2. (Q UASI -) DIVISIBLE CONVEX CONES Let us now turn to representations . ρ : π 1 ( Σ ) → SL(3, ℝ ) The connected component containing the Teichmüller space consists of the holonomies of convex real projective structures on ( Hitchin Σ representations ). These representations leave invariant a proper convex cone in ℝ 3 . The C projectivized domain has compact quotient and is said divisible . ℙ ( C ) When has punctures, we will consider finite-volume convex real Σ projective structures, which correspond to representations such that ρ is parabolic for every peripheral loop . ρ ( γ ) γ
2. (Q UASI -) DIVISIBLE CONVEX CONES Let us now turn to representations . ρ : π 1 ( Σ ) → SL(3, ℝ ) The connected component containing the Teichmüller space consists of the holonomies of convex real projective structures on ( Hitchin Σ representations ). These representations leave invariant a proper convex cone in ℝ 3 . The C projectivized domain has compact quotient and is said divisible . ℙ ( C ) When has punctures, we will consider finite-volume convex real Σ projective structures, which correspond to representations such that ρ is parabolic for every peripheral loop . 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 16 g − 16 + 6 n number of punctures of . Σ
A FFINE SPHERES
A FFINE SPHERES The action of on the proper convex cone can be understood ρ ( π 1 Σ ) C again in terms of foliations by invariant surfaces.
A FFINE SPHERES The action of on the proper convex cone can be understood ρ ( π 1 Σ ) C again in terms of foliations by invariant surfaces. Given a convex surface in ℝ 3 , there is a notion of affine normal vector N which is well-defined for the group SL(3, ℝ ) ⋉ ℝ 3 of equiaffine transformations.
A FFINE SPHERES The action of on the proper convex cone can be understood ρ ( π 1 Σ ) C again in terms of foliations by invariant surfaces. Given a convex surface in ℝ 3 , there is a notion of affine normal vector N which is well-defined for the group SL(3, ℝ ) ⋉ ℝ 3 of equiaffine transformations. Hyperbolic affine spheres are those convex surfaces for which the normal vectors pointing to the concave side meet at a single point.
A FFINE SPHERES The action of on the proper convex cone can be understood ρ ( π 1 Σ ) C again in terms of foliations by invariant surfaces. Given a convex surface in ℝ 3 , there is a notion of affine normal vector N which is well-defined for the group SL(3, ℝ ) ⋉ ℝ 3 of equiaffine transformations. 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 C a unique affine sphere asymptotic to . Its rescaled copies foliate ∂ C C .
A FFINE SPHERES The action of on the proper convex cone can be understood ρ ( π 1 Σ ) C again in terms of foliations by invariant surfaces. Given a convex surface in ℝ 3 , there is a notion of affine normal vector N which is well-defined for the group SL(3, ℝ ) ⋉ ℝ 3 of equiaffine transformations. 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 C a unique affine sphere asymptotic to . Its rescaled copies foliate ∂ C C . When is quasi-divisible, uniqueness implies that the affine sphere and C its rescaled copies are invariant under the action of . ρ ( π 1 Σ )
B ASIC EXAMPLE ( S ) AGAIN
B ASIC EXAMPLE ( S ) AGAIN When takes values in , the invariant cone is the usual ρ SO 0 (2,1) x 2 + y 2 < z 2 Minkowski cone . The invariant affine sphere is again the hyperboloid in this case.
B ASIC EXAMPLE ( S ) AGAIN When takes values in , the invariant cone is the usual ρ SO 0 (2,1) x 2 + y 2 < z 2 Minkowski cone . The invariant affine sphere is again the hyperboloid in this case.
B ASIC EXAMPLE ( S ) AGAIN When takes values in , the invariant cone is the usual ρ SO 0 (2,1) x 2 + y 2 < z 2 Minkowski cone . The invariant affine sphere is again the hyperboloid in this case. Another simple example is the Ț i ț eica affine sphere , which is xyz = 1 asymptotic to the boundary of the first octant.
3. T HE CASE SL(3, ℝ ) ⋉ ℝ 3
3. T HE CASE SL(3, ℝ ) ⋉ ℝ 3 We will now consider a representation where is the holonomy of a ρ τ ρ finite-volume convex real projective structure (i.e. quasi-divides a ρ τ ∈ Z 1 ρ ( π 1 ( Σ ), ℝ 3 ) proper convex cone ) and . C
3. T HE CASE SL(3, ℝ ) ⋉ ℝ 3 We will now consider a representation where is the holonomy of a ρ τ ρ finite-volume convex real projective structure (i.e. quasi-divides a ρ τ ∈ Z 1 ρ ( π 1 ( Σ ), ℝ 3 ) proper convex cone ) and . C C -spacelike C - null Since the linear part preserves a C proper convex cone , we can C define -spacelike and -null affine C C planes in terms of the intersection with (translates of) . C
3. T HE CASE SL(3, ℝ ) ⋉ ℝ 3 We will now consider a representation where is the holonomy of a ρ τ ρ finite-volume convex real projective structure (i.e. quasi-divides a ρ τ ∈ Z 1 ρ ( π 1 ( Σ ), ℝ 3 ) proper convex cone ) and . C C -spacelike C - null Since the linear part preserves a C proper convex cone , we can C define -spacelike and -null affine C C planes in terms of the intersection with (translates of) . C A ( future ) -regular domain is a non-empty C proper subset of ℝ 3 obtained as the intersection of (upward) half-spaces bounded by -lightlike planes. C
I NVARIANT DOMAINS
I NVARIANT DOMAINS Let us first state our result in the closed case, that is, suppose is closed Σ and divides . ρ : π 1 ( Σ ) → SL(3, ℝ ) C
I NVARIANT DOMAINS Let us first state our result in the closed case, that is, suppose is closed Σ and divides . ρ : π 1 ( Σ ) → SL(3, ℝ ) C τ ∈ Z 1 ρ ( π 1 ( Σ ), ℝ 3 ) Theorem(Nie-S.): For every affine deformation there exists a unique -invariant -regular domain, on which the C ρ τ action of is free and properly discontinuous. ρ τ
I NVARIANT DOMAINS Let us first state our result in the closed case, that is, suppose is closed Σ and divides . ρ : π 1 ( Σ ) → SL(3, ℝ ) C τ ∈ Z 1 ρ ( π 1 ( Σ ), ℝ 3 ) Theorem(Nie-S.): For every affine deformation there exists a unique -invariant -regular domain, on which the C ρ τ action of is free and properly discontinuous. ρ τ In the quasi-divisible case, invariant -regular domains exist if and only C if is admissible, namely if for every peripheral , is γ ∈ π 1 ( Σ ) τ ( γ ) τ contained in the plane preserved by . ρ ( γ )
I NVARIANT DOMAINS Let us first state our result in the closed case, that is, suppose is closed Σ and divides . ρ : π 1 ( Σ ) → SL(3, ℝ ) C τ ∈ Z 1 ρ ( π 1 ( Σ ), ℝ 3 ) Theorem(Nie-S.): For every affine deformation there exists a unique -invariant -regular domain, on which the C ρ τ action of is free and properly discontinuous. ρ τ In the quasi-divisible case, invariant -regular domains exist if and only C if is admissible, namely if for every peripheral , is γ ∈ π 1 ( Σ ) τ ( γ ) τ contained in the plane preserved by . ρ ( γ ) Theorem(Nie-S.): For every admissible affine deformation , there τ [0, ∞ ) n is a bijection between -invariant -regular domains and , C ρ τ and the action is free and properly discontinuous on each of them.
I NVARIANT DOMAINS Let us first state our result in the closed case, that is, suppose is closed Σ and divides . ρ : π 1 ( Σ ) → SL(3, ℝ ) C τ ∈ Z 1 ρ ( π 1 ( Σ ), ℝ 3 ) Theorem(Nie-S.): For every affine deformation there exists a unique -invariant -regular domain, on which the C ρ τ action of is free and properly discontinuous. ρ τ In the quasi-divisible case, invariant -regular domains exist if and only C if is admissible, namely if for every peripheral , is γ ∈ π 1 ( Σ ) τ ( γ ) τ contained in the plane preserved by . ρ ( γ ) Theorem(Nie-S.): For every admissible affine deformation , there τ [0, ∞ ) n is a bijection between -invariant -regular domains and , C ρ τ and the action is free and properly discontinuous on each of them. The bijection has the property that if and only if the D ⊆ D ′ corresponding n -tuples satisfy for all i . ( x 1 , …, x n ), ( x ′ 1 , …, x ′ n ) x i ≥ x ′ i
T HE MAXIMAL REGULAR DOMAIN
T HE MAXIMAL REGULAR DOMAIN In particular, there exists a -regular domain that contains all the others. C It is obtained by the following construction.
T HE MAXIMAL REGULAR DOMAIN In particular, there exists a -regular domain that contains all the others. C It is obtained by the following construction. It turns out that there exists a unique continuous -equivariant map ( ρ , ρ τ ) f : ∂ℙ ( C ) → { C − null planes} .
T HE MAXIMAL REGULAR DOMAIN In particular, there exists a -regular domain that contains all the others. C It is obtained by the following construction. It turns out that there exists a unique continuous -equivariant map ( ρ , ρ τ ) f : ∂ℙ ( C ) → { C − null planes} . The existence of such map is related to the Anosov property (Barbot, Danciger-Guéritaud-Kassel, Ghosh).
T HE MAXIMAL REGULAR DOMAIN In particular, there exists a -regular domain that contains all the others. C It is obtained by the following construction. It turns out that there exists a unique continuous -equivariant map ( ρ , ρ τ ) f : ∂ℙ ( C ) → { C − null planes} . The existence of such map is related to the Anosov property (Barbot, Danciger-Guéritaud-Kassel, Ghosh). Then the complement of the union ⋃ f ( x ) x ∈∂ℙ ( C ) has two connected components, that are the maximal regular and C − -regular domains invariant by the action of . ( − C ) ρ τ
A DMISSIBLE AFFINE DEFORMATIONS
A DMISSIBLE AFFINE DEFORMATIONS Geometrically, the admissibility condition is equivalent to the condition that for every peripheral , leaves invariant some affine γ ∈ π 1 ( Σ ) ρ τ ( γ ) plane.
A DMISSIBLE AFFINE DEFORMATIONS Geometrically, the admissibility condition is equivalent to the condition that for every peripheral , leaves invariant some affine γ ∈ π 1 ( Σ ) ρ τ ( γ ) plane. Before moving on, it is natural to ask how many admissible affine deformations exist up to conjugacy :
A DMISSIBLE AFFINE DEFORMATIONS Geometrically, the admissibility condition is equivalent to the condition that for every peripheral , leaves invariant some affine γ ∈ π 1 ( Σ ) ρ τ ( γ ) plane. 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 . 6 g − 6 + 2 n
A DMISSIBLE AFFINE DEFORMATIONS Geometrically, the admissibility condition is equivalent to the condition that for every peripheral , leaves invariant some affine γ ∈ π 1 ( Σ ) ρ τ ( γ ) plane. 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 . 6 g − 6 + 2 n The proposition holds true for the torus ( , ), showing that g = 1 n = 0 every affine deformation is trivial up to conjugacy. In this case is C known to be a triangular cone.
C ONSTANT CURVATURE FOLIATIONS
C ONSTANT CURVATURE FOLIATIONS Recall that for a convex surface in equiaffine space, there is a well- defined transversal vector field called affine normal . N
C ONSTANT 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 D X Y = ∇ X Y + h ( X , Y ) N D X N = B ( X ) + σ ( X ) N
C ONSTANT 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 D X Y = ∇ X Y + h ( X , Y ) N D X N = B ( X ) + σ ( X ) N is uniquely determined (up to sign) by the conditions that and N σ ≡ 0 the induced volume form coincides with the volume form of . ν = ι N det h
C ONSTANT 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 D X Y = ∇ X Y + h ( X , Y ) N D X N = B ( X ) + σ ( X ) N is uniquely determined (up to sign) by the conditions that and N σ ≡ 0 the induced volume form coincides with the volume form of . ν = ι N det h The convex surface has constant affine Gaussian curvature if . det B = k
C ONSTANT 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 D X Y = ∇ X Y + h ( X , Y ) N D X N = B ( X ) + σ ( X ) N is uniquely determined (up to sign) by the conditions that and N σ ≡ 0 the induced volume form coincides with the volume form of . ν = ι N det h The convex surface has constant affine Gaussian curvature if . det B = k Theorem(Nie-S.): Every invariant -regular domain is C D uniquely foliated by complete convex surfaces of CAGC asymptotic to . k ∈ (0, ∞ ) ∂ D
C ONSTANT 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 D X Y = ∇ X Y + h ( X , Y ) N D X N = B ( X ) + σ ( X ) N is uniquely determined (up to sign) by the conditions that and N σ ≡ 0 the induced volume form coincides with the volume form of . ν = ι N det h The convex surface has constant affine Gaussian curvature if . det B = k Theorem(Nie-S.): Every invariant -regular domain is C D uniquely foliated by complete convex surfaces of CAGC asymptotic to . 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. C ONVEX TUBE DOMAINS
4. C ONVEX TUBE DOMAINS The main idea of the proofs roughly consist in translating the statement into a “dual" projective geometry.
4. C ONVEX 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 ℝ 3 : for x ∈ ℝ 2 and , we define: ξ ∈ ℝ ( x , ξ ) ↦ graph of ( y ↦ x ⋅ y − ξ )
4. C ONVEX 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 ℝ 3 : for x ∈ ℝ 2 and , we define: ξ ∈ ℝ ( x , ξ ) ↦ graph of ( y ↦ x ⋅ y − ξ ) Given the proper convex cone , let C ⋆ be the dual C cone and be the section of C ⋆ at height 1. Then: Ω
4. C ONVEX 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 ℝ 3 : for x ∈ ℝ 2 and , we define: ξ ∈ ℝ ( x , ξ ) ↦ graph of ( y ↦ x ⋅ y − ξ ) Given the proper convex cone , let C ⋆ be the dual C cone and be the section of C ⋆ at height 1. Then: Ω -spacelike planes correspond to points in ; • C Ω × ℝ
4. C ONVEX 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 ℝ 3 : for x ∈ ℝ 2 and , we define: ξ ∈ ℝ ( x , ξ ) ↦ graph of ( y ↦ x ⋅ y − ξ ) Given the proper convex cone , let C ⋆ be the dual C cone and be the section of C ⋆ at height 1. Then: Ω -spacelike planes correspond to points in ; • C Ω × ℝ -null planes correspond to points in . • C ∂Ω × ℝ
4. C ONVEX 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 ℝ 3 : for x ∈ ℝ 2 and , we define: ξ ∈ ℝ ( x , ξ ) ↦ graph of ( y ↦ x ⋅ y − ξ ) Given the proper convex cone , let C ⋆ be the dual C cone and be the section of C ⋆ at height 1. Then: Ω -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 .
A UTOMORPHISM GROUPS
A UTOMORPHISM GROUPS The action of the automorphism group of on ℝ 3 then induces a C projective action preserving the convex tube domain as a subset Ω × ℝ of projective space.
A UTOMORPHISM GROUPS The action of the automorphism group of on ℝ 3 then induces a C projective action preserving the convex tube domain as a subset Ω × ℝ of projective space. The projective transformations obtained in this way are those that do not switch the two ends of and have eigenvalue at the fixed points ±1 Ω × ℝ at infinity. Concretely, the group homomorphism is: ( ρ , τ ) ↦ ( t ( ρ − 1 τ ) 1 ) t ρ − 1 0
A UTOMORPHISM GROUPS The action of the automorphism group of on ℝ 3 then induces a C projective action preserving the convex tube domain as a subset Ω × ℝ of projective space. The projective transformations obtained in this way are those that do not switch the two ends of and have eigenvalue at the fixed points ±1 Ω × ℝ at infinity. Concretely, the group homomorphism is: ( ρ , τ ) ↦ ( t ( ρ − 1 τ ) 1 ) t ρ − 1 0 For instance linear transformations ( ) induce automorphisms that τ = 0 preserve the slice . Ω × {0}
A UTOMORPHISM GROUPS The action of the automorphism group of on ℝ 3 then induces a C projective action preserving the convex tube domain as a subset Ω × ℝ of projective space. The projective transformations obtained in this way are those that do not switch the two ends of and have eigenvalue at the fixed points ±1 Ω × ℝ at infinity. Concretely, the group homomorphism is: ( ρ , τ ) ↦ ( t ( ρ − 1 τ ) 1 ) t ρ − 1 0 For instance linear transformations ( ) induce automorphisms that τ = 0 preserve the slice . Ω × {0} Under this duality, non-vertical planes in correspond to points of Ω × ℝ ℝ 3 (i.e. the set of planes going through a given point).
I NVARIANT GRAPHS
I NVARIANT GRAPHS From this perspective, -regular domains correspond to lower C semicontinuous functions : φ : ∂Ω → ℝ ∪ {+ ∞ }
I NVARIANT GRAPHS From this perspective, -regular domains correspond to lower C semicontinuous functions : φ : ∂Ω → ℝ ∪ {+ ∞ } Given , the corresponding l.s.c. function is • φ ( x ) = sup ( x ⋅ y − η ) D ( y , η ) ∈ D
I NVARIANT GRAPHS From this perspective, -regular domains correspond to lower C semicontinuous functions : φ : ∂Ω → ℝ ∪ {+ ∞ } Given , the corresponding l.s.c. function is • φ ( x ) = sup ( x ⋅ y − η ) D ( y , η ) ∈ D Given , the corresponding -regular domain is • C φ D = ⋂ {( y , η ) : η > x ⋅ y − φ ( x )} x ∈∂Ω
I NVARIANT GRAPHS From this perspective, -regular domains correspond to lower C semicontinuous functions : φ : ∂Ω → ℝ ∪ {+ ∞ } Given , the corresponding l.s.c. function is • φ ( x ) = sup ( x ⋅ y − η ) D ( y , η ) ∈ D Given , the corresponding -regular domain is • C φ D = ⋂ {( y , η ) : η > x ⋅ y − φ ( x )} x ∈∂Ω Given an affine deformation as before, ρ τ : π 1 ( Σ ) → SL(3, ℝ ) constructing a -invariant -regular domain is equivalent to finding a C ρ τ lower semicontinuous function whose graph is φ : ∂Ω → ℝ ∪ {+ ∞ } invariant for the action induced by on the convex tube domain. ρ τ
I NVARIANT GRAPHS From this perspective, -regular domains correspond to lower C semicontinuous functions : φ : ∂Ω → ℝ ∪ {+ ∞ } Given , the corresponding l.s.c. function is • φ ( x ) = sup ( x ⋅ y − η ) D ( y , η ) ∈ D Given , the corresponding -regular domain is • C φ D = ⋂ {( y , η ) : η > x ⋅ y − φ ( x )} x ∈∂Ω Given an affine deformation as before, ρ τ : π 1 ( Σ ) → SL(3, ℝ ) constructing a -invariant -regular domain is equivalent to finding a C ρ τ lower semicontinuous function whose graph is φ : ∂Ω → ℝ ∪ {+ ∞ } invariant for the action induced by on the convex tube domain. ρ τ Similarly, invariant convex surfaces correspond to convex l.s.c. functions whose graph is invariant in . u : Ω → ℝ ∪ {+ ∞ } Ω × ℝ
P ARABOLIC FIXED POINTS
P ARABOLIC 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 τ ρ τ ( γ ) fixed point in for every peripheral . In this case, an entire ∂Ω × ℝ γ vertical line is fixed pointwise.
P ARABOLIC 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 τ ρ τ ( γ ) fixed point in for every peripheral . In this case, an entire ∂Ω × ℝ γ vertical line is fixed pointwise. A “distinguished” fixed point corresponds to the affine plane in ℝ 3 that contains the line fixed by . ρ τ ( γ ) � ixed points � ixed planes distinguished � ixed point
P ARABOLIC 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 τ ρ τ ( γ ) fixed point in for every peripheral . In this case, an entire ∂Ω × ℝ γ vertical line is fixed pointwise. A “distinguished” fixed point corresponds to the affine plane in ℝ 3 that contains the con line fixed by . ρ τ ( γ ) The unique continuous function φ : ∂Ω → ℝ whose graph is -invariant contains all the ρ τ “distinguished” fixed points for peripheral . γ
P ARABOLIC 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 τ ρ τ ( γ ) fixed point in for every peripheral . In this case, an entire ∂Ω × ℝ γ vertical line is fixed pointwise. A “distinguished” fixed point corresponds to the affine plane in ℝ 3 that contains the con line fixed by . ρ τ ( γ ) The unique continuous function φ : ∂Ω → ℝ whose graph is -invariant contains all the ρ τ “distinguished” fixed points for peripheral . γ The other (non-continuous) invariant graphs are obtained by choosing a representative in each -orbit of the parabolic fixed points, and moving p π 1 ( Σ ) φ ( p ) below the distinguished fixed point (one real parameter for every puncture of ). 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 D 2 v = 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 D 2 v = v − 4 in Ω v | ∂Ω = 0 and the existence and uniqueness of solutions is essentially the theorem of 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 D 2 v = v − 4 in Ω v | ∂Ω = 0 and the existence and uniqueness of solutions is essentially the theorem of Cheng-Yau. In the general case, the equation is (for fixed ): t ∈ ℝ { det D 2 u = e − t v − 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 D 2 v = v − 4 in Ω v | ∂Ω = 0 and the existence and uniqueness of solutions is essentially the theorem of Cheng-Yau. In the general case, the equation is (for fixed ): t ∈ ℝ { det D 2 u = e − t v − 4 in Ω u | ∂Ω = φ where is the Cheng-Yau solution above and is the l.s.c. function v φ determining the invariant -regular domain. 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 D 2 v = v − 4 in Ω v | ∂Ω = 0 and the existence and uniqueness of solutions is essentially the theorem of Cheng-Yau. In the general case, the equation is (for fixed ): t ∈ ℝ { det D 2 u = e − t v − 4 in Ω u | ∂Ω = φ where is the Cheng-Yau solution above and is the l.s.c. function v φ determining the invariant -regular domain. C This two-step Monge-Ampère equation was studied by Li-Simon-Chen (1997), but with a boundary regularity too restrictive for our setting.
E UCLIDEAN COMPLETENESS : CLOSED CASE
E UCLIDEAN 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.
E UCLIDEAN 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 of along some (hence x 0 ∂Ω any) line segment in as we approach . Ω x 0
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