains , higher Teichmüller theory deals with the study of deformations of Fuchsian representations into and their associated geometric structures. For instance:

G SO0(2,1) ρ : π1(Σ) → SO0(2,1) G

• 1. For

, these deformations give rise to maximal globally hyperbolic Minkowski manifolds, introduced by Mess (1990).

G = SO0(2,1) ⋉ ℝ3

• 2. For

, these deformations give rise to convex real projective structures (Benoist, Choi, Goldman, Guichard, Kim, Labourie, Loftin, Marquis, Papadopoulos, Wienhard…)

G = SL(3,ℝ)

HIGHER TEICHMÜLLER THEORY

Let be a closed orientable surface with [with punctures].

Σ χ(Σ) < 0

Given a Lie group that contains , higher Teichmüller theory deals with the study of deformations of Fuchsian representations into and their associated geometric structures. For instance:

G SO0(2,1) ρ : π1(Σ) → SO0(2,1) G

• 1. For

, these deformations give rise to maximal globally hyperbolic Minkowski manifolds, introduced by Mess (1990).

G = SO0(2,1) ⋉ ℝ3

• 2. For

, these deformations give rise to convex real projective structures (Benoist, Choi, Goldman, Guichard, Kim, Labourie, Loftin, Marquis, Papadopoulos, Wienhard…)

G = SL(3,ℝ)

• 3. In this talk we will generalize the above two cases and consider

.

G = SL(3,ℝ) ⋉ ℝ3

• 1. AFFINE DEFORMATIONS OF FUCHSIAN REPS…

• 1. AFFINE DEFORMATIONS OF FUCHSIAN REPS…

Let be a Fuchsian representation.

ρ : π1(Σ) → SO0(2,1)

• 1. AFFINE DEFORMATIONS OF FUCHSIAN REPS…

Let be a Fuchsian representation.

ρ : π1(Σ) → SO0(2,1)

Let us consider affine deformations of the form

ρτ(γ) ⋅ x = ρ(γ)x + τ(γ)

• 1. AFFINE DEFORMATIONS OF FUCHSIAN REPS…

Let be a Fuchsian representation.

ρ : π1(Σ) → SO0(2,1)

Let us consider affine deformations of the form where satisfies the cocycle condition

τ : π1(Σ) → ℝ3 ρτ(γ) ⋅ x = ρ(γ)x + τ(γ) τ(γη) = ρ(γ)τ(η) + τ(γ)

• 1. AFFINE DEFORMATIONS OF FUCHSIAN REPS…

Let be a Fuchsian representation.

ρ : π1(Σ) → SO0(2,1)

Let us consider affine deformations of the form where satisfies the cocycle condition

τ : π1(Σ) → ℝ3

The vector space of such affine deformations up to conjugation is the first cohomology group .

H1

ρ(π1(Σ), ℝ3)

ρτ(γ) ⋅ x = ρ(γ)x + τ(γ) τ(γη) = ρ(γ)τ(η) + τ(γ)

• 1. AFFINE DEFORMATIONS OF FUCHSIAN REPS…

Let be a Fuchsian representation.

ρ : π1(Σ) → SO0(2,1)

Let us consider affine deformations of the form where satisfies the cocycle condition

τ : π1(Σ) → ℝ3

The vector space of such affine deformations up to conjugation is the first cohomology group .

H1

ρ(π1(Σ), ℝ3)

Using the isomorphism , this cohomology group is identified to .

ℝ3 ≅ 𝔱𝔭(2,1) H1

Adρ(π1(Σ), 𝔱𝔭(2,1)) ≅ T[ρ]𝒰(Σ)

ρτ(γ) ⋅ x = ρ(γ)x + τ(γ) τ(γη) = ρ(γ)τ(η) + τ(γ)

• 1. AFFINE DEFORMATIONS OF FUCHSIAN REPS…

Let be a Fuchsian representation.

ρ : π1(Σ) → SO0(2,1)

Let us consider affine deformations of the form where satisfies the cocycle condition

τ : π1(Σ) → ℝ3

The vector space of such affine deformations up to conjugation is the first cohomology group .

H1

ρ(π1(Σ), ℝ3)

Using the isomorphism , this cohomology group is identified to .

ℝ3 ≅ 𝔱𝔭(2,1) H1

Adρ(π1(Σ), 𝔱𝔭(2,1)) ≅ T[ρ]𝒰(Σ)

That is, the vector bundle of affine deformations of Fuchsian representations identifies with the tangent bundle .

T𝒰(Σ) ρτ(γ) ⋅ x = ρ(γ)x + τ(γ) τ(γη) = ρ(γ)τ(η) + τ(γ)

…AND MINKOWSKI MANIFOLDS

…AND MINKOWSKI MANIFOLDS

The work of Mess described the geometric structures associated to such affine deformations into .

ρτ SO0(2,1) ⋉ ℝ3

…AND MINKOWSKI MANIFOLDS

The work of Mess described the geometric structures associated to such affine deformations into .

ρτ SO0(2,1) ⋉ ℝ3

The essential notion is that of regular domain, that is, a convex domain that is obtained as the intersection of (at least 3) half-spaces bounded by non-parallel null planes (i.e. such that the restriction of the Minkowski metric is degenerate).

…AND MINKOWSKI MANIFOLDS

The work of Mess described the geometric structures associated to such affine deformations into .

ρτ SO0(2,1) ⋉ ℝ3

The essential notion is that of regular domain, that is, a convex domain that is obtained as the intersection of (at least 3) half-spaces bounded by non-parallel null planes (i.e. such that the restriction of the Minkowski metric is degenerate). For each affine deformation there exists a unique future-convex and a unique past-convex regular domain invariant under the action.

ρτ

…AND MINKOWSKI MANIFOLDS

The work of Mess described the geometric structures associated to such affine deformations into .

ρτ SO0(2,1) ⋉ ℝ3

The essential notion is that of regular domain, that is, a convex domain that is obtained as the intersection of (at least 3) half-spaces bounded by non-parallel null planes (i.e. such that the restriction of the Minkowski metric is degenerate). For each affine deformation there exists a unique future-convex and a unique past-convex regular domain invariant under the action.

ρτ

They give rise to maximal globally hyperbolic manifolds in the quotient.

…AND MINKOWSKI MANIFOLDS

The work of Mess described the geometric structures associated to such affine deformations into .

ρτ SO0(2,1) ⋉ ℝ3

The essential notion is that of regular domain, that is, a convex domain that is obtained as the intersection of (at least 3) half-spaces bounded by non-parallel null planes (i.e. such that the restriction of the Minkowski metric is degenerate). For each affine deformation there exists a unique future-convex and a unique past-convex regular domain invariant under the action.

ρτ

They give rise to maximal globally hyperbolic manifolds in the quotient. Moreover, from Barbot-Béguin-Zeghib (2011), each invariant regular domain is uniquely foliated by spacelike surfaces of constant intrinsic curvature in asymptotic to the boundary of the domain.

(−∞,0)

BASIC EXAMPLE(S)

BASIC EXAMPLE(S)

When , the regular domains are the timelike cones over the origin and the f o l i a t i o n b y c o n s t a n t curvature is given by hyperboloids.

τ = 0

BASIC EXAMPLE(S)

When , the regular domains are the timelike cones over the origin and the f o l i a t i o n b y c o n s t a n t curvature is given by hyperboloids.

τ = 0

The general case can be understood as “inserting flat pieces along an invariant measured geodesic lamination”.

• 2. (QUASI-)DIVISIBLE CONVEX CONES

• 2. (QUASI-)DIVISIBLE CONVEX CONES

Let us now turn to representations .

ρ : π1(Σ) → SL(3,ℝ)

• 2. (QUASI-)DIVISIBLE CONVEX CONES

Let us now turn to representations .

ρ : π1(Σ) → SL(3,ℝ)

The connected component containing the Teichmüller space consists of the holonomies of convex real projective structures on (Hitchin representations).

Σ

• 2. (QUASI-)DIVISIBLE CONVEX CONES

Let us now turn to representations .

ρ : π1(Σ) → SL(3,ℝ)

The connected component containing the Teichmüller space consists of the holonomies of convex real projective structures on (Hitchin representations).

Σ

These representations leave invariant a proper convex cone in . The projectivized domain has compact quotient and is said divisible.

C ℝ3 ℙ(C)

• 2. (QUASI-)DIVISIBLE CONVEX CONES

Let us now turn to representations .

ρ : π1(Σ) → SL(3,ℝ)

The connected component containing the Teichmüller space consists of the holonomies of convex real projective structures on (Hitchin representations).

Σ

These representations leave invariant a proper convex cone in . The projectivized domain has compact quotient and is said divisible.

C ℝ3 ℙ(C)

When has punctures, we will consider finite-volume convex real projective structures, which correspond to representations such that is parabolic for every peripheral loop .

Σ ρ ρ(γ) γ

• 2. (QUASI-)DIVISIBLE CONVEX CONES

Let us now turn to representations .

ρ : π1(Σ) → SL(3,ℝ)

The connected component containing the Teichmüller space consists of the holonomies of convex real projective structures on (Hitchin representations).

Σ

These representations leave invariant a proper convex cone in . The projectivized domain has compact quotient and is said divisible.

C ℝ3 ℙ(C)

When has punctures, we will consider finite-volume convex real projective structures, which correspond to representations such that is parabolic for every peripheral loop .

Σ ρ ρ(γ) γ

The deformation space of finite-volume convex real projective structures is a ball of dimension , where g is the genus and n the number of punctures of .

16g − 16 + 6n Σ

AFFINE SPHERES

AFFINE SPHERES

The action of

• n the proper convex cone

can be understood again in terms of foliations by invariant surfaces.

ρ(π1Σ) C

AFFINE SPHERES

The action of

• n the proper convex cone

can be understood again in terms of foliations by invariant surfaces.

ρ(π1Σ) C

Given a convex surface in , there is a notion of affine normal vector which is well-defined for the group

• f equiaffine

transformations.

ℝ3 N SL(3,ℝ) ⋉ ℝ3

AFFINE SPHERES

The action of

• n the proper convex cone

can be understood again in terms of foliations by invariant surfaces.

ρ(π1Σ) C

Given a convex surface in , there is a notion of affine normal vector which is well-defined for the group

• f equiaffine

transformations.

ℝ3 N SL(3,ℝ) ⋉ ℝ3

Hyperbolic affine spheres are those convex surfaces for which the normal vectors pointing to the concave side meet at a single point.

AFFINE SPHERES

The action of

• n the proper convex cone

can be understood again in terms of foliations by invariant surfaces.

ρ(π1Σ) C

Given a convex surface in , there is a notion of affine normal vector which is well-defined for the group

• f equiaffine

transformations.

ℝ3 N SL(3,ℝ) ⋉ ℝ3

Hyperbolic affine spheres are those convex surfaces for which the normal vectors pointing to the concave side meet at a single point. By a theorem of Cheng-Yau (1977), every proper convex cone admits a unique affine sphere asymptotic to . Its rescaled copies foliate

C ∂C C .

AFFINE SPHERES

The action of

• n the proper convex cone

can be understood again in terms of foliations by invariant surfaces.

ρ(π1Σ) C

Given a convex surface in , there is a notion of affine normal vector which is well-defined for the group

• f equiaffine

transformations.

ℝ3 N SL(3,ℝ) ⋉ ℝ3

Hyperbolic affine spheres are those convex surfaces for which the normal vectors pointing to the concave side meet at a single point. By a theorem of Cheng-Yau (1977), every proper convex cone admits a unique affine sphere asymptotic to . Its rescaled copies foliate

C ∂C C .

When is quasi-divisible, uniqueness implies that the affine sphere and its rescaled copies are invariant under the action of .

C ρ(π1Σ)

BASIC EXAMPLE(S) AGAIN

BASIC EXAMPLE(S) AGAIN

When takes values in , the invariant cone is the usual Minkowski cone . The invariant affine sphere is again the hyperboloid in this case.

ρ SO0(2,1) x2 + y2 < z2

BASIC EXAMPLE(S) AGAIN

When takes values in , the invariant cone is the usual Minkowski cone . The invariant affine sphere is again the hyperboloid in this case.

ρ SO0(2,1) x2 + y2 < z2

BASIC EXAMPLE(S) AGAIN

When takes values in , the invariant cone is the usual Minkowski cone . The invariant affine sphere is again the hyperboloid in this case.

ρ SO0(2,1) x2 + y2 < z2

Another simple example is the Țițeica affine sphere , which is asymptotic to the boundary of the first octant.

xyz = 1

• 3. THE

CASE

SL(3,ℝ) ⋉ ℝ3

• 3. THE

CASE

SL(3,ℝ) ⋉ ℝ3

We will now consider a representation where is the holonomy of a finite-volume convex real projective structure (i.e. quasi-divides a proper convex cone ) and .

ρτ ρ ρ C τ ∈ Z1

ρ(π1(Σ), ℝ3)

• 3. THE

CASE

SL(3,ℝ) ⋉ ℝ3

We will now consider a representation where is the holonomy of a finite-volume convex real projective structure (i.e. quasi-divides a proper convex cone ) and .

ρτ ρ ρ C τ ∈ Z1

ρ(π1(Σ), ℝ3)

Since the linear part preserves a proper convex cone , we can define -spacelike and -null affine planes in terms of the intersection with (translates of) .

C C C C

C-null

C-spacelike

C

• 3. THE

CASE

SL(3,ℝ) ⋉ ℝ3

We will now consider a representation where is the holonomy of a finite-volume convex real projective structure (i.e. quasi-divides a proper convex cone ) and .

ρτ ρ ρ C τ ∈ Z1

ρ(π1(Σ), ℝ3)

Since the linear part preserves a proper convex cone , we can define -spacelike and -null affine planes in terms of the intersection with (translates of) .

C C C C

C-null

C-spacelike

C

A (future) -regular domain is a non-empty proper subset of

• btained as the

intersection of (upward) half-spaces bounded by -lightlike planes.

C ℝ3 C

INVARIANT DOMAINS

INVARIANT DOMAINS

Let us first state our result in the closed case, that is, suppose is closed and divides .

Σ ρ : π1(Σ) → SL(3,ℝ) C

INVARIANT DOMAINS

Let us first state our result in the closed case, that is, suppose is closed and divides .

Σ ρ : π1(Σ) → SL(3,ℝ) C

Theorem(Nie-S.): For every affine deformation there exists a unique

• invariant -regular domain, on which the

action of is free and properly discontinuous.

τ ∈ Z1

ρ(π1(Σ), ℝ3)

ρτ C ρτ

INVARIANT DOMAINS

Let us first state our result in the closed case, that is, suppose is closed and divides .

Σ ρ : π1(Σ) → SL(3,ℝ) C

Theorem(Nie-S.): For every affine deformation there exists a unique

• invariant -regular domain, on which the

action of is free and properly discontinuous.

τ ∈ Z1

ρ(π1(Σ), ℝ3)

ρτ C ρτ

In the quasi-divisible case, invariant -regular domains exist if and only if is admissible, namely if for every peripheral , is contained in the plane preserved by .

C τ γ ∈ π1(Σ) τ(γ) ρ(γ)

INVARIANT DOMAINS

Let us first state our result in the closed case, that is, suppose is closed and divides .

Σ ρ : π1(Σ) → SL(3,ℝ) C

Theorem(Nie-S.): For every affine deformation there exists a unique

• invariant -regular domain, on which the

action of is free and properly discontinuous.

τ ∈ Z1

ρ(π1(Σ), ℝ3)

ρτ C ρτ

In the quasi-divisible case, invariant -regular domains exist if and only if is admissible, namely if for every peripheral , is contained in the plane preserved by .

C τ γ ∈ π1(Σ) τ(γ) ρ(γ)

Theorem(Nie-S.): For every admissible affine deformation , there is a bijection between

• invariant
• regular domains and

, and the action is free and properly discontinuous on each of them.

τ ρτ C [0,∞)n

INVARIANT DOMAINS

Let us first state our result in the closed case, that is, suppose is closed and divides .

Σ ρ : π1(Σ) → SL(3,ℝ) C

Theorem(Nie-S.): For every affine deformation there exists a unique

• invariant -regular domain, on which the

action of is free and properly discontinuous.

τ ∈ Z1

ρ(π1(Σ), ℝ3)

ρτ C ρτ

In the quasi-divisible case, invariant -regular domains exist if and only if is admissible, namely if for every peripheral , is contained in the plane preserved by .

C τ γ ∈ π1(Σ) τ(γ) ρ(γ)

Theorem(Nie-S.): For every admissible affine deformation , there is a bijection between

• invariant
• regular domains and

, and the action is free and properly discontinuous on each of them.

τ ρτ C [0,∞)n

The bijection has the property that if and only if the corresponding n-tuples satisfy for all i.

D ⊆ D′ (x1, …, xn), (x′

1, …, x′ n)

xi ≥ x′

i

THE MAXIMAL REGULAR DOMAIN

THE MAXIMAL REGULAR DOMAIN

In particular, there exists a -regular domain that contains all the others. It is obtained by the following construction.

C

THE MAXIMAL REGULAR DOMAIN

In particular, there exists a -regular domain that contains all the others. It is obtained by the following construction.

C

It turns out that there exists a unique continuous

• equivariant map

(ρ, ρτ)

f : ∂ℙ(C) → {C − null planes} .

THE MAXIMAL REGULAR DOMAIN

In particular, there exists a -regular domain that contains all the others. It is obtained by the following construction.

C

It turns out that there exists a unique continuous

• equivariant map

(ρ, ρτ)

The existence of such map is related to the Anosov property (Barbot, Danciger-Guéritaud-Kassel, Ghosh).

f : ∂ℙ(C) → {C − null planes} .

THE MAXIMAL REGULAR DOMAIN

In particular, there exists a -regular domain that contains all the others. It is obtained by the following construction.

C

It turns out that there exists a unique continuous

• equivariant map

(ρ, ρτ)

The existence of such map is related to the Anosov property (Barbot, Danciger-Guéritaud-Kassel, Ghosh). Then the complement of the union has two connected components, that are the maximal regular and

• regular domains invariant by the action of

.

C− (−C) ρτ

⋃

x∈∂ℙ(C)

f(x)

f : ∂ℙ(C) → {C − null planes} .

ADMISSIBLE AFFINE DEFORMATIONS

ADMISSIBLE AFFINE DEFORMATIONS

Geometrically, the admissibility condition is equivalent to the condition that for every peripheral , leaves invariant some affine plane.

γ ∈ π1(Σ) ρτ(γ)

ADMISSIBLE AFFINE DEFORMATIONS

Geometrically, the admissibility condition is equivalent to the condition that for every peripheral , leaves invariant some affine plane.

γ ∈ π1(Σ) ρτ(γ)

Before moving on, it is natural to ask how many admissible affine deformations exist up to conjugacy:

ADMISSIBLE AFFINE DEFORMATIONS

Geometrically, the admissibility condition is equivalent to the condition that for every peripheral , leaves invariant some affine plane.

γ ∈ π1(Σ) ρτ(γ)

Before moving on, it is natural to ask how many admissible affine deformations exist up to conjugacy:

Proposition(Nie-S.): The space of admissible affine deformations of holonomies of finite-volume convex real projective structures on is a topological vector bundle of rank .

Σ 6g − 6 + 2n

ADMISSIBLE AFFINE DEFORMATIONS

Geometrically, the admissibility condition is equivalent to the condition that for every peripheral , leaves invariant some affine plane.

γ ∈ π1(Σ) ρτ(γ)

Before moving on, it is natural to ask how many admissible affine deformations exist up to conjugacy:

Proposition(Nie-S.): The space of admissible affine deformations of holonomies of finite-volume convex real projective structures on is a topological vector bundle of rank .

Σ 6g − 6 + 2n

The proposition holds true for the torus ( , ), showing that every affine deformation is trivial up to conjugacy. In this case is known to be a triangular cone.

g = 1 n = 0 C

CONSTANT CURVATURE FOLIATIONS

CONSTANT CURVATURE FOLIATIONS

Recall that for a convex surface in equiaffine space, there is a well- defined transversal vector field called affine normal.

N

CONSTANT CURVATURE FOLIATIONS

Recall that for a convex surface in equiaffine space, there is a well- defined transversal vector field called affine normal.

N

Writing the equations

DXY = ∇XY + h(X, Y)N DXN = B(X) + σ(X)N

CONSTANT CURVATURE FOLIATIONS

Recall that for a convex surface in equiaffine space, there is a well- defined transversal vector field called affine normal.

N

Writing the equations is uniquely determined (up to sign) by the conditions that and the induced volume form coincides with the volume form of .

N σ ≡ 0 ν = ιN det h

DXY = ∇XY + h(X, Y)N DXN = B(X) + σ(X)N

CONSTANT CURVATURE FOLIATIONS

Recall that for a convex surface in equiaffine space, there is a well- defined transversal vector field called affine normal.

N

Writing the equations is uniquely determined (up to sign) by the conditions that and the induced volume form coincides with the volume form of .

N σ ≡ 0 ν = ιN det h

The convex surface has constant affine Gaussian curvature if .

det B = k

DXY = ∇XY + h(X, Y)N DXN = B(X) + σ(X)N

CONSTANT CURVATURE FOLIATIONS

Recall that for a convex surface in equiaffine space, there is a well- defined transversal vector field called affine normal.

N

Writing the equations is uniquely determined (up to sign) by the conditions that and the induced volume form coincides with the volume form of .

N σ ≡ 0 ν = ιN det h

The convex surface has constant affine Gaussian curvature if .

det B = k

DXY = ∇XY + h(X, Y)N DXN = B(X) + σ(X)N

Theorem(Nie-S.): Every invariant

• regular domain

is uniquely foliated by complete convex surfaces of CAGC asymptotic to .

C D k ∈ (0,∞) ∂D

CONSTANT CURVATURE FOLIATIONS

Recall that for a convex surface in equiaffine space, there is a well- defined transversal vector field called affine normal.

N

Writing the equations is uniquely determined (up to sign) by the conditions that and the induced volume form coincides with the volume form of .

N σ ≡ 0 ν = ιN det h

The convex surface has constant affine Gaussian curvature if .

det B = k

DXY = ∇XY + h(X, Y)N DXN = B(X) + σ(X)N

Theorem(Nie-S.): Every invariant

• regular domain

is uniquely foliated by complete convex surfaces of CAGC asymptotic to .

C D k ∈ (0,∞) ∂D

In the closed case, we recover a result of Labourie (2007) by independent methods, and in the Minkowski case, the foliation of Barbot-Béguin-Zeghib.

• 4. CONVEX TUBE DOMAINS

• 4. CONVEX TUBE DOMAINS

The main idea of the proofs roughly consist in translating the statement into a “dual" projective geometry.

• 4. CONVEX TUBE DOMAINS

The main idea of the proofs roughly consist in translating the statement into a “dual" projective geometry. Indeed, we can identify the dual affine space with the space of non- vertical affine planes in : for and , we define:

ℝ3 x ∈ ℝ2 ξ ∈ ℝ

(x, ξ) ↦ graph of (y ↦ x ⋅ y − ξ)

• 4. CONVEX TUBE DOMAINS

The main idea of the proofs roughly consist in translating the statement into a “dual" projective geometry. Indeed, we can identify the dual affine space with the space of non- vertical affine planes in : for and , we define:

ℝ3 x ∈ ℝ2 ξ ∈ ℝ

(x, ξ) ↦ graph of (y ↦ x ⋅ y − ξ)

Given the proper convex cone , let be the dual cone and be the section of at height 1. Then:

C C⋆ Ω C⋆

• 4. CONVEX TUBE DOMAINS

The main idea of the proofs roughly consist in translating the statement into a “dual" projective geometry. Indeed, we can identify the dual affine space with the space of non- vertical affine planes in : for and , we define:

ℝ3 x ∈ ℝ2 ξ ∈ ℝ

(x, ξ) ↦ graph of (y ↦ x ⋅ y − ξ)

Given the proper convex cone , let be the dual cone and be the section of at height 1. Then:

C C⋆ Ω C⋆

• spacelike planes correspond to points in

;

C Ω × ℝ

• 4. CONVEX TUBE DOMAINS

The main idea of the proofs roughly consist in translating the statement into a “dual" projective geometry. Indeed, we can identify the dual affine space with the space of non- vertical affine planes in : for and , we define:

ℝ3 x ∈ ℝ2 ξ ∈ ℝ

(x, ξ) ↦ graph of (y ↦ x ⋅ y − ξ)

Given the proper convex cone , let be the dual cone and be the section of at height 1. Then:

C C⋆ Ω C⋆

• spacelike planes correspond to points in

;

C Ω × ℝ

• null planes correspond to points in

.

C ∂Ω × ℝ

• 4. CONVEX TUBE DOMAINS

The main idea of the proofs roughly consist in translating the statement into a “dual" projective geometry. Indeed, we can identify the dual affine space with the space of non- vertical affine planes in : for and , we define:

ℝ3 x ∈ ℝ2 ξ ∈ ℝ

(x, ξ) ↦ graph of (y ↦ x ⋅ y − ξ)

Given the proper convex cone , let be the dual cone and be the section of at height 1. Then:

C C⋆ Ω C⋆

• spacelike planes correspond to points in

;

C Ω × ℝ

• null planes correspond to points in

.

C ∂Ω × ℝ

We call such region the convex tube domain.

AUTOMORPHISM GROUPS

AUTOMORPHISM GROUPS

The action of the automorphism group of

• n

then induces a projective action preserving the convex tube domain as a subset

• f projective space.

C ℝ3 Ω × ℝ

AUTOMORPHISM GROUPS

The action of the automorphism group of

• n

then induces a projective action preserving the convex tube domain as a subset

• f projective space.

C ℝ3 Ω × ℝ

The projective transformations obtained in this way are those that do not switch the two ends of and have eigenvalue at the fixed points at infinity. Concretely, the group homomorphism is:

Ω × ℝ ±1

(ρ, τ) ↦ (

tρ−1 t(ρ−1τ) 1)

AUTOMORPHISM GROUPS

The action of the automorphism group of

• n

then induces a projective action preserving the convex tube domain as a subset

• f projective space.

C ℝ3 Ω × ℝ

The projective transformations obtained in this way are those that do not switch the two ends of and have eigenvalue at the fixed points at infinity. Concretely, the group homomorphism is:

Ω × ℝ ±1

(ρ, τ) ↦ (

tρ−1 t(ρ−1τ) 1)

For instance linear transformations ( ) induce automorphisms that preserve the slice .

τ = 0 Ω × {0}

AUTOMORPHISM GROUPS

The action of the automorphism group of

• n

then induces a projective action preserving the convex tube domain as a subset

• f projective space.

C ℝ3 Ω × ℝ

The projective transformations obtained in this way are those that do not switch the two ends of and have eigenvalue at the fixed points at infinity. Concretely, the group homomorphism is:

Ω × ℝ ±1

(ρ, τ) ↦ (

tρ−1 t(ρ−1τ) 1)

For instance linear transformations ( ) induce automorphisms that preserve the slice .

τ = 0 Ω × {0}

Under this duality, non-vertical planes in correspond to points of (i.e. the set of planes going through a given point).

Ω × ℝ ℝ3

INVARIANT GRAPHS

INVARIANT GRAPHS

From this perspective,

• regular domains correspond to lower

semicontinuous functions :

C φ : ∂Ω → ℝ ∪ {+∞}

INVARIANT GRAPHS

From this perspective,

• regular domains correspond to lower

semicontinuous functions :

C φ : ∂Ω → ℝ ∪ {+∞}

• Given , the corresponding l.s.c. function is

D

φ(x) = sup

(y,η)∈D

(x ⋅ y − η)

INVARIANT GRAPHS

From this perspective,

• regular domains correspond to lower

semicontinuous functions :

C φ : ∂Ω → ℝ ∪ {+∞}

• Given , the corresponding l.s.c. function is

D

• Given , the corresponding -regular domain is

φ C

φ(x) = sup

(y,η)∈D

(x ⋅ y − η)

D = ⋂

x∈∂Ω

{(y, η) : η > x ⋅ y − φ(x)}

INVARIANT GRAPHS

From this perspective,

• regular domains correspond to lower

semicontinuous functions :

C φ : ∂Ω → ℝ ∪ {+∞}

• Given , the corresponding l.s.c. function is

D

• Given , the corresponding -regular domain is

φ C

Given an affine deformation as before, constructing a

• invariant
• regular domain is equivalent to finding a

lower semicontinuous function whose graph is invariant for the action induced by

• n the convex tube domain.

ρτ : π1(Σ) → SL(3,ℝ) ρτ C φ : ∂Ω → ℝ ∪ {+∞} ρτ

φ(x) = sup

(y,η)∈D

(x ⋅ y − η)

D = ⋂

x∈∂Ω

{(y, η) : η > x ⋅ y − φ(x)}

INVARIANT GRAPHS

From this perspective,

• regular domains correspond to lower

semicontinuous functions :

C φ : ∂Ω → ℝ ∪ {+∞}

• Given , the corresponding l.s.c. function is

D

• Given , the corresponding -regular domain is

φ C

Given an affine deformation as before, constructing a

• invariant
• regular domain is equivalent to finding a

lower semicontinuous function whose graph is invariant for the action induced by

• n the convex tube domain.

ρτ : π1(Σ) → SL(3,ℝ) ρτ C φ : ∂Ω → ℝ ∪ {+∞} ρτ

Similarly, invariant convex surfaces correspond to convex l.s.c. functions whose graph is invariant in .

u : Ω → ℝ ∪ {+∞} Ω × ℝ

φ(x) = sup

(y,η)∈D

(x ⋅ y − η)

D = ⋂

x∈∂Ω

{(y, η) : η > x ⋅ y − φ(x)}

PARABOLIC FIXED POINTS

PARABOLIC FIXED POINTS

In terms of the action on the action on the convex tube domain, the admissibility condition on is equivalent to requiring that has a fixed point in for every peripheral . In this case, an entire vertical line is fixed pointwise.

τ ρτ(γ) ∂Ω × ℝ γ

PARABOLIC FIXED POINTS

In terms of the action on the action on the convex tube domain, the admissibility condition on is equivalent to requiring that has a fixed point in for every peripheral . In this case, an entire vertical line is fixed pointwise.

τ ρτ(γ) ∂Ω × ℝ γ

A “distinguished” fixed point corresponds to the affine plane in that contains the line fixed by .

ℝ3 ρτ(γ)

ixed planes distinguished ixed point ixed points

PARABOLIC FIXED POINTS

In terms of the action on the action on the convex tube domain, the admissibility condition on is equivalent to requiring that has a fixed point in for every peripheral . In this case, an entire vertical line is fixed pointwise.

τ ρτ(γ) ∂Ω × ℝ γ

A “distinguished” fixed point corresponds to the affine plane in that contains the line fixed by .

ℝ3 ρτ(γ)

con The unique continuous function whose graph is

• invariant contains all the

“distinguished” fixed points for peripheral .

φ : ∂Ω → ℝ ρτ γ

PARABOLIC FIXED POINTS

In terms of the action on the action on the convex tube domain, the admissibility condition on is equivalent to requiring that has a fixed point in for every peripheral . In this case, an entire vertical line is fixed pointwise.

τ ρτ(γ) ∂Ω × ℝ γ

A “distinguished” fixed point corresponds to the affine plane in that contains the line fixed by .

ℝ3 ρτ(γ)

con The unique continuous function whose graph is

• invariant contains all the

“distinguished” fixed points for peripheral .

φ : ∂Ω → ℝ ρτ γ

The other (non-continuous) invariant graphs are obtained by choosing a representative in each

• orbit of the parabolic fixed points, and moving

below the distinguished fixed point (one real parameter for every puncture of ).

p π1(Σ) φ(p) Σ

CAGC FOLIATIONS

CAGC FOLIATIONS

Let us now move on to the proof of the foliation result.

CAGC FOLIATIONS

Let us now move on to the proof of the foliation result. Under the convex duality, the problem is translated to an equation of Monge-Ampère type. For affine spheres (i.e. ), the equation is:

τ = 0

CAGC FOLIATIONS

Let us now move on to the proof of the foliation result. Under the convex duality, the problem is translated to an equation of Monge-Ampère type. For affine spheres (i.e. ), the equation is:

τ = 0 { det D2v = v−4 in Ω v|∂Ω = 0

CAGC FOLIATIONS

Let us now move on to the proof of the foliation result. Under the convex duality, the problem is translated to an equation of Monge-Ampère type. For affine spheres (i.e. ), the equation is:

τ = 0 { det D2v = v−4 in Ω v|∂Ω = 0

and the existence and uniqueness of solutions is essentially the theorem

• f Cheng-Yau. In the general case, the equation is (for fixed

):

t ∈ ℝ

CAGC FOLIATIONS

Let us now move on to the proof of the foliation result. Under the convex duality, the problem is translated to an equation of Monge-Ampère type. For affine spheres (i.e. ), the equation is:

τ = 0 { det D2v = v−4 in Ω v|∂Ω = 0

and the existence and uniqueness of solutions is essentially the theorem

• f Cheng-Yau. In the general case, the equation is (for fixed

):

t ∈ ℝ { det D2u = e−tv−4 in Ω u|∂Ω = φ

CAGC FOLIATIONS

Let us now move on to the proof of the foliation result. Under the convex duality, the problem is translated to an equation of Monge-Ampère type. For affine spheres (i.e. ), the equation is:

τ = 0 { det D2v = v−4 in Ω v|∂Ω = 0

and the existence and uniqueness of solutions is essentially the theorem

• f Cheng-Yau. In the general case, the equation is (for fixed

):

t ∈ ℝ { det D2u = e−tv−4 in Ω u|∂Ω = φ

where is the Cheng-Yau solution above and is the l.s.c. function determining the invariant -regular domain.

v φ C

CAGC FOLIATIONS

Let us now move on to the proof of the foliation result. Under the convex duality, the problem is translated to an equation of Monge-Ampère type. For affine spheres (i.e. ), the equation is:

τ = 0 { det D2v = v−4 in Ω v|∂Ω = 0

and the existence and uniqueness of solutions is essentially the theorem

• f Cheng-Yau. In the general case, the equation is (for fixed

):

t ∈ ℝ { det D2u = e−tv−4 in Ω u|∂Ω = φ

where is the Cheng-Yau solution above and is the l.s.c. function determining the invariant -regular domain.

v φ C

This two-step Monge-Ampère equation was studied by Li-Simon-Chen (1997), but with a boundary regularity too restrictive for our setting.

EUCLIDEAN COMPLETENESS: CLOSED CASE

EUCLIDEAN COMPLETENESS: CLOSED CASE

However, the trickiest point is to show that the solution u corresponds to a complete surface of CAGC, or equivalently, that it is an entire graph over the horizontal plane.

EUCLIDEAN COMPLETENESS: CLOSED CASE

However, the trickiest point is to show that the solution u corresponds to a complete surface of CAGC, or equivalently, that it is an entire graph over the horizontal plane. This is equivalent to showing that u has infinite inner derivatives, that is, the slope goes to infinity for every point

• f

along some (hence any) line segment in as we approach .

x0 ∂Ω Ω x0

EUCLIDEAN COMPLETENESS: CLOSED CASE

However, the trickiest point is to show that the solution u corresponds to a complete surface of CAGC, or equivalently, that it is an entire graph over the horizontal plane. This is equivalent to showing that u has infinite inner derivatives, that is, the slope goes to infinity for every point

• f

along some (hence any) line segment in as we approach .

x0 ∂Ω Ω x0

In the closed case, this is not hard, because (using cocompactness) one can estimate

EUCLIDEAN COMPLETENESS: CLOSED CASE

However, the trickiest point is to show that the solution u corresponds to a complete surface of CAGC, or equivalently, that it is an entire graph over the horizontal plane. This is equivalent to showing that u has infinite inner derivatives, that is, the slope goes to infinity for every point

• f

along some (hence any) line segment in as we approach .

x0 ∂Ω Ω x0

In the closed case, this is not hard, because (using cocompactness) one can estimate

u ≤ φ + ϵv

EUCLIDEAN COMPLETENESS: CLOSED CASE

However, the trickiest point is to show that the solution u corresponds to a complete surface of CAGC, or equivalently, that it is an entire graph over the horizontal plane. This is equivalent to showing that u has infinite inner derivatives, that is, the slope goes to infinity for every point

• f

along some (hence any) line segment in as we approach .

x0 ∂Ω Ω x0

In the closed case, this is not hard, because (using cocompactness) one can estimate

u ≤ φ + ϵv

where is the convex envelope of . Since the Cheng-Yau solution v has infinite inner derivatives, so does u.

φ φ

EUCLIDEAN COMPLETENESS: PUNCTURED CASE

EUCLIDEAN COMPLETENESS: PUNCTURED CASE

In the punctured case, an adaptation of this argument proves the inner infinite derivative condition except at the parabolic fixed points in .

∂Ω

EUCLIDEAN COMPLETENESS: PUNCTURED CASE

In the punctured case, an adaptation of this argument proves the inner infinite derivative condition except at the parabolic fixed points in .

∂Ω

Although is not , for every parabolic fixed point there exists an invariant disc through that contains .

∂Ω C2 x0 Δ x0 Ω

EUCLIDEAN COMPLETENESS: PUNCTURED CASE

In the punctured case, an adaptation of this argument proves the inner infinite derivative condition except at the parabolic fixed points in .

∂Ω

Although is not , for every parabolic fixed point there exists an invariant disc through that contains .

∂Ω C2 x0 Δ x0 Ω

By an application of the maximum principle, the Cheng-Yau solutions satisfy the inequality ( is the center of , r its radius):

x Δ

vΩ ≥ vΔ = − r2 − (x − x)2

EUCLIDEAN COMPLETENESS: PUNCTURED CASE

In the punctured case, an adaptation of this argument proves the inner infinite derivative condition except at the parabolic fixed points in .

∂Ω

Although is not , for every parabolic fixed point there exists an invariant disc through that contains .

∂Ω C2 x0 Δ x0 Ω

By an application of the maximum principle, the Cheng-Yau solutions satisfy the inequality ( is the center of , r its radius):

x Δ

vΩ ≥ vΔ = − r2 − (x − x)2

Hence we have

det D2u = cv−4

Ω ≥ c(r2 − (x − x)2)−2 ≅ c|x − x0|−2

EUCLIDEAN COMPLETENESS: PUNCTURED CASE

In the punctured case, an adaptation of this argument proves the inner infinite derivative condition except at the parabolic fixed points in .

∂Ω

Although is not , for every parabolic fixed point there exists an invariant disc through that contains .

∂Ω C2 x0 Δ x0 Ω

By an application of the maximum principle, the Cheng-Yau solutions satisfy the inequality ( is the center of , r its radius):

x Δ

vΩ ≥ vΔ = − r2 − (x − x)2

Hence we have

det D2u = cv−4

Ω ≥ c(r2 − (x − x)2)−2 ≅ c|x − x0|−2

and this is the rate of growth necessary to prove that u has infinite inner derivative as we approach x, using a supersolution method.

THANKS FOR YOUR ATTENTION!