Affine deformations of quasi-divisible convex cones
AAFFINE DEFORMATIONS OF QUASI-DIVISIBLE CONVEX CONES
XIN NIE AND ANDREA SEPPIA
BSTRACT . For any subgroup of SL(3, (cid:82) ) (cid:110) (cid:82) obtained by adding a translation part to a subgroup of SL(3, (cid:82) )which is the fundamental group of a finite-volume convex projective surface, we first show that under a naturalcondition on the translation parts of parabolic elements, the affine action of the group on (cid:82) has convex domainsof discontinuity that are regular in a certain sense, generalizing a result of Mess for globally hyperbolic flatspacetimes. We then classify all these domains and show that the quotient of each of them is an affine manifoldfoliated by convex surfaces with constant affine Gaussian curvature. The proof is based on a correspondencebetween the geometry of an affine space endowed with a convex cone and the geometry of a convex tube domain.As an independent result, we show that the moduli space of such groups is a vector bundle over the modulispace of finite-volume convex projective structures, with rank equaling the dimension of the Teichmüller space. C ONTENTS
1. Introduction 12. Correspondence between the two geometries 63. C -regular domains and affine ( C , k )-surfaces 114. Affine deformations of quasi-divisible convex cones 205. Proof of main results 29References 351. I NTRODUCTION
Given a Lie group G containing SO (2,1) = Isom + ( (cid:72) ) and a closed hyperbolic surface S with fun-damental group π ( S ) identified as a Fuchsian group in SO (2,1), representations π ( S ) → G that aredeformations of the inclusion are objects of study in higher Teichmüller theory . We study in this paper thecase where G is the group SL(3, (cid:82) ) (cid:110) (cid:82) of special affine transformations of the real affine space (cid:65) . Thiscan be viewed as the combination of two well studied cases: • the isometry group SO(2,1) (cid:110) (cid:82) of the Minkowski space (cid:82) , where the deformations give rise to maximal globally hyperbolic flat spacetimes [Mes07] (see also [BG01, Bar05, Bon05, KS07, Bel14,BS16, FS20]); • the special linear group SL(3, (cid:82) ), where the deformations yield convex real projective structures(see e.g. [Gol90, Ben08, KP14, CLM18]).We also extend the setting by allowing S to have punctures. Affine deformations, regular domains and CAGC surfaces.
Given a proper convex cone C in (cid:82) (see §2.1 for the definition), we let Aut ( C ) < SL(3, (cid:82) ) denote the group of special linear transformationspreserving C , which is also the group of orientation-preserving projective automorphisms of the convexdomain (cid:80) ( C ) ⊂ (cid:82)(cid:80) .Following [Ben08, Mar14], (cid:80) ( C ) is said to be divisible (resp. quasi-divisible ) by a group Γ < SL(3, (cid:82) )if Γ is discrete, contained in Aut ( C ), and the quotient (cid:80) ( C )/ Γ is compact (resp. has finite volume withrespect to the Hilbert metric). Furthermore, we always assume Γ is torsion-free, so that the quotient is a r X i v : . [ m a t h . DG ] S e p XIN NIE AND ANDREA SEPPI a closed (resp. finite-volume) convex projective surface . Abusing the terminology, we also say that C is(quasi-)divisible by Γ if (cid:80) ( C ) is.Given a map τ : Γ → (cid:82) , a subgroup in SL(3, (cid:82) ) (cid:110) (cid:82) of the form Γ τ : = (cid:169) ( A , τ ( A )) ∈ SL(3, (cid:82) ) (cid:110) (cid:82) (cid:175)(cid:175) A ∈ Γ (cid:170) is called an affine deformation of Γ . The group relation forces τ to be an element in the space Z ( Γ , (cid:82) ) ofcocycles. We call τ admissible if for every parabolic element A ∈ Γ , τ ( A ) is contained in the 2-dimensionalsubspace of (cid:82) preserved by A . This condition is vacuous if C is divisible by Γ , in which case there is noparabolic element.In [NS19], we generalized standard notions in Minkowski geometry, such as spacelike/null planes andregular domains, to C-spacelike/C-null planes and
C-regular domains in (cid:65) , defined with respect to aproper convex cone C in the underlying vector space (cid:82) ( c.f. §2.1 and §3.3 below). Our first result is: Theorem A.
Let C ⊂ (cid:82) be a proper convex cone quasi-divisible by a torsion-free group Γ < SL(3, (cid:82) ) andlet τ ∈ Z ( Γ , (cid:82) ) . Then(1) There exists a C-regular domain in (cid:65) preserved by Γ τ if and only if τ is admissible. In thiscase, there is a unique continuous map f from ∂ (cid:80) ( C ) to the space of C-null planes in (cid:65) which isequivariant in the sense that f ( A . x ) = ( A , τ ( A )). f ( x ) for all x ∈ ∂ (cid:80) ( C ) and A ∈ Γ . The complementof the union of planes (cid:83) x ∈ ∂ (cid:80) ( C ) f ( x ) in (cid:65) has two connected components D + and D − , which areC-regular and ( − C ) -regular domains preserved by Γ τ , respectively.(2) If C is divisible by Γ , then D + is the unique C-regular domain preserved by Γ τ . Otherwise, assumethe surface S : = (cid:80) ( C )/ Γ has n ≥ punctures and τ is admissible, then all the C-regular domainspreserved by Γ τ form a family ( D µ ) parameterized by µ ∈ (cid:82) n ≥ , such that D (0, ··· ,0) = D + and we haveD µ ⊂ D µ (cid:48) if and only if µ is coordinate-wise larger than or equal to µ (cid:48) .(3) Γ τ acts freely and properly discontinuously on every C-regular domain preserved by it, with quo-tient homeomorphic to S × (cid:82) . When C is the future light cone C ⊂ (cid:82) , , the divisible case of this theorem is part of the seminalwork of Mess [Mes07]. Brunswic [Bru16a, Bru16b] has obtained results in the quasi-divisible case for C as well. For general C , in the divisible case, the equivariant continuous map given by Part (1) isrelated to the Anosov property of Γ τ , studied by Barbot [Bar10] and Danciger-Guéritaud-Kassel [DGK17]in different but related settings. Remark.
For the trivial deformation τ =
0, we have D + = C , and any other C -regular domain preservedby Γ τ = Γ is obtained by first choosing a Γ -invariant family of C -null planes intersecting C , such that eachplane in the family is preserved by some parabolic element, and then trimming C along these planes.Each puncture of S corresponds to a conjugacy class of parabolic elements and gives rise to an (cid:82) ≥ -worthof choices. For a general admissible τ , D µ is obtained from the maximal C -regular domain D + preservedby Γ τ by the same construction, which explains Part (2) of Theorem A. Note that although D ± is themaximal ( ± C )-regular domain preserved by Γ τ , it is observed in [Mes07] that D + ∪ D − might not be themaximal domain of discontinuity for the Γ τ -action on (cid:65) , even in the divisible and C = C case.Our second result establishes a canonical “time function” on each C -regular domain in (cid:65) preservedby Γ τ , whose level surfaces have Constant Affine Gaussian Curvature (CAGC): Theorem B.
Let C ⊂ (cid:82) be a proper convex cone quasi-divisible by a torsion-free group Γ < SL(3, (cid:82) ) , τ ∈ Z ( Γ , (cid:82) ) be an admissible cocycle and D be a C-regular domain preserved by Γ τ . Then for any k > , Dcontains a unique complete affine ( C , k ) -surface Σ k generating ∂ D. This surface is preserved by Γ τ and isasymptotic to the boundary of D. Moreover, ( Σ k ) k > is a foliation of D, and the function K : D → (cid:82) given byK | Σ k = log k is convex. Here, affine ( C , k ) -surfaces are a particular class of convex surfaces with CAGC k , whose supportingplanes are C -spacelike (see §3.5 for details). The main result of [NS19] is a statement similar to Theorem FFINE DEFORMATIONS OF QUASI-DIVISIBLE CONVEX CONES 3
B, for C -regular domains without any group action assumed, but instead assuming that the planar convexdomain (cid:80) ( C ) satisfies the interior circle condition at every boundary point. When C is quasi-divisible, ∂ (cid:80) ( C ) is known to have at most C α -regularity [Ben04, Gui05], and the condition is not satisfied.Our main motivation for establishing Theorems A and B is to produce affine 3-manifolds which gener-alize maximal globally hyperbolic flat spacetimes: Corollary C.
Let C ⊂ (cid:82) be a proper convex cone quasi-divisible by a torsion-free group Γ < SL(3, (cid:82) ) anddenote the surface (cid:80) ( C )/ Γ by S. Let τ ∈ Z ( Γ , (cid:82) ) be an admissible cocycle and D be a C-regular domainpreserved by Γ τ . Then there is a homeomorphism from the affine manifold M : = D / Γ τ to S × (cid:82) + , such thateach slice S × { k } is a locally strongly convex surface with CAGC k with respect to the affine structure on M,and the projection to the (cid:82) + -factor is a locally log -convex function on M with respect to the affine structure. For the future light cone C , an affine ( C , k )-surface is just a spacelike, future-convex surface in (cid:82) with classical Gaussian curvature k (or intrinsic curvature − k ; c.f. [NS19, Prop. 3.7]). When S isclosed ( i.e. when C is divisible by Γ ), some of the statements in Corollary C are contained in the worksBarbot-Béguin-Zeghib [BBZ11] for C and Labourie [Lab07, §8] for general C . Moduli space of admissible deformations.
Two natural questions that one might ask while lookingat the above results are: what are all the quasi-divisible proper convex cones in (cid:82) , and what are all theiradmissible affine deformations?It follows from results of Marquis [Mar12] that in the above setting, the orientable surface S = (cid:80) ( C )/ Γ is homeomorphic to either the torus or the surface S g , n of negative Euler characteristic with genus g and n punctures. Since the case of torus is simple (see Remark 4.11), we will only look into the abovequestions for S g , n .The first question essentially asks for a description of the moduli space P g , n of finite-volume convexprojective structures on S g , n . For n =
0, Goldman [Gol90] first provided a Fenchel-Nielsen type descrip-tion, then Labourie [Lab07] and Loftin [Lof01] obtained a holomorphic one. The two descriptions aregeneralized by Marquis [Mar10] and Benoist-Hulin [BH13], respectively, to n ≥
1. These results implythat P g , n is homeomorphic to a ball of dimension 16 g − + n .The second question is concerned with the moduli space (cid:99) P g , n of representations ρ : π ( S g , n ) → SL(3, (cid:82) ) (cid:110) (cid:82) such that the SL(3, (cid:82) )-component of ρ defines a finite-volume convex projective structure and the (cid:82) -component is given by an admissible cocycle. With elementary arguments, we show: Proposition D.
For any g , n ≥ with − g − n < , (cid:99) P g , n is a topological vector bundle over P g , n of rank g − + n. Note that the rank equals the dimension of the Teichmüller space T g , n . In fact, T g , n is naturallycontained in P g , n , and the part of (cid:99) P g , n over T g , n can be identified with the tangent bundle of T g , n . WhileMess [Mes07] has introduced several new ideas to study this part, generalization of his methods to (cid:99) P g , n is an interesting task not yet undertaken. Affine space with a cone vs. convex tube domain.
The main tool in the proof of Theorems A and B,also used implicitly in [NS19], is a correspondence between the following two geometries: • The geometry of (cid:65) with respect to the group Aut ( C ) (cid:110) (cid:82) of special affine transformations whoselinear parts preserve a given proper convex cone C ⊂ (cid:82) . • The geometry of a convex tube domain in (cid:82) , i.e. an open set of the form T = Ω × (cid:82) with Ω a boundedplanar convex domain, with respect a group Aut ( T ) of certain projective transformations, whichwe call the automorphisms of T .We refer to §2.4 below for the precise definition of automorphisms of T , only mentioning here that theycan be roughly understood as the projective transformations Φ ∈ PGL(4, (cid:82) ) preserving T which are givenby matrices of the form (cid:195) B t Y (cid:33) , B ∈ SL(3, (cid:82) ), Y ∈ (cid:82) XIN NIE AND ANDREA SEPPI (multiplying B and the lower-right 1 by different constants yields a projective transformation preserving T but not in Aut ( T )), and the groups in the two geometries are isomorphic to each other through (cid:195) A X (cid:33) ←→ (cid:195) t A − ( A − X ) 1 (cid:33) ,where the first matrix represents the element ( A , X ) in Aut ( C ) (cid:110) (cid:82) . As one might guess from the appear-ance of the inverse transpose t A − , the convex domain Ω in the second geometry can be identified with asection of the cone C ∗ ⊂ (cid:82) ∗ dual to the cone C from the first geometry.When C is the future light cone C ⊂ (cid:82) , the first geometry is just that of the Minkowski space (cid:82) ,whereas the second is the co-Minkowski geometry of the round tube (see e.g. [SS04, Dan13, BF18, FS19,DMS20]). We proceed to give more details for general C .A polarization of the affine space (cid:65) is a choice of a point p ∞ on the plane at infinity P ∞ : = (cid:82)(cid:80) \ (cid:65) ∼= (cid:82)(cid:80) . Given p ∞ , we define the dual polarized affine space (cid:65) ∗ as the space of affine planes in (cid:65) not con-taining p ∞ at infinity, which is an affine chart in the dual projective space (cid:82)(cid:80) ∗ = { affine planes in (cid:65) } ∪ { P ∞ } , with polarization just given by P ∞ .Given a proper convex cone C ⊂ (cid:82) whose projectivization (cid:80) ( C ) ⊂ P ∞ contains p ∞ , every C -spacelikeplane in (cid:65) can be viewed as a point in (cid:65) ∗ . We will show: Proposition E.
The set S C : = (cid:169) C-spacelike planes in (cid:65) (cid:170) is a convex tube domain in (cid:65) ∗ , whose underly-ing planar convex domain is a section of the dual cone C ∗ . The natural action of Aut ( C ) (cid:110) (cid:82) on S C inducesan isomorphism between Aut ( C ) (cid:110) (cid:82) and the automorphism group of S C as a convex tube domain. objects in (cid:65) objects in (cid:65) ∗ point plane not containing P ∞ at infinityplane not containing p ∞ at infinity point C -spacelike plane point in Ω × (cid:82) C -null plane point in ∂ Ω × (cid:82) affine transformation in Aut ( C ) (cid:110) (cid:82) automorphism of Ω × (cid:82) subgroup of Aut ( C ) (cid:110) (cid:82) of the form Γ τ ,where Γ quasi-divides C and τ is admissible subgroup of Aut ( Ω × (cid:82) ) projecting bijectively to agroup quasi-dividing Ω s.t. every element withparabolic projection has fixed point in ∂ Ω × (cid:82) C -regular (resp. ( − C )-regular) domain D graph in ∂ Ω × (cid:82) of a lower (resp. upper)semicontinuous function ϕ on ∂ Ω smooth, strongly convex, complete C -convexsurface generating a C -regular domain D ( c.f. §3.3) graph in Ω × (cid:82) of a function u ∈ S ( Ω ) ( c.f. §3.3) whose boundary value ϕ corresponds to D ( c.f. the last row)affine ( C , k )-surface ( c.f. §3.5) graph in Ω × (cid:82) of some u ∈ S ( Ω ) satisfyingdet D u = k − w − Ω (Prop. 3.12)T ABLE
1. Dictionary between the two geometries.In summary, we have obtained the first few rows of Table 1 (where we write the convex tube domain S C as Ω × (cid:82) ). The rest of the table will be explained in §3. This dictionary enables us to deduce TheoremsA and B from the following dual results about convex tube domains: FFINE DEFORMATIONS OF QUASI-DIVISIBLE CONVEX CONES 5
Theorem F.
Let Ω be a bounded convex domain in (cid:82) ⊂ (cid:82)(cid:80) quasi-divisible by a torsion-free group Λ ofprojective transformations, and Λ (cid:48) < Aut ( Ω × (cid:82) ) be a group of automorphisms of the convex tube domain Ω × (cid:82) which projects to Λ bijectively. Then(1) The following conditions are equivalent to each other:(a) every element of Λ (cid:48) with parabolic projection in Λ has a fixed point in ∂ Ω × (cid:82) ;(b) there exists a continuous function ϕ ∈ C ( ∂ Ω ) with graph gr( ϕ ) ⊂ ∂ Ω × (cid:82) preserved by Λ (cid:48) ;(c) there exists a lower semicontinuous function (cid:98) ϕ : Ω → (cid:82) ∪ { +∞ } , which is not constantly +∞ ,with graph preserved by Λ (cid:48) ;(2) Suppose these conditions are fulfilled. Then the function ϕ in (b) is unique. On the other hand,the function (cid:98) ϕ in (c) is unique and equals ϕ only if Ω is divisible by Λ . Otherwise, all the (cid:98) ϕ ’scan be described as follows. Let F ⊂ ∂ Ω be the set of fixed points of parabolic elements in Λ andpick p , ··· , p n ∈ F such that F is the disjoint union of the orbits Λ . p j , j = ··· , n. For each µ = ( µ , ··· , µ n ) ∈ (cid:82) n ≥ , let ϕ µ : ∂ Ω → (cid:82) be the function with graph preserved by Λ (cid:48) such that ϕ µ ( p j ) = ϕ ( p j ) − µ j for all j; ϕ µ = ϕ on ∂ Ω \ F (with ϕ from (b)). Then ϕ µ is lower semicontinuous, and every (cid:98) ϕ in (c) equals some ϕ µ .(3) Let w Ω ∈ C ( Ω ) ∩ C ∞ ( Ω ) be the unique convex solution (established by Cheng-Yau [CY77] , see Thm.3.10 below) to the Dirichlet problem of Monge-Ampère equation (cid:40) det D w = w − w | ∂ Ω = Then for any ϕ µ from Part (2) and any t > , the Dirichlet problem (cid:40) det D u = e − t w − Ω u | ∂ Ω = ϕ µ has a unique convex solution u t ∈ C ∞ ( Ω ) . It has the following properties: • (cid:107) D u t (cid:107) tends to +∞ on the boundary of Ω ; • the graph gr( u t ) ⊂ Ω × (cid:82) is preserved by Λ (cid:48) ; • for every fixed x ∈ Ω , t (cid:55)→ u t ( x ) is a strictly increasing concave function, with value tending to −∞ and ϕ µ ( x ) as t tends to −∞ and +∞ , respectively. In the last part, the function ϕ µ is the convex envelope of ϕ µ . We refer to §3.1 for its definition and forthe precise meaning of the boundary value u | ∂ Ω when u is a convex function on Ω .Theorem F gives a picture similar to the familiar ones from quasi-Fuchsian hyperbolic manifolds andglobally hyperbolic anti-de Sitter spacetimes, see Figure 1.1. Namely, the group Λ < Aut ( Ω ), viewed as convex hull of F IGURE Ω × (cid:82) , is analogous to a Fuchsian group acting on (cid:72) or AdS , which preserves a slicein the 3-space as well as the boundary circle of the slice. After a perturbation, we obtain the group Λ (cid:48) which preserves the Jordan curve gr( ϕ ) on the boundary. The graphs of the family of convex functions ( u t ) XIN NIE AND ANDREA SEPPI produced by Part (3) then gives a canonical foliation for the lower part T − of Ω × (cid:82) outside of the convexhull of the curve, while the upper part is foliated in the same way by graphs of concave functions. A newphenomenon here is that when the surface Ω / Λ has n punctures, for each µ ∈ (cid:82) n ≥ , there is a subdomain T − µ of T − preserved by Λ (cid:48) , namely the part of Ω × (cid:82) underneath gr( ϕ µ ), which is obtained by modifyingthe Jordan curve at parabolic fixed points and making it discontinuous. The theorem asserts that every T − µ is foliated in the same way as T − . Remark.
The correspondence between the two geometries holds in any dimension d ≥
2, despite of ourrestriction to d = d = (cid:82) d − is problematic when d ≥
4, due to the existence of Pogorelov-type non-strictly convexsolutions to the underlying Monge-Ampère problem (see also [NS19, Remark 8.1]). Therefore, Theorem Bcannot be generalized to higher dimensions.
Organization of the paper.
We first give more details about the correspondence between the two ge-ometries and prove Prop. E in §2, then we explain the last three rows of Table 1 in §3. In §4, we reviewsome backgrounds about quasi-divisible convex cone and affine deformations, then prove Prop. D. Finally,in §5, we prove Thm. F using results from [NS19] and explain how the other main results of this paper,namely Thm. A, B and Cor. C, are deduced from Thm. F.
Acknowledgments.
We are deeply indebted to Thierry Barbot for the discussions related to the subjectof this paper and for his interest and encouragement. The first author would like to thank Yau Mathe-matics Sciences Center for the hospitality during the preparation of the paper.2. C
ORRESPONDENCE BETWEEN THE TWO GEOMETRIES
In this section, we prove Proposition E after giving details about the constructions involved, whichcorrespond to the first few rows of Table 1.2.1.
From C -spacelike planes to convex tube domain. A convex domain in a vector space or an affinespace is said to be proper if it does not contain any entire straight line. In a vector space, a convex cone isby definition a convex domain invariant under positive scalings. We henceforth fix a proper convex cone C ⊂ (cid:82) and a splitting (cid:82) = (cid:82) × (cid:82) such that C = { t ( x ,1) | x ∈ Ω ∗ , t > } for a bounded convex domain Ω ∗ in (cid:82) containing the origin . A point in (cid:82) will often be written in theform ( x , ξ ), where x ∈ (cid:82) and ξ ∈ (cid:82) are the “horizontal” and “vertical” coordinates, respectively. Also fix aninner product “ · ” on (cid:82) .Using the splitting, we endow the affine space (cid:65) ∼= (cid:82) with the polarization given by the point atinfinity corresponding to the vertical lines { x } × (cid:82) (see the introduction for the notions of polarization anddual affine space). We then also identify the dual affine space (cid:65) ∗ with (cid:82) in the following specific way: (cid:82) ∼ → (cid:65) ∗ : = (cid:169) non-vertical affine planes in (cid:65) (cid:170) ,(2.1) ( x , ξ ) (cid:55)→ graph of the affine function y (cid:55)→ x · y − ξ .Following [NS19], we introduce: The reason for the asterisk in the notation is that Ω ∗ will be the dual of another convex domain Ω introduced later. Although itseems more natural here to switch the notations for Ω ∗ and Ω , this would be inconvenient for our purpose because we mainly workwith Ω in this paper. FFINE DEFORMATIONS OF QUASI-DIVISIBLE CONVEX CONES 7
Definition 2.1.
A 2-dimensional subspace P ⊂ (cid:82) is said to be C-spacelike if P meets the closure C of C only at the origin 0 ∈ (cid:82) , and is said to be C-null if P ∩ C is a subset of ∂ C that is not the single point0. An affine plane P ⊂ (cid:65) is said to be C -spacelike/ C -null if the vector subspace underlying P ( i.e. thetranslation of P to 0) is. We let S C and N C denote the set of all C -spacelike and C -null planes in (cid:65) ,respectively.Since a C -spacelike or C -null plane does not contain any vertical line, we can view it as a point in (cid:65) ∗ ,hence view S C and N C as subset of (cid:65) ∗ . In order to describe these subsets, we consider the following set Ω ⊂ (cid:82) derived from the above Ω ∗ : Ω : = { x ∈ (cid:82) | x · y < y ∈ Ω ∗ } .It can be shown that Ω is also a bounded convex domain containing the origin. It is in fact the dual of Ω ∗ in the sense of Sasaki [Sas85], which explains the notation.Now the 3rd and 4th rows of Table 1 can be stated precisely as: Proposition 2.2.
The bijection (2.1) identifies the convex tube domain Ω × (cid:82) ⊂ (cid:82) and its boundary ∂ Ω × (cid:82) with the subsets S C and N C of (cid:65) ∗ , respectively. The proof is straightforward and we omit it here.2.2. Ω as a section of − C ∗ . The convex domain Ω can be interpreted geometrically as follows. Recallthat the dual cone C ∗ of C is defined as the convex cone in the dual vector space (cid:82) ∗ (the space of alllinear forms on (cid:82) ) consisting of linear forms with positive values on C \ { } . We extend the inner producton (cid:82) to (cid:82) by setting ( x , ξ ) · ( y , η ) : = x · y + ξη ,and use it to identify the dual vector space (cid:82) ∗ with (cid:82) itself. Then it is easy to check that Ω is exactlythe section of the opposite dual cone − C ∗ ⊂ (cid:82) ∗ ∼= (cid:82) by the horizontal plane (cid:82) × { − } . In other words, wecan write(2.2) − C ∗ = { t ( x , − | x ∈ Ω , t > } .The significance of this interpretation is that it identifies Ω projectively with the convex domain (cid:80) ( C ∗ )in (cid:82)(cid:80) ∗ : = (cid:80) ( (cid:82) ∗ ). We let Aut ( Ω ) denote the group of orientation-preserving projective transformationsof Ω , which identifies with the subgroup Aut ( C ∗ ) of SL(3, (cid:82) ) that preserves the cone C ∗ . Using (2.2), onechecks that the image of x ∈ Ω by the projective action of B ∈ Aut ( Ω ) has the expression B . x : = (cid:183) B (cid:181) x − (cid:182)(cid:184) , (cid:183) − B (cid:181) x − (cid:182)(cid:184) ,where [ X ] ∈ (cid:82) and [ X ] ∈ (cid:82) denote the horizontal and vertical components of X ∈ (cid:82) = (cid:82) × (cid:82) , respec-tively. Note that the natural isomorphism Aut ( C ) ∼= Aut ( C ∗ ) given by the inverse transpose A ↔ t A − induces an Aut ( C )-action on Ω , sending x ∈ Ω to t A − . x .2.3. The action of
Aut ( C ) (cid:110) (cid:82) on C -spacelike and C -null planes. We always view the semi-directproduct SL(3, (cid:82) ) (cid:110) (cid:82) as the group of special affine transformations of (cid:65) ∼= (cid:82) in such a way that ( A , X ) ∈ SL(3, (cid:82) ) (cid:110) (cid:82) represents the transformation Y (cid:55)→ AY + X . The components A and X are called the linearpart and translation part of the affine transformation, respectively.The subgroup Aut ( C ) (cid:110) (cid:82) ⊂ SL(3, (cid:82) ) (cid:110) (cid:82) of special affine transformations with linear part preserving C naturally acts on the spaces S C and N C of C -spacelike and C -null planes. The identifications in Propo-sition 2.2 translate these actions to a natural action on Ω × (cid:82) , with the following coordinate expression: XIN NIE AND ANDREA SEPPI
Proposition 2.3.
The action of
Aut ( C ) (cid:110) (cid:82) on S C ∪ N C ∼= Ω × (cid:82) is given by ( A , X ).( x , ξ ) = (cid:183) − t A − (cid:181) x − (cid:182)(cid:184) (cid:195)(cid:183) t A − (cid:181) x − (cid:182)(cid:184) , ξ + A − X · (cid:181) x − (cid:182)(cid:33) for any ( A , X ) ∈ Aut ( C ) (cid:110) (cid:82) and ( x , ξ ) ∈ Ω × (cid:82) . Comparing with the last paragraph of the previous subsection, one sees that the horizontal componentof ( A . X ).( x , ξ ) is exactly t A − . x , namely the image of x by the projective action of A . The reason for thiswill be clear in §2.5 below. Proof.
We need to determine ( x (cid:48) , ξ (cid:48) ) : = ( A , X ).( x , ξ ). Under the identification (2.1), ( x , ξ ) corresponds to theplane P that is the graph of the affine function y (cid:55)→ x · y − ξ . Equivalently, we have P = (cid:189)(cid:181) y η (cid:182) ∈ (cid:82) (cid:175)(cid:175)(cid:175)(cid:175) (cid:181) y η (cid:182) · (cid:181) x − (cid:182) = ξ (cid:190) .Let us now compute the image of P by ( A , X ):( A , X ). P = (cid:189) A (cid:181) y η (cid:182) + X (cid:175)(cid:175)(cid:175)(cid:175) (cid:181) y η (cid:182) · (cid:181) x − (cid:182) = ξ (cid:190) = (cid:189) A (cid:181) y η (cid:182) + X (cid:175)(cid:175)(cid:175)(cid:175) (cid:181) A (cid:181) y η (cid:182) + X (cid:182) · (cid:181) t A − (cid:181) x − (cid:182)(cid:182) = ξ + X · t A − (cid:181) x − (cid:182)(cid:190) = (cid:189)(cid:181) y (cid:48) η (cid:48) (cid:182) (cid:175)(cid:175)(cid:175)(cid:175) (cid:181) y (cid:48) η (cid:48) (cid:182) · (cid:181) t A − (cid:181) x − (cid:182)(cid:182) = ξ + A − X · (cid:181) x − (cid:182)(cid:190) .Dividing by (cid:183) − t A − (cid:181) x − (cid:182)(cid:184) , this shows that ( A , X ). P corresponds to the pair ( x (cid:48) , ξ (cid:48) ) ∈ Ω × (cid:82) with the re-quired expression x (cid:48) = (cid:183) t A − (cid:181) x − (cid:182)(cid:184) , (cid:183) − t A − (cid:181) x − (cid:182)(cid:184) , ξ (cid:48) = ξ + A − X · (cid:181) x − (cid:182)(cid:183) − t A − (cid:181) x − (cid:182)(cid:184) . (cid:3) Automorphisms of convex tube domains.
We refer to a subset of (cid:82) of the form T = Ω × (cid:82) as a convex tube domain if Ω is a bounded convex domain in (cid:82) . Viewing (cid:82) as an affine chart in (cid:82)(cid:80) , we areinterested in projective transformations Φ ∈ PGL(4, (cid:82) ) of (cid:82)(cid:80) preserving T . Note that such a Φ fixes thepoint at infinity p T : = T ∩ ( (cid:82)(cid:80) \ (cid:82) )of T (where T denotes the closure of T in (cid:82)(cid:80) ), which is just the common point at infinity of the verticallines { x } × (cid:82) ⊂ Ω × (cid:82) .The space of vertical lines in T naturally identifies with Ω . Since Φ sends one vertical line to another,it induces a self-mapping of this space, which is actually a projective transformation of Ω because Φ sendsa line in the space ( i.e. the set of those { x } × (cid:82) with x belonging to a line in Ω ) to another line. This gives aprojection π : { Φ ∈ PGL(4, (cid:82) ) | Φ ( T ) = T } → Aut ± ( Ω ),where we let Aut ± ( Ω ) denote the group of projective transformations of Ω , and reserve the notation Aut ( Ω )for orientation-preserving projective transformations. Definition 2.4.
A projective transformation Φ is said to be an automorphism of a convex tube domain T if it preserves T and satisfies the following extra conditions: FFINE DEFORMATIONS OF QUASI-DIVISIBLE CONVEX CONES 9 (i) Φ does not switch the two ends of T . In other words, if we endow each vertical line in T with theupward orientation, then Φ sends one line to another in an orientation-preserving way.(ii) The projection π ( Φ ) is in Aut ( Ω ) ( i.e. π ( Φ ) is orientation-preserving).(iii) The eigenvalue of Φ at p T is ± T by Aut ( T ). Remark . In Condition (iii), by an eigenvalue of Φ , we mean an eigenvalue of a representative (cid:101) Φ ∈ GL(4, (cid:82) ) of Φ ∈ PGL(4, (cid:82) ) with det( (cid:101) Φ ) = ±
1. Since there are two such (cid:101) Φ ’s opposite to each other andneither of them is privileged over the other, this eigenvalue is well defined only up to sign. Condition (iii)rules out, for example, dilations of the (cid:82) -factor.We will study graphs of (extended-real-valued) functions on Ω or ∂ Ω as geometric objects in the convextube domain Ω × (cid:82) or its boundary ∂ Ω × (cid:82) , and will freely use the following basic facts, for any Φ ∈ Aut ( Ω × (cid:82) ): • for any function u on Ω or ∂ Ω , the image of the graph gr( u ) by Φ is again the graph of somefunction (cid:101) u on Ω or ∂ Ω ; • if u is lower/upper semicontinuous, convex or smooth, so is (cid:101) u ; • if two functions u and u satisfy u ≤ u , then we also have (cid:101) u ≤ (cid:101) u .Here and below, we denote the graph of any extended-real-valued function f on a set E ⊂ (cid:82) bygr( f ) : = { ( x , ξ ) ∈ E × (cid:82) | f ( x ) = ξ } .2.5. The isomorphism
Aut ( C ) (cid:110) (cid:82) ∼= Aut ( Ω × (cid:82) ) . We now return to the setting of §2.1 ∼ Ω isinduced from a proper convex cone C ⊂ (cid:82) . Precisely, Ω is a section of − C ∗ , so that Aut ( Ω ) identifies with Aut ( C ∗ ) (see §2.2).From the expression in Prop. 2.3, one can check that Aut ( C ) (cid:110) (cid:82) acts on S C ∼= Ω × (cid:82) by automorphismsof the convex tube domain Ω × (cid:82) , hence gives a homomorphism(2.3) Aut ( C ) (cid:110) (cid:82) → Aut ( Ω × (cid:82) ).We proceed to show that (2.3) is an isomorphism, which, together with the framework built in the previoussubsections, implies Prop. E in the introduction. Proposition 2.6.
Let C and Ω be as in §2.1. Then there is a natural isomorphism Aut ( Ω × (cid:82) ) ∼= (cid:189)(cid:181) B t Y (cid:182) (cid:175)(cid:175)(cid:175)(cid:175) B ∈ Aut ( C ∗ ), Y ∈ (cid:82) (cid:190) , through which the homomorphism (2.3) can be written as (2.4) ( A , X ) (cid:55)−→ (cid:195) t A − ( A − X ) 1 (cid:33) = t (cid:195) A − X (cid:33) − . As a result, (2.3) is an isomorphism, and form the following commutative diagram together with the iso-morphism
Aut ( C ) ∼ → Aut ( C ∗ ) = Aut ( Ω ) , A (cid:55)→ t A − and the natural projections: Aut ( C ) (cid:110) (cid:82) Aut ( Ω × (cid:82) ) Aut ( C ) Aut ( Ω ) ∼∼ Proof.
Since Ω is the section of − C ∗ by (cid:82) × { − } , we can view Ω × (cid:82) ∼= Ω × { − } × (cid:82) as the section of thecone ( C ∗ ∪ ( − C ∗ )) × (cid:82) in (cid:82) by the affine plane (cid:82) × { − } × (cid:82) in (cid:82) . Thus, a projective transformation Φ ∈ PGL(4, (cid:82) ) preserves Ω × (cid:82) if and only if the linear transformation (cid:101) Φ ∈ GL(4, (cid:82) ) representing it (defined up to multiplication by scalar matrices) preserves this cone. In this case, the image ( x (cid:48) , ξ (cid:48) ) = Φ .( x , ξ ) of( x , ξ ) ∈ Ω × (cid:82) is determined by the condition(2.5) (cid:101) Φ x − ξ (cid:203) x (cid:48) − ξ (cid:48) ,where “ (cid:203) ” denotes the colinear relation of vectors. But it is elementary to check that (cid:101) Φ preserves thatcone if and only if it has the form (cid:101) Φ = (cid:195) B t Y λ (cid:33) , with B ( C ∗ ) = ± C ∗ .Here B ∈ GL(3, (cid:82) ), Y ∈ (cid:82) and λ (cid:54)= Φ ( x , ξ ) = (cid:183) − B (cid:181) x − (cid:182)(cid:184) (cid:195)(cid:183) B (cid:181) x − (cid:182)(cid:184) , λξ + t Y (cid:181) x − (cid:182)(cid:33) .Each such Φ has a unique representative (cid:101) Φ as above with B ∈ SL(3, (cid:82) ), so we henceforth let (cid:101) Φ onlydenote this representative. From the expression (2.6) of Φ .( x , ξ ), we see that the projection π ( Φ ) ∈ Aut ± ( Ω )is the projective transformation of Ω given by B ( c.f. the last paragraph of §2.2). So the three definingconditions for Φ to be an automorphisms of Ω × (cid:82) are reflected in the components λ and B of (cid:101) Φ as follows: • Condition (i) is equivalent to λ > • Condition (ii) is equivalent to B ∈ Aut ( C ∗ ) = Aut ( Ω ) (otherwise, we have B ( C ∗ ) = − C ∗ , and B givesan orientation-reversing projective transformation of Ω ). • Condition (iii) is equivalent to λ = ± Aut ( Ω × (cid:82) ) → (cid:189)(cid:181) B t Y (cid:182) (cid:175)(cid:175)(cid:175)(cid:175) B ∈ Aut ( C ∗ ), Y ∈ (cid:82) (cid:190) , Φ (cid:55)→ (cid:101) Φ is an isomorphism. Then, comparing (2.6) with the expression of the Aut ( C ) (cid:110) (cid:82) -action on Ω × (cid:82) in Prop.2.3, we see that the homomorphism (2.3) resulting from the action has the required expression (2.4). Theproof is completed by the elementary fact that (2.4) does give an isomorphism and fit into the requireddiagram. (cid:3) The discussions till now are based on defining (cid:65) ∗ as the space of non-vertical affine planes in (cid:65) . Butthe construction of dual polarized affine spaces is involutive in the sense that we can equally identify (cid:65) with the space of non-vertical affine planes in (cid:65) ∗ , which are exactly the affine planes crossing the convextube domain Ω × (cid:82) ∼= S C ⊂ (cid:65) ∗ . This identification has the following basic property: Lemma 2.7.
The identification (cid:65) ∼= (cid:169) non-vertical affine planes in (cid:65) ∗ } is equivariant with respect to theactions of Aut ( C ) (cid:110) (cid:82) ∼= Aut ( Ω × (cid:82) ) in the sense that if p , p ∈ (cid:65) correspond to the affine planes P , P ⊂ (cid:65) ∗ , respectively, and Φ ∈ Aut ( Ω × (cid:82) ) sends the section P ∩ ( Ω × (cid:82) ) to P ∩ ( Ω × (cid:82) ) , then the element of Aut ( C ) (cid:110) (cid:82) corresponding to Φ sends p to p . Note that if we let P i be the closure of P i in (cid:82)(cid:80) ( i.e. the projective plane formed by P i and its line atinfinity), then the assumption Φ (cid:161) P ∩ ( Ω × (cid:82) ) (cid:162) = P ∩ ( Ω × (cid:82) ) in the lemma is equivalent to Φ ( P ) = P , butdoes not imply Φ ( P ) = P . Proof.
Since (cid:65) ∗ is the space of non-vertical affine planes in (cid:65) and Ω × (cid:82) ⊂ (cid:65) ∗ is the subset of C -spacelikeplanes, we have P i ∩ ( Ω × (cid:82) ) = (cid:169) C -spacelike planes in (cid:65) passing through p i (cid:170) FFINE DEFORMATIONS OF QUASI-DIVISIBLE CONVEX CONES 11 for i = Aut ( C ) (cid:110) (cid:82) ∼= Aut ( Ω × (cid:82) ), the condition Φ (cid:161) P ∩ ( Ω × (cid:82) ) (cid:162) = P ∩ ( Ω × (cid:82) ) means that the affine transformation ( A , X ) ∈ Aut ( C ) (cid:110) (cid:82) corresponding to Φ sends each C -spacelike plane passing through p to one passing through p . It follows that ( A , X ) sends p to p . (cid:3) Aut ( Ω ) as a subgroup of Aut ( Ω × (cid:82) ) . Although the linear transformation group
Aut ( C ) is a quotientof the affine transformation group Aut ( C ) (cid:110) (cid:82) in a canonical way, the inclusion of the former into thelatter depends on our ad hoc choice of the identification (cid:65) ∼= (cid:82) in §2.1, or more precisely, choice of anorigin point in (cid:65) , so that Aut ( C ) is the subgroup fixing the point. Different choices give rise to conjugatecopies of Aut ( C ) in Aut ( C ) (cid:110) (cid:82) .The same thing can be said about Aut ( Ω ) and Aut ( Ω × (cid:82) ) through the isomorphism in Prop. 2.6. In fact,our coordinates on Ω × (cid:82) are so-chosen that the image of Aut ( C ) < Aut ( C ) (cid:110) (cid:82) in Aut ( Ω × (cid:82) ) is the copyof Aut ( Ω ) which consists of the automorphisms of Ω × (cid:82) preserving the zero horizontal slice Ω × { } . Moreexplicitly, by the expression (2.6) of the Aut ( Ω × (cid:82) )-action on Ω × (cid:82) given in the proof of Prop. 2.6, the actionof A ∈ Aut ( Ω ) is(2.7) A ( x , ξ ) = (cid:183) − A (cid:181) x − (cid:182)(cid:184) (cid:195)(cid:183) A (cid:181) x − (cid:182)(cid:184) , ξ (cid:33) .Observe that this action commutes with the involution ( x , ξ ) → ( x , − ξ ).We will need the following lemma about images of vertically aligned points in Ω × (cid:82) by the action of Aut ( Ω × (cid:82) ) and the subgroup Aut ( Ω ): Lemma 2.8.
Let ( x , ξ ) and ( x , ξ ) be points in Ω × (cid:82) with the same projection x ∈ Ω , and Φ ∈ Aut ( Ω × (cid:82) ) be an automorphism of Ω × (cid:82) projecting to A : = π ( Φ ) ∈ Aut ( Ω ) . Suppose Φ ( x , ξ i ) = ( x (cid:48) , ξ (cid:48) i ) , i = , wherex (cid:48) = Ax. Then(1) For any s ∈ (cid:82) , we have Φ ( x ,(1 − s ) ξ + s ξ ) = ( x (cid:48) ,(1 − s ) ξ (cid:48) + s ξ (cid:48) ). (2) Viewing Aut ( Ω ) as the subgroup of Aut ( Ω × (cid:82) ) preserving Ω × { } , we haveA ( x , ξ − ξ ) = ( x (cid:48) , ξ (cid:48) − ξ (cid:48) ). As a consequence, if u , u : Ω → (cid:82) are functions such that the graph gr( u ) ⊂ Ω × (cid:82) is preserved by Φ , then gr( u ) is also preserved by Φ if and only if gr( u − u ) is preserved by A.Proof. The map from the vertical line { x } × (cid:82) to { x (cid:48) } × (cid:82) induced by Φ is an affine transformations becauseit is a projective transformation fixing the point at infinity, or alternatively, because of the expression Φ ( x , ξ ) = (cid:183) − A (cid:181) x − (cid:182)(cid:184) (cid:195)(cid:183) A (cid:181) x − (cid:182)(cid:184) , ξ + t Y (cid:181) x − (cid:182)(cid:33) (see the proof of Prop. 2.6). Part (1) follows as a consequence because for any affine transformation f : (cid:82) → (cid:82) we have f (cid:161) (1 − s ) ξ + s ξ (cid:162) = (1 − s ) f ( ξ ) + s f ( ξ ). Part (2) can be verified directly by comparing thisexpression of Φ ( x , ξ ) and that of A ( x , ξ ) given above in (2.7). (cid:3) C - REGULAR DOMAINS AND AFFINE ( C , k )- SURFACES
In the section, we first give some background materials on convex analysis, then we review the theoryof C -regular domains, C -convex surfaces and affine ( C , k )-surfaces developed in [NS19], and explain thecorrespondences in the last three rows of Table 1. Convex functions.
In this paper, a lower semicontinuous function is assumed to take values in (cid:82) ∪ { +∞ } if not otherwise specified. Let LC ( (cid:82) ) denote the space of lower semicontinuous, convex functionson (cid:82) that are not constantly +∞ . Given u ∈ LC ( (cid:82) ), if the effective domain dom ( u ) : = { x | u ( x ) < +∞ } of u has nonempty interior U : = intdom ( u ), then the values of u on ∂ U (hence the values on the whole (cid:82) )are determined by the restriction u | U , because given any x ∈ ∂ U , it can be shown that(3.1) u ( x ) = liminf U (cid:51) x → x u ( x ) = lim s → + u ((1 − s ) x + sx ) ∈ ( −∞ , +∞ ]for any x ∈ U (see [NS19, §4.1]).Therefore, given a convex domain U ⊂ (cid:82) and a convex function u : U → (cid:82) , we define the boundaryvalue of u as the function on ∂ U whose value at x ∈ ∂ U is the liminf or limit in (3.1), which are equal andindependent of x ∈ U . We slightly abuse the notation for restrictions and denote this function by u | ∂ U .By [NS19, Prop. 4.1], the extension of u to (cid:82) given by u | ∂ U and by setting the value to be +∞ outsideof U is an element of LC ( (cid:82) ). This gives a canonical way of viewing every convex function on a convexdomain as an element of LC ( (cid:82) ), which also explains the notation u | ∂ U .In this setting, given x ∈ ∂ U , we say that u has infinite inner derivatives at x if either u ( x ) = +∞ or u ( x ) is finite but(3.2) lim s → + u ( x + s ( x − x )) − u ( x ) s = −∞ for any x ∈ U . Note that the fraction is an increasing function in s ∈ (0,1] by convexity of u , hence thelimit exists in [ −∞ , +∞ ). We refer to [NS19, §4] for more properties of this definition, especially the factthat if (3.2) holds for one x ∈ U , then it holds for all x ∈ U .Now fix a bounded convex domain Ω ⊂ (cid:82) . For any function ψ : ∂ Ω → (cid:82) ∪ { +∞ } that is bounded frombelow and is not constantly +∞ , we define the convex envelope ψ of ψ as the function on (cid:82) given by ψ ( x ) : = sup { a ( x ) | a : (cid:82) → (cid:82) is an affine function with a | ∂ Ω ≤ ψ } .The reason for the assumptions on ψ in the definition is that otherwise it would yield constant functions −∞ or +∞ , which are not interesting.The convex envelope ψ has the following fundamental properties. • ψ belongs to LC ( (cid:82) ). • ψ | ∂ Ω ≤ ψ on ∂ Ω , and the equality holds everywhere if and only if ψ is lower semicontinuous andrestricts to a convex function on any line segment in ∂ Ω (see [NS19, Lemma 4.6]). • dom ( ψ ) is the convex hull of { x ∈ ∂ Ω | ψ ( x ) < +∞ } in (cid:82) (see [NS19, Prop. 4.8]). In particular, if ψ only takes values in (cid:82) , then dom ( ψ ) = Ω . • ψ is pointwise no less than any function in LC ( (cid:82) ) majorized by ψ on ∂ Ω (see [NS19, Cor. 4.5]). Inparticular, for any convex u : Ω → (cid:82) with boundary value u | ∂ Ω ≤ ψ , we have u ≤ ψ in Ω .By the first two properties, the subset of LC ( (cid:82) ) formed by all convex envelopes can be understood asfollows. Denote LC ( ∂ Ω ) : = (cid:189) u : ∂ Ω → (cid:82) ∪ { +∞ } (cid:175)(cid:175)(cid:175)(cid:175) u is lower semicontinuous and not constantly +∞ ; therestriction of u to any line segment in ∂ Ω is convex (cid:190) .Then the assignment ϕ (cid:55)→ ϕ is a bijection from LC ( ∂ Ω ) to the subset of LC ( (cid:82) ) consisting of all functionsof the form ψ , and its inverse is just the restriction map u (cid:55)→ u | ∂ Ω .We will freely use the fact that the constructions of boundary values and convex envelopes are covariantwith respect to automorphisms of the convex tube domain Ω × (cid:82) (see §2.4) in the sense that • if u , u : Ω → (cid:82) are convex functions such that Φ ∈ Aut ( Ω × (cid:82) ) brings gr( u ) to gr( u ), then theaction of Φ on ∂ Ω × (cid:82) brings gr( u | ∂ Ω ) to gr( u | ∂ Ω ); • if ψ , ψ : ∂ Ω → (cid:82) ∪ { +∞ } are bounded from below and not constantly +∞ , such that the action of Φ ∈ Aut ( Ω × (cid:82) ) on ∂ Ω × (cid:82) brings gr( ψ ) to gr( ψ ), then Φ also brings gr( ψ ) to gr( ψ ). FFINE DEFORMATIONS OF QUASI-DIVISIBLE CONVEX CONES 13
The first bullet point can be seen from definition (3.1) of boundary values. The second follows from thedefinition of convex envelopes and the fact that automorphisms of Ω × (cid:82) send graphs of affine functions tographs of affine functions.3.2. Legendre transform.
The
Legendre transform of u ∈ LC ( (cid:82) ) is by definition the function u ∗ ∈ LC ( (cid:82) ) given by u ∗ ( y ) : = sup x ∈ (cid:82) ( x · y − u ( x )).It is a fundamental fact that the Legendre transformation u (cid:55)→ u ∗ is an involution on LC ( (cid:82) ) (see [NS19,§4.5]). If u is an (cid:82) -valued convex function only defined on a convex domain U ⊂ (cid:82) , we define its Legendretransform u ∗ by viewing u as an element of LC ( (cid:82) ) via the canonical extension described in §3.1. Notethat we have u ≤ u if and only if u ∗ ≥ u ∗ .We now give a geometric interpretation of u ∗ using the notion of dual polarized affine space from theintroduction. As in §2.1, we first identify (cid:65) with (cid:82) = (cid:82) × (cid:82) and endow it with the polarization givenby the point at infinity of vertical lines { x } × (cid:82) , so that the dual affine space (cid:65) ∗ is the set of non-verticalaffine planes in (cid:65) , which can be identified with (cid:82) as well through the map (2.1). We then view thegraphs or epigraphs of u and u ∗ as subsets of (cid:65) and (cid:65) ∗ , respectively, through the two identifications.Under this setup, we have: Proposition 3.1.
Let u ∈ LC ( (cid:82) ) be such that dom ( u ) is bounded. Then dom ( u ∗ ) = (cid:82) . In this case, given ( y , η ) ∈ (cid:82) , we have u ∗ ( y ) = η if and only if the graph of the affine function x (cid:55)→ x · y − η is a supporting planeof the graph gr( u ) ⊂ (cid:82) of u.As a consequence, if we view the graph and epigraph of u (resp. u ∗ ) as subsets of (cid:65) (resp. (cid:65) ∗ ) in theaforementioned way, then gr( u ∗ ) is exactly the set of non-vertical supporting planes of gr( u ) . Similarly, ep ◦ ( u ∗ ) ⊂ (cid:65) ∗ is the set of non-vertical affine planes in (cid:65) disjoint from ep( u ) . Here and below, by a supporting plane of a set E ⊂ (cid:82) , we mean an affine plane P ⊂ (cid:82) such that E ∩ P (cid:54)= (cid:59) and E is contained in one of the two closed half-spaces of (cid:82) with boundary P . For any extended-real-valued function f on (cid:82) , we letep( f ) : = (cid:169) ( x , ξ ) ∈ (cid:82) (cid:175)(cid:175) f ( x ) ≤ ξ (cid:170) , ep ◦ ( f ) : = (cid:169) ( x , ξ ) ∈ (cid:82) (cid:175)(cid:175) f ( x ) < ξ (cid:170) = ep( f ) \ gr( f )denote the epigraph and strict epigraph of f , respectively. Proof.
The boundedness of dom ( u ) implies that the supremum in the definition of u ∗ ( y ) is actually amaximum. More precisely, given y ∈ (cid:82) , since the upper semicontinuous function x (cid:55)→ x · y − u ( x ) is −∞ outside of the bounded set dom ( u ), it attains its maximum at some x ∈ dom ( u ), which means exactly that u ∗ ( y ) = x · y − u ( x ). In particular, we have u ∗ ( y ) < +∞ for any y ∈ (cid:82) , hence dom ( u ∗ ) = (cid:82) .As a consequence, the condition “( y , η ) ∈ gr( u ∗ )” is equivalent to “ η = x · y − u ( x ), where x is a maximalpoint of x (cid:55)→ x · y − u ( x )”. The latter condition can be written alternatively as “ u ( x ) ≥ x · y − η for all x ∈ (cid:82) ,with equality achieved at x ”, which means exactly that gr( x (cid:55)→ x · y − η ) is a supporting plane of gr( u ) at x . It follows that gr( u ∗ ) is the set of non-vertical supporting planes of ep( u ), because the identification (cid:82) ∼= (cid:65) ∗ is so defined that ( y , η ) ∈ (cid:82) corresponds to the non-vertical affine plane gr( x (cid:55)→ x · y − η ). The laststatement is similar. (cid:3) Example 3.2.
Figure 3.1 shows the graphs of some radially symmetric convex functions u on the unitdisk (cid:68) : = { x ∈ (cid:82) | | x | < } and their Legendre transforms u ∗ . Each example of u is pointwise smallerthan the next one. In the first and the last examples, where u = u ( x ) = (cid:112) − | x | , respectively,the graph of u ∗ is the boundary of the future light cone C : = (cid:169) ( x , ξ ) ∈ (cid:82) | | x | < ξ (cid:170) and the hyperboloid (cid:72) : = (cid:169) ( x , ξ ) ∈ (cid:82) | | x | + = ξ (cid:170) in C , respectively.The following lemma relates the Legendre transform of a convex C -function with the gradient, andgive useful interpretations of the gradient-blowup property: Lemma 3.3.
Let U ⊂ (cid:82) be a convex domain and u : U → (cid:82) be a strictly convex C -function. Then F IGURE (cid:82) . (1) The gradient map x (cid:55)→ D u ( x ) is a homeomorphism from U to the convex domain D u ( U ) : = (cid:169) D u ( x ) | x ∈ U (cid:170) ⊂ (cid:82) (2) The restriction of the Legendre transform u ∗ ∈ LC ( (cid:82) ) to D u ( U ) is given byu ∗ ( D u ( x )) = x · D u ( x ) − u ( x ), ∀ x ∈ U . (3) The following conditions are equivalent to each other: • u has infinite inner derivative at every point of ∂ U (see §3.1); • (cid:107) D u ( x ) (cid:107) tends to +∞ as x ∈ U tends to ∂ U.Furthermore, if U is bounded, both conditions are equivalent to D u ( U ) = (cid:82) . See Example 3.2 again for illustrations of these results. Among the four instances of u in Figure 3.1,the 2nd and the 4th are smooth and strictly convex in (cid:68) , and only the 4th has infinite inner derivativesat boundary points. In the 2nd one, the graph of u ∗ over D u ( (cid:68) ) is the exactly the part of gr( u ∗ ) in C . Proof. (1) Since x (cid:55)→ D u ( x ) is continuous in U , by Brouwer Invariance of Domain, it suffices to show theinjectivity. Suppose by contradiction that D u ( x ) = D u ( x ) for x , x ∈ U . Then the graph of the convexfunction x (cid:55)→ u ( x ) − x · D u ( x ) has horizontal supporting planes at both x and x , which implies that thefunction is a constant on the line segment joining x and x . It follows that u is an affine function on thatsegment, contradicting the strict convexity.(2) Since u is C at every x ∈ U , the only supporting plane of the graph gr( u ) at ( x , u ( x )) is thetangent plane, which is the graph of the affine function x (cid:55)→ u ( x ) + ( x − x ) · D u ( x ) = x · D u ( x ) − (cid:161) x · D u ( x ) − u ( x ) (cid:162) .It follows from Prop. 3.1 that u ∗ ( D u ( x )) = x · D u ( x ) − u ( x ), as required.(3) We view u as an element of LC ( (cid:82) ) by extending it to the whole (cid:82) in the way described in §3.1.By [NS19, Prop. 4.13], for any x ∈ ∂ U , the following conditions are equivalent to each over (Condition (c)below is formulated as (iii) in [NS19, Prop. 4.13] using the notion of subgradients ):(a) u has finite inner derivative at x ;(b) there is a sequence of points ( x i ) i = ··· in U tending to x such that (cid:107) D u ( x i ) (cid:107) does not tend to +∞ ;(c) the graph gr( u ) has a non-vertical supporting plane at ( x , u ( x )).The first (resp. second) bullet point in the required statement means exactly that there does not exist x ∈ ∂ U satisfying (a) (resp. (b)), so the two bullet points are equivalent to each other.To complete the proof, we now assume U is bounded and only need to show that D u ( U ) (cid:54)= (cid:82) if and onlyif gr( u ) has a non-vertical supporting plane at ( x , u ( x )) for some x ∈ ∂ U .Suppose gr( u ) has a non-vertical supporting plane P at ( x , u ( x )). We have P = gr( x (cid:55)→ x · y − η ) forsome y ∈ (cid:82) , η ∈ (cid:82) . With the same reasoning as in the proof of Part (1), we see that D u ( x ) (cid:54)= y for any FFINE DEFORMATIONS OF QUASI-DIVISIBLE CONVEX CONES 15 x ∈ U , otherwise u would be an affine function on the line segment joining x and x , contradicting thestrict convexity. This shows the “if” part.Conversely, suppose D u ( U ) (cid:54)= (cid:82) and pick y ∈ (cid:82) \ D u ( U ). By Prop. 3.1, we have η : = u ∗ ( y ) < +∞ , hence P : = gr( x (cid:55)→ x · y − η ) is a supporting plane of gr( u ). But P cannot be the supporting plane of gr( u ) at anypoint in U because y ∉ D u ( U ), hence must be a supporting plane at some x ∈ ∂ U . This shows the “onlyif” part and completes the proof. (cid:3) Now fix a proper convex cone C ⊂ (cid:82) and let Ω ⊂ (cid:82) be the corresponding convex domain as in §2, sothat there is an isomorphism Aut ( C ) (cid:110) (cid:82) ∼= Aut ( Ω × (cid:82) ) (see Prop. 2.6). We henceforth only consider convexfunctions u ∈ LC ( (cid:82) ) with dom ( u ) ⊂ Ω . We shall view the graph of u over Ω and the graph gr( u ∗ ) of theLegendre transform of u as geometric objects in the convex tube domain Ω × (cid:82) ⊂ (cid:65) ∗ and the affine space (cid:65) , respectively, which are acted upon by Aut ( Ω × (cid:82) ) and Aut ( C ) (cid:110) (cid:82) , respectively (comparing to Prop.3.1, the roles of (cid:65) and (cid:65) ∗ are switched here).It follows from Prop. 3.1 that the two actions are actually intertwined. We now give a formal statementonly for convex C -functions: Lemma 3.4.
Let C and Ω be as in §2.1, Φ be an element of Aut ( Ω × (cid:82) ) , and u , u ∈ C ( Ω ) be convexfunctions such that Φ brings the graph gr( u ) to gr( u ) . Let Σ i be the graph over D u i ( Ω ) of the Legendretransform u ∗ i (i = ), i.e. Σ i : = gr( u ∗ i | D u i ( Ω ) ) = (cid:169)(cid:161) D u i ( x ), x · D u i ( x ) − u i ( x ) (cid:162)(cid:175)(cid:175) x ∈ U i (cid:170) ⊂ (cid:82) ∼= (cid:65) (see Lemma 3.3 (2)). Then Σ is the image of Σ by the element of Aut ( C ) (cid:110) (cid:82) which corresponds to Φ underthe isomorphism Aut ( C ) (cid:110) (cid:82) ∼= Aut ( Ω × (cid:82) ) . Note that by Lemma 3.3 (3), Σ i is an entire graph ( i.e. D u i ( Ω ) = (cid:82) ) only when u i has the gradient-blowup property. Proof.
If we identify (cid:65) as at the set of non-vertical affine planes in (cid:65) ∗ (see the last part of §2.5), then byProp. 3.1, Σ i ⊂ (cid:65) consists exactly of the tangent planes to the graph gr( u i ) ⊂ Ω × (cid:82) ⊂ (cid:65) ∗ . So the requiredstatement follows from Lemma 2.7. (cid:3) C -regular domains and C -convex surfaces. In view of the notions of C -null and C -spacelikeplanes introduced in §2.1, we call a closed half-space H ⊂ (cid:65) C -null (resp. C -spacelike) if H contains atranslation of the cone C and the boundary plane ∂ H is C -null (resp. C -spacelike). In other words, a C -null/spacelike half-space is the upper half of (cid:65) cut out by a C -null/spacelike plane. We further define: • A C-regular domain is a nonempty, open, proper subset D (cid:40) (cid:65) which is the interior of the in-tersection of a collection S of C -null half-spaces . If S consists of all the C -null half-spacescontaining a set E ⊂ (cid:65) , we call D the C -regular domain generated by E . • A C-convex surface is an open subset Σ of the boundary of some convex domain U ⊂ (cid:82) such thatthe supporting half-space of U at every point of Σ is C -spacelike. Σ is said to be complete if it isproperly embedded, or equivalent, it is the entire boundary of U .Here, by a supporting half-space of a convex domain U ⊂ (cid:82) at a boundary point p ∈ ∂ U , we mean a closedhalf-space H ⊂ (cid:82) containing U such that p ∈ ∂ H . Remark . For the future light cone C of the Minkowski space (cid:82) , C -regular domains are classicallyknown as regular domains or domains of dependence , whereas C -convex surfaces are just future-convex ,spacelike surfaces. See [NS19, §3.1] for more discussions about these definitions and the backgrounds.In order to identify the space of C -convex surfaces, we introduced the following subset of LC ( (cid:82) ) ( c.f. §3.1): let S ( Ω ) denote the set of all u ∈ LC ( (cid:82) ) satisfying The definitions in [NS19] and this paper have two inessential differences: first, the empty set and the whole (cid:65) are bothconsidered as C -regular domains in [NS19], but not here; second, in [NS19], the function ϕ ≡ +∞ is included as an element of LC ( ∂ Ω ) (which corresponds to D = (cid:65) in the sense of Thm. 3.6 (1) below), but not here. - U : = intdom ( u ) is nonempty and contained in Ω (see §3.1 for the notation);- u is smooth and locally strongly convex in U ;- the norm of gradient (cid:107) D u ( x ) (cid:107) tends to +∞ as x ∈ U tends to ∂ U .Here, a C -function is said to be locally strongly convex if its Hessian is positive definite (which is strongerthan being locally strictly convex). Although a general u ∈ S ( Ω ) may take the value +∞ in Ω , in applica-tions later on, we mainly consider those u ’s such that u | ∂ Ω is (cid:82) -valued. In this case, u is (cid:82) -valued in Ω aswell and can be viewed as an element of C ∞ ( Ω ).Using the notations introduced in the previous subsections, we can formulate the correspondences inthe 2nd-to-last and 3rd-to-last rows of Table 1 as: Theorem 3.6 ([NS19, Thm. 5.2 and 5.13]) . Let C and Ω be as in §2.1. Then the following statements hold.(1) The assignment ϕ (cid:55)→ D = ep ◦ ( ϕ ∗ ) gives a bijection from LC ( ∂ Ω ) to the space of all C-regular domainsin (cid:65) . Moreover, D is proper (see §2.1) if and only if the convex hull of dom ( ϕ ) : = { x ∈ ∂ Ω | ϕ ( x ) < +∞ } in (cid:82) has nonempty interior.(2) The assignment u (cid:55)→ Σ = gr( u ∗ ) gives a bijection from S ( Ω ) to the space of all smooth, stronglyconvex, complete, C-convex surfaces in (cid:65) .(3) Let u ∈ S ( Ω ) and suppose ϕ : = u | ∂ Ω is not constantly +∞ (hence belongs to LC ( ∂ Ω ) ). Then theC-regular domain D = ep ◦ ( ϕ ∗ ) is generated by the C-convex surface Σ = gr( u ∗ ) .(4) Suppose u ∈ S ( Ω ) and ϕ ∈ LC ( ∂ Ω ) . Then the C-convex surface Σ = gr( u ∗ ) is asymptotic to theboundary of the C-regular domain D = ep ◦ ( ϕ ∗ ) if and only if the following conditions are satisfied: • dom ( u ) coincides with dom ( ϕ ) (it is shown in §3.1 that the latter set equals the convex hull of dom ( ϕ ) ⊂ ∂ Ω in (cid:82) ); • ϕ ( x ) − u ( x ) tends to as x ∈ intdom ( u ) tends to the boundary of dom ( u ) .In particular, we have ϕ = u | ∂ Ω in this case. In the last part, “ Σ is asymptotic to ∂ D ” means the distance from p ∈ Σ to ∂ D tends to 0 as p goes toinfinity in Σ . The last two parts of the theorem actually imply that this condition is in general strictlystronger than “ Σ generates D ” . Example 3.7.
The first and the last examples in Example 3.2 are the simplest cases of Parts (1) and (2)of the above theorem, respectively, which yield the light cone C itself as a C -regular domain and thehyperboloid (cid:72) as a complete C -convex surface. In contrast, in the 2nd and 3rd examples, the convexfunction u is neither a convex envelope nor contained in S ( Ω ), hence gr( u ∗ ) is neither the boundary of a C -regular domain nor a complete C -convex surface. Nevertheless, a part of gr( u ∗ ) is still an incomplete C -convex surface in both examples. See Remark 3.8 below. Remark . By Lemma 3.3 (2), the C -convex surface in Part (2) of the theorem can be written asgr( u ∗ ) = (cid:169)(cid:161) D u ( x ), x · D u ( x ) − u ( x ) (cid:162)(cid:175)(cid:175) x ∈ intdom ( u ) (cid:170) ,and the completeness of the surface corresponds to the gradient blowup condition in the definition of S ( Ω ) via Lemma 3.3 (3). Putting aside the completeness condition, one can also show that for any convex C -function u on an open set U ⊂ Ω , the surface (cid:169)(cid:161) D u ( x ), x · D u ( x ) − u ( x ) (cid:162)(cid:175)(cid:175) x ∈ U (cid:170) , i.e. the graph of u ∗ over D u ( U ), is a C -convex surface, although the whole graph gr( u ∗ ) might be not. Remark . In [NS19], we also considered a set of functions S ( Ω ) ⊂ LC ( (cid:82) ) containing S ( Ω ), whosedefinition does not assume any differentiability or strict convexity, and showed in [NS19, Thm. 5.2] that S ( Ω ) identifies with the space of all complete C -convex surfaces (not necessarily smooth or strictly convex)in (cid:65) . This generalizes Thm. 3.6 (2) above. See also [NS19, Example 5.14] for an example of a function in S ( Ω ) \ S ( Ω ) and the corresponding C -convex surface, which is not strictly convex. If ϕ ∈ C ( ∂ Ω ), the two conditions are equivalent (see [NS19, p.30]). In contrast, an example of a complete C -convex surfacegenerating a C -regular domain but not asymptotic to the boundary of the domain is given in [NS19, Example 5.14], where we have ϕ = +∞ on a part of ∂ Ω . There also exist examples where ϕ is (cid:82) -valued and bounded. FFINE DEFORMATIONS OF QUASI-DIVISIBLE CONVEX CONES 17
The geometric interpretation of Legendre transformation in Prop. 3.1 provides the following moredetailed descriptions of the bijections (1) and (2) (the roles of (cid:65) and (cid:65) ∗ in Prop. 3.1 are switched here inorder to be consistent with our convention that C -regular domains live in (cid:65) and convex tube domains in (cid:65) ∗ , so we are actually viewing (cid:65) as the space of non-vertical affine planes in (cid:65) ∗ ): • Given ϕ ∈ LC ( ∂ Ω ), points in the C -regular domain D = ep ◦ ( ϕ ∗ ) ⊂ (cid:65) are exactly the non-verticalaffine planes in (cid:65) ∗ disjoint from the epigraph ep( ϕ ) ⊂ ∂ Ω × (cid:82) ⊂ (cid:65) ∗ , or in other words, the affineplanes crossing Ω × (cid:82) from below of gr( ϕ ). • Given u ∈ S ( Ω ), points on the C -convex surface Σ = gr( u ∗ ) are exactly the non-vertical supportingplanes of the surface gr( u ) ⊂ Ω × (cid:82) .3.4. Hyperbolic affine spheres.
The theory of
Affine Differential Geometry studies properties of sur-faces in the affine space (cid:65) that are invariant under special affine transformations. A crucial ingredi-ent in the theory is the fact that any smooth locally strongly convex surface Σ ⊂ (cid:65) carries a canonicaltransversal vector field N Σ , called the affine normal field , pointing towards the convex side of Σ . Using N Σ , one can define the affine shape operator S Σ of Σ , which is a smooth section of End( T Σ ), in the sameway as in classical surface theory, and then define the affine Gaussian curvature κ Σ : Σ → (cid:82) to be thedeterminant det( S Σ ) (see e.g. [NS94] for details).We are interested in certain Σ ’s with Constant Affine Gaussian Curvature (CAGC). A simple yet crucialsub-class of them are hyperbolic affine spheres , which are by definition those Σ with S Σ = id. This conditionis equivalent to the existence of a center o ∈ (cid:65) of Σ such that at any p ∈ Σ , the affine normal N Σ ( p ) equalsthe vector −→ op .The above discussion is local in nature. We henceforth restrict ourselves to “global” hyperbolic affinespheres, i.e. properly embedded ones, of which the simplest example is the hyperboloid (cid:72) in the lightcone C ⊂ (cid:82) , . These affine spheres are classified by a theorem of Cheng and Yau [CY77], stating thatthey are in 1-to-1 correspondence with proper convex cones in (cid:82) , just in the way how (cid:72) corresponds to C . Namely, each cone C contains a unique properly embedded hyperbolic affine sphere, denoted by Σ C ,which is asymptotic to ∂ C , and conversely every properly embedded affine sphere centered at 0 is some Σ C for a unique C .In the proof of Cheng-Yau’s theorem, one encodes a convex surface Σ by a function on a bounded convexdomain, and translates the geometrical condition that Σ is an affine sphere to a PDE on that function.Using the framework in §3.3, we can formulate this translation process and the statement of the theoremitself as: Theorem 3.10 ([CY77]) . For any bounded convex domain Ω ⊂ (cid:82) , there exists a unique convex functionw Ω ∈ C ( Ω ) ∩ C ∞ ( Ω ) solving the Dirichlet problem of Monge-Ampère equation (cid:40) det D w = w − , w | ∂ Ω = Furthermore, w Ω has the following properties: • the norm of gradient (cid:107) D w Ω ( x ) (cid:107) tends to +∞ as x ∈ Ω tends to ∂ Ω ; • the graph gr( w Ω ) ⊂ Ω × (cid:82) is invariant under the action of Aut ( Ω ) , viewed as the subgroup of Aut ( Ω × (cid:82) ) preserving the slice Ω × { } (see §2.6).Moreover, if C and Ω are as in §2.1, then the graph Σ C = gr( w ∗ Ω ) of the Legendre transform w ∗ Ω is the uniquehyperbolic affine sphere asymptotic to ∂ C. The second bullet point is not included in [CY77] but is equivalent to the fact that the affine sphere Σ C = gr( w ∗ Ω ) is invariant under automorphisms of C , which is in turn a consequence of the uniqueness of Σ C , or equivalently, the uniqueness of w Ω . See Lemma 3.4 for a precise explanation of these equivalences.We refer to w Ω as the Cheng-Yau support function of Ω . The simplest example is already given inExample 3.2: we have w (cid:68) ( x ) = (cid:112) − | x | for the unique disk (cid:68) , and the corresponding affine sphere is just Σ C = (cid:72) . Affine ( C , k ) -surfaces. We proceed to consider a wider sub-class of smooth strongly convex CAGCsurfaces. By [NS19, §3.2], a sufficient (but not necessary) condition for κ Σ to be a positive constant k isthat the surface N Σ ( Σ ) in the vector space (cid:82) ( i.e. the surface formed by all affine normal vectors of Σ ) iscontained in a scaled affine sphere of the form k Σ (cid:48) , where Σ (cid:48) is some hyperbolic affine sphere centeredat 0. If in particular Σ (cid:48) is the Cheng-Yau affine sphere Σ C for a proper convex cone C , we call Σ an affine ( C , k ) -surface . Remark . While most of the properties studied in Affine Differential Geometry, such as the propertyof having CAGC, are invariant under special affine transformations, the property of being an affine ( C , k )-surface is only invariant under Aut ( C ) (cid:110) (cid:82) . In fact, a general special affine transformation ( A , X ) ∈ SL(3, (cid:82) ) (cid:110) (cid:82) sends an affine ( C , k )-surface to an affine ( A ( C ), k )-surface.It is easy to see that an affine ( C , k )-surface is C -convex in the sense of §3.3 (see [NS19, §3.2] fordetails). Therefore, by Theorem 3.6 (2), every complete affine ( C , k )-surface can be written as gr( u ∗ ) forsome u ∈ S ( Ω ). It is essentially shown in [LSC97] that the exact condition on such a u for gr( u ∗ ) to bean affine ( C , k )-surface is a Monge-Ampère equation involving the function w Ω from Thm. 3.10, whichis itself the solution to a Monge-Ampère equation (hence called a “two-step Monge-Ampère equation” in[LSC97]). By [NS19, §7.2], we can formulate this result as follows, also covering incomplete pieces of C -convex surfaces discussed in Remark 3.8: Proposition 3.12.
Let C and Ω be as in §2.1, U ⊂ Ω be an open set and u ∈ C ∞ ( U ) be strictly convex. Let Σ denote the graph over D u ( U ) of the Legendre transform u ∗ , i.e. Σ : = gr( u ∗ | D u ( U ) ) = (cid:169)(cid:161) x , x · D u ( x ) − u ( x ) (cid:162)(cid:175)(cid:175) x ∈ U (cid:170) ⊂ (cid:82) ∼= (cid:65) (see Lemma 3.3 (2) and Remark 3.8). Fix k > . Then Σ is an affine ( C , k ) -surface if and only if u satisfiesthe Monge-Ampère equation det D u = k − w − Ω in U. As a consequence, the -to- correspondence in Thm. 3.6 (2) restricts to a correspondence betweenall complete affine ( C , k ) -surfaces and those u ∈ S ( Ω ) which satisfies the above equation in the interior of dom ( u ) . This proposition is a more detailed version of [NS19, Cor. 7.5] (which only deals with complete affine( C , k )-surfaces) and the proof is the same.3.6. Foliation by C -convex surfaces. In view of Theorem 3.6 (2), we further ask the following question:Let ( u t ) t ∈ (cid:82) be a one-parameter family of functions in S ( Ω ) with common boundary value ϕ ∈ LC ( ∂ Ω ), sothat Σ t = gr( u ∗ t ) ( t ∈ (cid:82) ) are a family of complete C -convex surfaces generating the same C -regular domain D = ep ◦ ( ϕ ∗ ), ϕ ∈ LC ( ∂ Ω ). Then when is ( Σ t ) t ∈ (cid:82) a foliation of D ?In [NS19, Thm. 5.15], we gave a necessary and sufficient condition on ( u t ) for ( Σ t ) to be a convex foliation in the sense that the leaves are the level surfaces of a convex function on D . Restricting to thecase where ϕ only takes values in (cid:82) , we have: Proposition 3.13 ([NS19, Thm. 5.15]) . Let ϕ : ∂ Ω → (cid:82) be a lower semicontinuous function and let ( u t ) t ∈ (cid:82) ⊂ S ( Ω ) be such that u t | ∂ Ω = ϕ for every t. Then the following conditions are equivalent: • for every fixed x ∈ Ω , t (cid:55)→ u t ( x ) is a strictly increasing concave function on (cid:82) , with value tending to −∞ and ϕ ( x ) as t tends to −∞ and +∞ , respectively; • there is a convex function K : ep ◦ ( ϕ ∗ ) → (cid:82) such that gr( u ∗ t ) = K − ( t ) for every t ∈ (cid:82) . Since concave functions from (cid:82) to (cid:82) are continuous, the first bullet point implies that the graphs of the u t ’s themselves form a foliation of the strict lower epigraph T − : = (cid:169) ( x , ξ ) ∈ Ω × (cid:82) (cid:175)(cid:175) ξ < ϕ ( x )) (cid:170) of ϕ . Therefore, we can define a map F : T − → D FFINE DEFORMATIONS OF QUASI-DIVISIBLE CONVEX CONES 19 in the following way: given p ∈ T − , pick the leaf passing through p in the foliation (gr( u t )), then let F ( p )be the supporting plane of this leaf at p , which can be understood as a point in D ( c.f. the last paragraphof §3.3). See Figure 3.2.F IGURE F .By Prop. 3.1 and Lemma 3.3, F sends each leaf gr( u t ) in T − to the leaf gr( u ∗ t ) in D , and has theexpression F ( x , u t ( x )) = (cid:161) D u t ( x ), x · D u t ( x ) − u t ( x ) (cid:162) .As an ingredient in the proof of Corollary C, we shall show: Proposition 3.14.
Suppose ( u t ) satisfies the conditions in Prop. 3.13 and further assume that the commonboundary value ϕ of the u t ’s is a bounded function. Then the map F defined above is a homeomorphism.Proof. By the above expression of F , we can write F = F ◦ F − with the maps F and F defined as follows: F : Ω × (cid:82) → T − , F ( x , t ) : = (cid:161) x , u t ( x ) (cid:162) F : Ω × (cid:82) → D , F ( x , t ) : = (cid:161) D u t ( x ), x · D u t ( x ) − u t ( x ) (cid:162) .We shall prove the proposition by showing that F and F are both homeomorphisms.The assumption clearly implies that F is bijective. F is also bijective because on one hand, by Lemma3.3, F sends each slice Ω × { t } bijectively to the graph gr( u ∗ t ); on the other hand, (gr( u ∗ t )) t ∈ (cid:82) is a foliationof D by Prop. 3.13.We proceed to show that F and F are continuous via the following two claims. Claim 1: For any t ∈ (cid:82) and compact set Ω ⊂ Ω , u t converges to u t uniformly on Ω as t → t . Suppose by contradiction that it is not the case. Then there exists a sequence ( t n ) n = , , ··· in (cid:82) convergingto t and a sequence ( x n ) n = , , ··· in Ω such that | u t n ( x n ) − u t ( x n ) | ≥ δ for all n and some fixed δ >
0. Byrestricting to a subsequence, we may assume that x n converges to some x ∈ Ω as n → ∞ , and that either t n > t for all n or t n < t for all n . Moreover, since u t n ( x ) → u t ( x ) as by assumption, we may furtherassume that x n (cid:54)= x for all n .We first treat the t n > t case, in which we have u t n ( x n ) − u t ( x n ) ≥ δ because u t ( x ) is increasing in t .Since u t ( x n ) → u t ( x ) by continuity of u t and u t n ( x ) → u t ( x ) by assumption, we have u t n ( x n ) − u t n ( x ) = (cid:161) u t n ( x n ) − u t ( x n ) (cid:162) + (cid:161) u t ( x n ) − u t ( x ) (cid:162) + (cid:161) u t ( x ) − u t n ( x ) (cid:162) ≥ δ n is large enough. Let x (cid:48) n and x (cid:48)(cid:48) n be the boundary points of Ω such that x (cid:48) n , x , x n and x (cid:48)(cid:48) n lie on thesame line in this order. We shall compare the restriction of u t n to this line with the affine function h n onthe line which takes the same values as u t n at x and x n . Namely, h n is given by h n ( x + s ( x n − x )) = u t n ( x ) + s (cid:161) u t n ( x n ) − u t n ( x ) (cid:162) for all s ∈ (cid:82) , and we have u t n ( x (cid:48)(cid:48) n ) ≥ h n ( x (cid:48)(cid:48) n ) by convexity of u t n . It follows that ϕ ( x (cid:48)(cid:48) n ) = u t n ( x (cid:48)(cid:48) n ) ≥ h n ( x (cid:48)(cid:48) n ) = u t n ( x ) + x (cid:48)(cid:48) n − x x n − x (cid:161) u t n ( x n ) − u t n ( x ) (cid:162) . This leads to a contradiction: The right-hand side tends to +∞ as n → ∞ because u t n ( x ) → u t ( x ), x (cid:48)(cid:48) n − x x n − x → +∞ and u t n ( x n ) − u t n ( x ) ≥ δ , whereas the left-hand side is bounded by assumption.In the t n < t case, we can replace x (cid:48)(cid:48) n by x (cid:48) n in the above argument (note that x (cid:48) n − x x n − x → −∞ ) and arriveat a contradiction in the same way. This proves Claim 1. Claim 2: The map ( x , t ) → D u t ( x ) is continuous. Pick ( x , t ) ∈ Ω × (cid:82) . Since modifying ( u t ) byadding the same affine function to every u t does not affect the statement, we may assume u t ( x ) = D u t ( x ) =
0. As a consequence, there is a constant C >
0, such that when r > | u t ( x ) | ≤ Cr , ∀ x ∈ B ( x , r )Given such an r , by Claim 1, we find δ r > | u t ( x ) | ≤ | u t ( x ) − u t ( x ) | + | u t ( x ) | ≤ Cr , ∀ x ∈ B ( x , r ), t ∈ ( t − δ r , t + δ r ).We now use (3.3) to show that for any r as above, we have(3.4) | D u t ( x ) | ≤ Cr , ∀ ( x , t ) ∈ B ( x , r /2) × ( t − δ r , t + δ r ),which would imply the claim. To this end, we fix ( x , t ) ∈ B ( x , r /2) × ( t − δ r , t + δ r ) and consider thesupporting affine function h of u t at x , i.e.h ( x ) : = u t ( x ) + D u t ( x ) · ( x − x ).We have h ≤ u t by convexity of u t , hence h ≤ Cr on B ( x , r ) by (3.3). On the other hand, we have h ( x ) = u t ( x ) ≥ − Cr also by (3.3). Therefore, (3.4) follows from the lemma below, and the proof of Claim2 is finished.The two claims imply that the maps F and F are continuous. Since F and F are already shown tobe bijective, by Brouwer Invariance of Domain, they are indeed homeomorphisms, as required. (cid:3) Lemma 3.15.
Let (cid:178) , r > and x ∈ (cid:82) . If an affine function h ( x ) = x · y + η (y ∈ (cid:82) , η ∈ (cid:82) ) on (cid:82) satisfies • h ≤ (cid:178) on the boundary of the disk B ( x , r ) , • h ( x ) ≥ − (cid:178) for some x ∈ B ( x , r /2) ,then the gradient y of h satisfies | y | ≤ (cid:178) r . Proof.
Since shifting h by a translation of (cid:82) does not affect the statement, we may assume x =
0. Theassumptions imply (cid:178) ≥ h (cid:161) r | y | y (cid:162) = r | y | + η , − (cid:178) ≤ h ( x ) = x · y + η ≤ | x || y | + η ≤ r | y | + η .Subtracting the first inequality by the second, we get 2 (cid:178) ≥ r | y | , which implies the required inequality. (cid:3)
4. A
FFINE DEFORMATIONS OF QUASI - DIVISIBLE CONVEX CONES
In this section, we first review backgrounds on parabolic automorphisms of convex cones, then wedefine admissible cocycles and prove Prop. D in the introduction.4.1.
Quasi-divisibility and parabolic elements.
For the future light cone C in (cid:82) , the automor-phism group Aut ( C ) is the identity component SO (2,1) of SO(2,1) and identifies with the group Isom + ( (cid:72) )of orientation-preserving isometries of the hyperboloid (cid:72) ⊂ C . It is well known that non-identity ele-ments in this group are classified into three types, namely elliptic, hyperbolic and parabolic ones, andthat the fundamental group of a complete hyperbolic surface with finite volume, viewed as a Fuchsiangroup in SO (2,1), only contains the last two types of elements, with parabolic ones corresponding to thecusps of the surface. FFINE DEFORMATIONS OF QUASI-DIVISIBLE CONVEX CONES 21
More generally, an element A (cid:54)= I in SL(3, (cid:82) ) is said to be parabolic if it is conjugate to a parabolicelement of SO(2,1), or equivalently, if A has the Jordan form ,whereas A is said to be hyperbolic if it is diagonalizable. Both types of elements can occur in the automor-phism group Aut ( C ) of a proper convex cone C ⊂ (cid:82) , and their action can be visualized in the projectivizedpicture as Figure 4.1. hyperbolic parabolic F IGURE
Aut ( C ).For a parabolic element A ∈ Aut ( C ), the eigenline and the 2-dimensional generalized eigenspace are,respectively, the unique line in ∂ C fixed by A and the unique C -null subspace of (cid:82) ( c.f. Definition 2.1)fixed by A . In the projectivized picture, they correspond to a point p A ∈ ∂ (cid:80) ( C ) and the tangent line of ∂ (cid:80) ( C ) at p A , respectively. As shown by Benoist-Hulin [BH13], there is a pencil of conics in (cid:82)(cid:80) based at p A , in which every conic is preserved by A , and ∂ (cid:80) ( C ) is pinched between two of them (see Figure 4.1).We formulated this result as: Lemma 4.1 ([BH13, Prop. 3.6 (3)]) . Suppose C ⊂ (cid:82) is a proper convex cone and A ∈ Aut ( C ) is parabolic.Then there are projective disks D , D in (cid:82)(cid:80) preserved by A, such that D ⊂ (cid:80) ( C ) ⊂ D , and the boundaries ∂ D and ∂ D only touch at the fixed point of A. The definition of divisible and quasi-divisible convex cones are briefly reviewed in the introduction,and we refer to [Ben08, Mar10] for details. However, instead of the definition itself, we will rather usethe following characterization of quasi-divisibility due to Marquis, which generalizes the aforementionedfact for Fuchsian groups:
Theorem 4.2 ([Mar12]) . Let C ⊂ (cid:82) be a proper convex cone and Γ be a torsion-free discrete subgroup of SL(3, (cid:82) ) contained in Aut ( C ) . Then C is quasi-divisible by Γ if and only if S : = (cid:80) ( C )/ Γ is homeomorphicto a closed surface with finitely many (possibly zero) punctures and every puncture has a neighborhood ofthe form D / 〈 A 〉 , where D ⊂ (cid:80) ( C ) is a projective disk and A ∈ Γ is a parabolic element preserving D. In thiscase, the holonomy of every non-peripheral loop on S is hyperbolic. We collect some other well known results about quasi-divisible convex cones that will be used later on:
Proposition 4.3.
Let C ⊂ (cid:82) be a proper convex cone quasi-divisible by a torsion-free group Γ < SL(3, (cid:82) ) .Then the following statements hold.(1) ∂ (cid:80) ( C ) is strictly convex and C .(2) In ∂ (cid:80) ( C ) , every Γ -orbit is dense.(3) Suppose γ , γ ∈ Γ \ { I } are primitive (i.e. γ i is not a positive power of any other element of Γ ). Then γ and γ share a fixed point if and only if γ = γ ± . (4) Let Γ ∗ < SL(3, (cid:82) ) be the image of Γ under the isomorphism Aut ( C ) ∼= Aut ( C ∗ ) (see §2.2). Then thedual cone C ∗ of C is quasi-divisible by Γ ∗ . Parts (1) and (4) are Thm. 0.5 and Thm. 0.4, respectively, of the aforementioned work of Marquis[Mar12], whereas (2) and (3) are generalizations of well known facts for Fuchsian groups and can beshown with the same argument as in the Fuchsian case (see e.g. [Bea95, §5])4.2.
Affine transformation with parabolic linear part.
In order to study affine deformations ofquasi-divisible convex cones, let us consider affine transformations ( A , X ) ∈ Aut ( C ) (cid:110) (cid:82) such that A isparabolic. The following fundamental property of such ( A , X )’s is the origin of our definition of admissiblecocycles: Proposition 4.4.
Let C ⊂ (cid:82) a proper convex cone and let ( A , X ) ∈ Aut ( C ) (cid:110) (cid:82) be such that A is parabolic.Let L ⊂ (cid:82) be the eigenline of A and P ⊂ (cid:82) be the C-null subspace preserved by A. Then the followingconditions are equivalent to each other:(a) the vector X lies in P ;(b) ( A , X ) is conjugate to A through a translation;(c) ( A , X ) preserves an affine plane in (cid:65) .When these conditions are satisfied, the affine planes preserved by ( A , X ) are exactly those parallel to P ,and there is a distinguished one P among these planes, such that the set of fixed points of ( A , X ) in (cid:65) isan affine line lying on P parallel to L . Here, in Condition (b),
Aut ( C ) and (cid:82) are both viewed as subgroups of Aut ( C ) (cid:110) (cid:82) , and the normalsubgroup (cid:82) is referred to as the group of translations .The core of Prop. 4.4 is the following basic property of the Jordan form of A : Lemma 4.5.
The following statements hold for the linear transformationA : = . (1) The affine planes in (cid:82) preserved by A are exactly the horizontal planes (cid:82) × { ξ } .(2) Let X ∈ (cid:82) be a vector not in the generalized eigenspace (cid:82) × { } of A . Then the affine transforma-tion ( A , X ) ∈ SL(3, (cid:82) ) (cid:110) (cid:82) does not preserve any affine plane.Proof. It is an elementary fact that if an affine transformation ( A , X ) ∈ SL(3, (cid:82) ) (cid:110) (cid:82) preserves an affineplane P , then the linear part A preserves the 2-dimensional subspace of (cid:82) parallel to P .One readily checks that every horizontal plane (cid:82) × { ξ } is preserved by A . Conversely, by the abovefact, these are the only affine planes preserved because the generalized eigenspace (cid:82) × { } is the only2-dimensional subspace preserved. This proves Part (1).The above fact also implies that an affine plane preserved by ( A , X ) can only has the form (cid:82) × { ξ } aswell. So we obtain Part (2) by noting that if X is not horizontal then (cid:82) × { ξ } is not preserved. (cid:3) Proof of Prop. 4.4.
Since A is conjugate to the A in Lemma 4.5, by the lemma, the affine planes preservedby A are exactly those parallel to P . Also, the fixed points of A are exactly the points of the eigenline L .Therefore, when Condition (b) is satisfied, the last two statements of the proposition holds. This provesthe “Moreover” part. It remain to show the equivalence between (a), (b) and (c).The conjugate of A in Aut ( C ) (cid:110) (cid:82) by a translation X ∈ (cid:82) is ( A ,( I − A ) X ), because it sends Y ∈ (cid:82) to A ( Y − X ) + X = AY + ( I − A ) X . Therefore, the implication “(a) ⇒ (b)” follows from the fact that (cid:169) ( I − A ) X | X ∈ (cid:82) (cid:170) = P ,which can be easily checked using the Jordan form A . The implication “(b) ⇒ (c)” is obvious. Finally,“(c) ⇒ (a)” follows immediately from Lemma 4.5. (cid:3) FFINE DEFORMATIONS OF QUASI-DIVISIBLE CONVEX CONES 23
Remark . In terms of fixed planes, affine transformations with hyperbolic linear part behave differentlyfrom those with parabolic linear part. In fact, given a hyperbolic A ∈ Aut ( C ), one readily checks that • if the middle eigenvalue of A is 1, then every ( A , X ) ∈ Aut ( C ) (cid:110) (cid:82) is conjugate through a transla-tion to some ( A , X ) with X in the middle eigenspace; • otherwise, ( A , X ) must be conjugate through a translate to A itself.In any case, ( A , X ) has two fixed C -null planes, parallel to the two C -null subspaces fixed by A .Now let C and Ω be as in §2, so that we have isomorphisms Aut ( C ) ∼= Aut ( C ∗ ) = Aut ( Ω ) and Aut ( C ) (cid:110) (cid:82) ∼= Aut ( Ω × (cid:82) ) (see Prop. 2.6). The former isomorphism sends a parabolic A ∈ Aut ( C ) to the inversetranspose t A − ∈ Aut ( Ω ), which is again parabolic. Therefore, by the commutative diagram in Prop. 2.6,an element ( A , X ) of Aut ( C ) (cid:110) (cid:82) with A parabolic corresponds, under the latter isomorphism, to some Φ ∈ Aut ( Ω × (cid:82) ) whose projection π ( Φ ) ∈ Aut ( Ω ) is parabolic. Thus, the correspondence between the twogeometries explained in the introduction and the previous section translates Prop. 4.4 into the followingdual result about Φ : Corollary 4.7.
Let Φ be an automorphism of the convex tube domain Ω × (cid:82) ⊂ (cid:82) whose projection π ( Φ ) ∈ Aut ( Ω ) is parabolic with fixed point p ∈ ∂ Ω . Then we have the following dichotomy: either • Φ does not have any fixed point in (cid:82) , or • the set of fixed points of Φ is the vertical line { p } × (cid:82) .In the latter case, there is a distinguished fixed point ( p , ξ ) and a non-vertical line L ⊂ (cid:82) tangent to Ω × (cid:82) at ( p , ξ ) such that the affine planes in (cid:82) preserved by Φ are exactly those containing L (see Figure 4.2). � ixed planes distinguished � ixed point � ixed points F IGURE Φ ∈ Aut ( Ω × (cid:82) ) with π ( Φ ) parabolic,when Φ does have fixed points.In particular, if we view a parabolic element A of Aut ( Ω ) with fixed point p ∈ ∂ Ω as an automorphismsof Ω × (cid:82) preserving the slice Ω × { } (see §2.6), then the distinguished fixed point of A is just ( p ,0). We willneed the following result about the existence of certain smooth convex functions with A -invariant graph: Lemma 4.8.
Let A be a parabolic element in
Aut ( Ω ) fixing p ∈ ∂ Ω . Then for any µ ≥ , there is a convexf ∈ C ∞ ( Ω ) such that the boundary value of f at p (in the sense of §3.1) is − µ and the graph of f is preservedby A (as an automorphisms of Ω × (cid:82) ). Note that the µ = f to be an affine function in this case. Proof.
By Lemma 4.1, Ω is contained in an A -invariant projective disk whose boundary passes through p .By choosing the coordinates of (cid:82) appropriately, we may assume that p = ( − (cid:68) : = { x ∈ (cid:82) | | x | < } . Let ψ be the lower semicontinuous function on ∂ (cid:68) such that ψ = ∂ (cid:68) \ { p } and ψ ( p ) = − µ .We view A as a projective transformation of the round tube domain (cid:68) × (cid:82) preserving both the subdo-main Ω × (cid:82) and the slice (cid:68) × { } . Its action on ∂ (cid:68) × (cid:82) preserves the punctured circle ( ∂ (cid:68) \ { p } ) × { } and pointwise fixes the vertical line { p } × (cid:82) . It follows that the graph of ψ is preserved by A , hence so is thegraph of the convex envelop ψ .Let u denote the function on (cid:68) whose restriction to any line segment I joining p and another point x ∈ ∂ (cid:68) is the affine function on I interpolating the values − µ and 0 at p and x . By elementary calculations,we find the expression of u in (cid:68) to be u ( x ) = µ (cid:195) ( x + + x x + − (cid:33) for any x = ( x , x ) ∈ (cid:68) ,and a further calculation shows that the Hessian of u is positive semi-definite, hence u is convex. Also,the boundary value of u is exactly ψ .We claim that ψ = u in (cid:68) . In fact, on one hand, we have ψ ≤ u because by convexity of ψ and the factthat ψ = u on ∂ (cid:68) , the restriction of ψ to the aforementioned segment I is less than or equal to the affinefunction described; on the other hand, ψ ≥ u because ψ is pointwise no less than any convex function withboundary value ψ .As a consequence of the claim, ψ = u is smooth in (cid:68) by the above expression. Also, ψ has A -invariantgraph and has boundary value − µ at p . Thus, the restriction of ψ to Ω gives the required function f . (cid:3) Remark . In contrast to Lemma 4.8, there is no convex function f : Ω → (cid:82) with A -invariant graphwhose boundary value at p is strictly positive. This can be proved by reducing again to the case Ω = (cid:68) andshowing that for any iteration sequence ( A n ( x )) n = , , ··· in ∂ (cid:68) \{ p } converges to p , we have lim n →+∞ f ( A n ( x )) = Φ ∈ Aut ( Ω × (cid:82) ) such that π ( Φ ) is parabolic and Φ has fixed points, we deduces from Prop. 4.4 that Φ is conjugate to π ( Φ ). So the above results imply thatif ( p , ξ ) is the distinguished fixed point of Φ in the sense of Cor. 4.7, then there exists a convex functionwith Φ -invariant graph and with boundary value ξ at p if and only if ξ ≤ ξ .4.3. Admissible cocycles.
Given a group Γ < SL(3, (cid:82) ) and a map τ : Γ → (cid:82) , denote Γ τ : = (cid:169) ( A , τ ( A )) ∈ SL(3, (cid:82) ) (cid:110) (cid:82) | A ∈ Γ (cid:170) .The following facts are well known and easy to verify: • Γ τ is a subgroup of SL(3, (cid:82) ) (cid:110) (cid:82) if and only if τ is a 1 -cocycle on Γ with values in the Γ -module (cid:82) ,which means more precisely that τ belongs to the vector space Z ( Γ , (cid:82) ) : = (cid:169) τ : Γ → (cid:82) (cid:175)(cid:175) τ ( AB ) = τ ( A ) + A τ ( B ), ∀ A , B ∈ Γ (cid:170) .In this case, we call Γ τ an affine deformation of Γ . • Two affine deformations Γ τ and Γ τ are said to be equivalent if there is X ∈ (cid:82) such that( A , τ ( A )) is conjugate to ( A , τ ( A )) through the translation X for any A ∈ Γ . It is the case ifand only if τ − τ lies in the vector space of 1 -coboundariesB ( Γ , (cid:82) ) : = (cid:169) τ X (cid:175)(cid:175) X ∈ (cid:82) (cid:170) , where τ X ( A ) : = ( I − A ) X .Therefore, the first cohomology of Γ with values in (cid:82) , i.e. the vector space H ( Γ , (cid:82) ) : = Z ( Γ , (cid:82) ) (cid:177) B ( Γ , (cid:82) ),is the space of equivalence classes of affine deformations. In view of Prop. 4.4, we further define: Definition 4.10.
Let C ⊂ (cid:82) be a proper convex cone quasi-divisible by a torsion-free group Γ < SL(3, (cid:82) ).Then a cocycle τ ∈ Z ( Γ , (cid:82) ) is said to be admissible if for every parabolic element A in Γ , the vector τ ( A )is in the C -null subspace preserved by A .In particular, since ( I − A ) X is contained in the C -null subspace preserved by A for any parabolic A ∈ Aut ( C ) and X ∈ (cid:82) , every 1-coboundary τ X is admissible. FFINE DEFORMATIONS OF QUASI-DIVISIBLE CONVEX CONES 25
Although the above definitions are made for a subgroup Γ < SL(3, (cid:82) ), one can readily adapt them torepresentations ρ ∈ Hom( Π ,SL(3, (cid:82) )), where Π is an abstract group. Precisely, the space of 1-cocycles and1-coboundaries for ρ are defined as Z ρ ( Π , (cid:82) ) : = (cid:169) τ : Π → (cid:82) (cid:175)(cid:175) τ ( αβ ) = τ ( α ) + ρ ( α ) τ ( β ), ∀ α , β ∈ Π (cid:170) , B ρ ( Π , (cid:82) ) : = (cid:169) τ X (cid:175)(cid:175) X ∈ (cid:82) (cid:170) , where τ X ( γ ) : = ( I − ρ ( γ )) X ,so that every representation of Π in SL(3, (cid:82) ) (cid:110) (cid:82) can be written as ρ τ ( γ ) : = ( ρ ( γ ), τ ( γ ))for some ρ ∈ Hom( Π ,SL(3, (cid:82) )) and τ ∈ Z ρ ( Π , (cid:82) ), and two such representations are conjugate to each otherthrough a translation if and only if they have the form ρ τ and ρ τ , with τ − τ ∈ B ρ ( Π , (cid:82) ). Moreover,when there is a proper convex cone C ⊂ (cid:82) quasi-divisible by the image ρ ( Π ), we call τ ∈ Z ρ ( Π , (cid:82) ) admis-sible if τ ( γ ) is in the C -null subspace preserved by ρ ( γ ) whenever ρ ( γ ) is parabolic.4.4. Moduli spaces.
Let C ⊂ (cid:82) be a proper convex cone quasi-divisible by a torsion-free group Γ < SL(3, (cid:82) ). Since (cid:80) ( C ) is topologically a disk and Γ acts on it properly discontinuously by orientation preserv-ing homeomorphism, Theorem 4.2 implies that the surface S : = (cid:80) ( C )/ Γ is homeomorphic to the orientablesurface S g , n with genus g and n punctures, where the nonnegative integers g and n satisfy: • ( g , n ) (cid:54)= (0,0) or (0,1), as S cannot be homeomorphic to the sphere or the disk; • ( g , n ) (cid:54)= (0,2), since otherwise Γ is the cyclic group generated by a single element in Aut ( C ), andit would be impossible that both punctures of S ≈ S have the property in the conclusion ofTheorem 4.2.Therefore, we have either ( g , n ) = (1,0) or 2 − g − n <
0. Namely, S is homeomorphic to either the torus orsome S g , n with negative Euler characteristic.For each allowed ( g , n ), the moduli space of convex projective structures with finite volume on S g , n isdefined as the topological quotient P g , n : = Hom ( π ( S g , n ),SL(3, (cid:82) )) (cid:177) SL(3, (cid:82) ),where SL(3, (cid:82) ) acts by conjugation on the space of representationsHom ( π ( S g , n ),SL(3, (cid:82) )): = (cid:189) ρ ∈ Hom( π ( S g , n ),SL(3, (cid:82) )) (cid:175)(cid:175)(cid:175)(cid:175) ρ is faithful and there is a proper convexcone C ⊂ (cid:82) quasi-divisible by ρ ( π ( S g , n )) (cid:190) Similarly, we define the moduli space of admissible affine deformations of convex projective structureswith finite volume on S g , n as the quotient (cid:99) P g , n : = Hom ( π ( S g , n ),SL(3, (cid:82) ) (cid:110) (cid:82) ) (cid:177) SL(3, (cid:82) ) (cid:110) (cid:82) ,where SL(3, (cid:82) ) (cid:110) (cid:82) acts by conjugation onHom ( π ( S g , n ),SL(3, (cid:82) ) (cid:110) (cid:82) ): = (cid:189) ρ τ ∈ Hom( π ( S g , n ),SL(3, (cid:82) ) (cid:110) (cid:82) ) (cid:175)(cid:175)(cid:175)(cid:175) ρ is in Hom ( π ( S g , n ),SL(3, (cid:82) )), τ ∈ Z ρ ( π ( S g , n ), (cid:82) ) is admissible (cid:190) .The natural projection(4.1) Hom ( π ( S g , n ),SL(3, (cid:82) ) (cid:110) (cid:82) ) → Hom ( π ( S g , n ),SL(3, (cid:82) )), ρ τ (cid:55)→ ρ induces a projection (cid:99) P g , n → P g , n , which is a surjective continuous map. We now prove Proposition Din the introduction by showing that when 2 − g − n <
0, the latter projection is a vector bundle of rank6 g − + n . Proof of Prop. D.
We view (cid:99) P g , n a two-step quotient of Hom ( π ( S g , n ),SL(3, (cid:82) ) (cid:110) (cid:82) ):(4.2) (cid:99) P g , n = (cid:161) Hom ( π ( S g , n ),SL(3, (cid:82) ) (cid:110) (cid:82) ) (cid:177) (cid:82) (cid:162)(cid:177) SL(3, (cid:82) ).Namely, first quotienting by the normal subgroup of translations (cid:82) (cid:67) SL(3, (cid:82) ) (cid:110) (cid:82) , then by the quotientgroup SL(3, (cid:82) ) = (SL(3, (cid:82) ) (cid:110) (cid:82) )/ (cid:82) . We claim that • the projection (4.1) is a vector bundle of rank 6 g − + n ; • the first quotient in (4.2) is the quotient of this vector bundle by a sub-bundle of rank 3.To prove the claim, we pick a standard set (cid:169) α , ··· , α g , β , ··· , β g , γ , ··· , γ n (cid:170) of generators of π ( S g , n ),with generating relation [ α , β ] ··· [ α g , β g ] γ ··· γ n = id,and consider the mapHom ( π ( S g , n ),SL(3, (cid:82) ) (cid:110) (cid:82) ) → Hom ( π ( S g , n ),SL(3, (cid:82) )) × ( (cid:82) ) g + n (4.3) ρ τ (cid:55)→ (cid:161) ρ , τ ( α ), ··· , τ ( γ n ) (cid:162) ,which assigns to each ρ τ its projection ρ and the values of the admissible cocycle τ at the generators. View-ing the target of the map (4.3) as the trivial vector bundle of rank 6 g + n over Hom ( π ( S g , n ),SL(3, (cid:82) )),we shall prove the first part of the claim by showing that the map identifies the source with a sub-bundleof rank 6 g − + n in the target.To this end, fix a representation ρ ∈ Hom ( π ( S g , n ),SL(3, (cid:82) )) and let C ⊂ (cid:82) be the cone divisible bythe image of ρ . Any admissible cocycle τ ∈ Z ρ ( π ( S g , n ), (cid:82) ) is completely determined by its values at thegenerators because of the cocycle condition τ ( αβ ) = τ ( α ) + ρ ( α ) τ ( β );whereas the only constraints on these values are- τ ( γ j ) belongs to the C -null subspace V j ( ρ ) ⊂ (cid:82) preserved by ρ ( γ j ) for j = ··· , n ;- if we use the cocycle condition to expand τ ([ α , β ] ··· [ α g , β g ] γ ··· γ n ) into an expression only in-volving the values of ρ and τ on the generators, then the expression gives 0.By a calculation, we find the expansion to be τ ([ α , β ] ··· [ α g , β g ] γ ··· γ n ) = (cid:161) I − ρ ( α ) ρ ( β ) ρ ( α ) − (cid:162) τ ( α ) + ρ ( α ) (cid:161) I − ρ ( β ) ρ ( α ) − ρ ( β ) − (cid:162) τ ( β ) + • τ ( α ) + ··· + • τ ( γ n ),where the bullet in front of each τ ( α i ), τ ( β i ) with i ≥ τ ( γ j ) is a specific linear transformationof (cid:82) given by the ρ ( α i )’s, ρ ( β i )’s and ρ ( γ j )’s. So the two constraints together require ( τ ( α ), ··· , τ ( γ n )) tobe in the kernel of the linear map L ρ : ( (cid:82) ) g ⊕ V ( ρ ) ⊕ ··· ⊕ V n ( ρ ) → (cid:82) L ρ ( X , ··· X g , Y , ··· , Y g , Z , ··· , Z n ) : = (cid:161) I − ρ ( α ) ρ ( β ) ρ ( α ) − (cid:162) X + ρ ( α ) (cid:161) I − ρ ( β ) ρ ( α ) − ρ ( β ) − (cid:162) Y + • X + ··· + • Z n .In order to show that L ρ is surjective, we define the axis of any hyperbolic element A in Aut ( C ) tobe the projective line Axis ( A ) ⊂ (cid:82)(cid:80) which is the projectivization of the 2-subspace of (cid:82) spanned by theeigenlines of the largest and smallest eigenvalues of A . In particular, if C is the future light cone C ,then (cid:80) ( C ) is the Klein model of the hyperbolic plane and the geodesic Axis ( A ) ∩ (cid:80) ( C ) is the axis of A in the sense of hyperbolic geometry. In general, if the middle eigenvalue of A is 1, then the 2-subspaceprojecting to Axis ( A ) is exactly the image of the linear map I − A , hence Axis ( A ) = (cid:80) (cid:169) ( I − A ) X (cid:175)(cid:175) X ∈ (cid:82) (cid:170) .By Thm. 4.2 and Prop. 4.3 (3), ρ ( α ) and ρ ( β ) are hyperbolic elements with different axes. So we canshow the surjectivity of L ρ in the following cases separately: FFINE DEFORMATIONS OF QUASI-DIVISIBLE CONVEX CONES 27 • If the middle eigenvalue of ρ ( β ) is not 1, then we have (cid:169)(cid:161) I − ρ ( α ) ρ ( β ) ρ ( α ) − (cid:162) X (cid:175)(cid:175) X ∈ (cid:82) (cid:170) = (cid:82) ,hence L ρ is surjective. • If the middle eigenvalue of ρ ( α ) is not 1, L ρ is also surjective because (cid:169) ρ ( α ) (cid:161) I − ρ ( β ) ρ ( α ) − ρ ( β ) − (cid:162) Y (cid:175)(cid:175) Y ∈ (cid:82) (cid:170) = (cid:82) . • If both ρ ( α ) and ρ ( β ) have middle eigenvalue 1, the two projective lines (cid:80) (cid:169)(cid:161) I − ρ ( α ) ρ ( β ) ρ ( α ) − (cid:162) X (cid:175)(cid:175) X ∈ (cid:82) (cid:170) = ρ ( α ) Axis ( ρ ( β )), (cid:80) (cid:169) ρ ( α ) (cid:161) I − ρ ( β ) ρ ( α ) − ρ ( β ) − (cid:162) Y (cid:175)(cid:175) Y ∈ (cid:82) (cid:170) = ρ ( α ) ρ ( β ) Axis ( ρ ( α ))are different because Axis ( ρ ( α )) (cid:54)= Axis ( ρ ( β )). As a result, we have (cid:169)(cid:161) I − ρ ( α ) ρ ( β ) ρ ( α ) − (cid:162) X + ρ ( α ) (cid:161) I − ρ ( β ) ρ ( α ) − ρ ( β ) − (cid:162) Y (cid:175)(cid:175) X , Y ∈ (cid:82) (cid:170) = (cid:82) .Therefore, L ρ is surjective in this case as well.The surjectivity of L ρ and the obvious fact that V j ( ρ ) and L ρ depend continuously on ρ imply that theimage of the map (4.3), namely the set V : = (cid:169) ( ρ ,ker L ρ ) (cid:175)(cid:175) ρ ∈ Hom ( π ( S g , n ),SL(3, (cid:82) )) (cid:170) ,is the total space of a vector bundle over Hom ( π ( S g , n ),SL(3, (cid:82) )) of rank 6 g − + n , whose fiber at ρ isker L ρ . Since the map is a homeomorphism from Hom ( π ( S g , n ),SL(3, (cid:82) ) (cid:110) (cid:82) ) to V , this implies the firstpart of the claim.For the second part of the claim, note that by the discussion of coboundaries in the previous subsection,the action of X ∈ (cid:82) on Hom ( π ( S g , n ),SL(3, (cid:82) ) (cid:110) (cid:82) ) sends ρ τ to ρ τ + τ X , where τ X ( γ ) : = ( I − ρ ( γ )) X ( ∀ γ ∈ π ( S g , n )). Therefore, if we identify Hom ( π ( S g , n ),SL(3, (cid:82) ) (cid:110) (cid:82) ) with V via the map (4.3) as above, thenits quotient by (cid:82) is given by quotienting every fiber ker L ρ of V by the image of the linear map (cid:82) → ker L ρ X (cid:55)→ (cid:161) ( I − ρ ( α )) X , ··· ,( I − ρ ( α g )) X ,( I − ρ ( β ) X , ··· ,( I − ρ ( β g )) X , ( I − ρ ( γ )) X , ··· ,( I − ρ ( γ n )) X (cid:162) .With an argument similar to the above proof for surjectivity of L ρ , one can show that this map is injectiveby looking at the components ( I − ρ ( α )) X and ( I − ρ ( β )) X and considering the three cases as aboveaccording to middle eigenvalues of ρ ( α ) and ρ ( β ). The map also depends continuously on ρ , hence theimages for all ρ together form a vector sub-bundle of V of rank 3. This completes the proof of the claim.The claim implies that the first quotient E : = Hom ( π ( S g , n ),SL(3, (cid:82) ) (cid:110) (cid:82) ) (cid:177) (cid:82) in (4.2) is the totalspace of a vector bundle of rank 6 g − + n over the manifold M : = Hom ( π ( S g , n ),SL(3, (cid:82) )). One readilychecks that the action of G : = SL(3, (cid:82) ) on M lifts to an action on E which sends fibers to fibers by linearisomorphisms. Therefore, by the two lemmas given in the next subsection, (cid:99) P g , n = E / G is a vector bundleof rank 6 g − + n over P g , n = M / G , as required. (cid:3) Remark . The above argument also works in the case ( g , n ) = (1,0) and shows that if C ⊂ (cid:82) is aproper convex cone quasi-divisible by a group Γ isomorphic to (cid:90) , then we have Z ( Γ , (cid:82) ) = B ( Γ , (cid:82) ),which means that every affine deformation of Γ is conjugate to Γ itself by a translation. In this case, it iswell known that C is a triangular cone.4.5. Appendix: two technical lemmas.Lemma 4.12.
The conjugation action of
SL(3, (cid:82) ) on Hom ( π ( S g , n ),SL(3, (cid:82) )) is free and proper.Proof. A continuous action of a topological group G on a Hausdorff space M is proper if and only if forany compact set K ⊂ M , { g ∈ G | gK ∩ K (cid:54)= (cid:59) } is relatively compact in G . With this criterion, it is easy tocheck that if M and N are Hausdorff G -spaces such that the action on N is free and proper, and there isan equivariant continuous map M → N , then the action on M is free and proper as well. Now set G = SL(3, (cid:82) ) and pick a standard set of generators α , ··· , α g , β , ··· , β g , γ , ··· , γ n of π ( S g , n ).Given ρ ∈ Hom ( π ( S g , n ), G ), we let C be the corresponding quasi-divisible convex cone and consider ρ ( α ) and ρ ( β ), which are hyperbolic elements in Aut ( C ) without common fixed points by Thm. 4.2 andProp. 4.3 (3). For any hyperbolic A ∈ G , let p ( A ), p ( A ) and p ( A ) denote the fixed points of A in (cid:82)(cid:80) corresponding to the largest, middle and smallest eigenvalues of A , respectively. Note that if A ∈ Aut ( C ),then two of the three points, namely p ( A ) and p ( A ), are on ∂ (cid:80) ( C ), and the projective lines p ( A ) p ( A )and p ( A ) p ( A ) are tangent to ∂ (cid:80) ( C ) at the two points, respectively (see Figure 4.1). Therefore, the strictconvexity of ∂ (cid:80) ( C ) (see Prop. 4.3 (1)) implies that the six points p i ( ρ ( α )), p i ( ρ ( β )) ( i = i.e. there is no line passing through any three of them).As a consequence, the assignment ρ (cid:55)→ ( ρ ( α ), ρ ( β )) gives an equivariant map from Hom ( π ( S g , n ), G )to the G -space X : = (cid:189) ( A , B ) ∈ G × G (cid:175)(cid:175)(cid:175) A and B are hyperbolic and the six points p i ( A ), p i ( B ) ( i = (cid:82)(cid:80) are in general position (cid:190) (with the G -action by conjugation). In light of the fact mentioned at the beginning, we now only need toshow that the G -action on X is free and proper.The freeness is easy to check and we omit the details. To show the properness, suppose by contradic-tion that the G -action on X is not proper, then there exist a convergent sequence ( A n , B n ) in X and anunbounded sequence ( C n ) in G such that ( A (cid:48) n , B (cid:48) n ) : = ( C n A n C − n , C n B n C − n ) also converges in X . Using theCartan decomposition of SL(3, (cid:82) ), we write C n = P n T n Q n for each n , where P n , Q n ∈ SO(3) and ( T n ) is anunbounded sequence in the space λ λ λ (cid:175)(cid:175)(cid:175)(cid:175) λ ≥ λ ≥ λ > λ λ λ = ⊂ SL(3, (cid:82) ).By restricting to a subsequence, we may assume that both sequences ( P n ) and ( Q n ) converge in SO(3).Suppose e , e , e ∈ (cid:82)(cid:80) correspond to the coordinate axes in (cid:82) . We claim that for any sequence( q n ) in (cid:82)(cid:80) converging to a point not on the projective line e e , all the limit points of the sequence( T n q n ) must be on the line e e . To show this, we may assume by contradiction that q n → q ∞ ∉ e e and T n q n → q (cid:48)∞ ∉ e e as n → ∞ . Write T n = diag ( T (1) n , T (2) n , T (3) n ) and q n = [ q (1) n : q (2) n : q (3) n ]. The condition q (cid:48)∞ ∉ e e means q (cid:48)∞ = [ x : x : 1] for some x , x ∈ (cid:82) , hence the convergence T n q n → q (cid:48)∞ implies that q (3) n (cid:54)= n large enough and that T (1) n T (3) n q (1) n q (3) n → x , T (2) n T (3) n q (2) n q (3) n → x ,Since ( T n ) is unbounded and T (1) n ≥ T (2) n ≥ T (3) n by assumption, we have T (1) n / T (3) n → +∞ and T (2) n / T (3) n ≥ q (1) n / q (3) n → q (2) n / q (3) n is bounded. As a result, we have q ∞ ∈ e e , a contradiction. Thisproves the claim.Now suppose A (cid:48) n → A (cid:48)∞ as n → ∞ . Then we have p i ( A (cid:48) n ) = P n T n Q n p i ( A n ) → p i ( A (cid:48)∞ )for i = A n → A ∞ , we have Q n p i ( A n ) → Q ∞ p i ( A ∞ ). Therefore, the above claimresults in the implication(4.4) p i ( A ∞ ) ∉ Q − ∞ e e =⇒ p i ( A (cid:48)∞ ) ∈ P ∞ e e .The same argument applies to ( B (cid:48) n ) and yields the implication(4.5) p i ( B ∞ ) ∉ Q − ∞ e e =⇒ p i ( B (cid:48)∞ ) ∈ P ∞ e e .Since ( A ∞ , B ∞ ) ∈ X , at most two of the six points p i ( A ∞ ), p i ( B ∞ ) ( i = Q − ∞ e e .This means that at least four of the six points satisfy the condition on the left-hand side of (4.4) or (4.5). Itfollows that as least four of the six points p i ( A (cid:48)∞ ), p i ( B (cid:48)∞ ) ( i = P ∞ e e , contradictingthe fact that ( A (cid:48)∞ , B (cid:48)∞ ) ∈ X . This completes the proof. (cid:3) FFINE DEFORMATIONS OF QUASI-DIVISIBLE CONVEX CONES 29
Lemma 4.13.
Let G be a locally compact topological group, M be a Hausdorff G-space and E → M be atopological vector bundle of rank r such that the G-action on M is free and proper, and lifts to an action onE which sends fibers to fibers by linear isomorphisms. Then E / G → M / G is also a vector bundle of rank r.Proof.
H. Cartan’s axiom (FP) for principal bundles [Car50] is equivalent to the statement that M → M / G is a principal G -bundle (where G is a locally compact group and M a Hausdorff G -space) if and only ifthe action is free and every point of M has a neighborhood U such that { g ∈ G | gU ∩ U (cid:54)= (cid:59) } is relativelycompact in G (this condition is weaker than properness). See also [Pal61].Therefore, under the assumption of the lemma, we can pick an open cover U of M / G such that thepreimage π − ( U ) ⊂ M of each U ∈ U identifies with the product U × G , on which G acts by multiplicationon the G -factor.On the other hand, there is an open cover W of M such that for each W ∈ W , we have a bundle chart f W : E | W ∼ → W × (cid:82) r . After refining the open cover U if necessary, we may find g U ∈ G for each U ∈ U ,such that the slice U × { g U } in U × G ∼= π − ( U ) is contained in some W ∈ W (see Figure 4.3). By restrictingF IGURE f W to this slice, we get an identification h U : E | U × { g U } ∼ → U × (cid:82) r . Meanwhile, E | U × { g U } can be identified with ( E / G ) | U (the preimage of U by the map E / G → M / G ) because every G -orbit in E | π − ( U ) passes through E | U × { g U } exactly once. It is routine to check that the family of maps h U : ( E / G ) | U ∼= E | U × { g U } ∼ → U × (cid:82) r , U ∈ U form a bundle atlas for E / G , which completes the proof. (cid:3)
5. P
ROOF OF MAIN RESULTS
In this section, we first give a proof of Parts (1) and (2) of Theorem F using the framework set up inthe previous sections and a lemma proved in §5.1 below, then we deduce Theorem F (3) from our earlierwork [NS19], and explain how Theorems A, B and Corollary C follow from Theorem F.5.1.
Continuous boundary function with Λ (cid:48) -invariant graph. We first show that Condition (a) inTheorem F (1) implies the existence of certain smooth functions on Ω with Λ (cid:48) -invariant graph: Lemma 5.1.
Let Ω ⊂ (cid:82) be a bounded convex domain quasi-divisible by a torsion-free group Λ < Aut ( Ω ) .Suppose Λ (cid:48) < Aut ( Ω × (cid:82) ) projects to Λ bijectively, and every element with parabolic projection has a fixedpoint in ∂ Ω × (cid:82) . For each puncture of the surface S : = Ω / Λ , we take a neighborhood homeomorphic toa punctured disk, assume these neighborhoods have disjoint closures, and let U be their union. Thenthere exists a function v ∈ C ∞ ( Ω ) with Λ (cid:48) -invariant graph, such that the restriction of v to each connectedcomponent of the lift (cid:101) U ⊂ Ω of U is an affine function. If Ω is divisible by Λ , then S is a closed surface and the lemma just asserts the existence of a smoothfunction with Λ (cid:48) -invariant graph in this case. Proof.
Since the connected components of U have disjoint closures, we can enlarge U to a bigger openset U containing the closure U , such that U is still a disjoint union of neighborhoods of the punctureshomeomorphic to a punctured disk. We then take simply connected open sets U , ··· , U N in S disjointfrom U to form an open cover of S together with U .Let ( ι i ) be a C ∞ -partition of unity subordinate to this open cover. Namely, each ι i is a C ∞ -function on S taking values in [0,1], with support contained in U i , such that (cid:80) Ni = ι i = S . Note that ι = U because U is disjoint from any U i with i ≥ (cid:101) U i denote the lift of U i , i.e. the pre-image of U i by the covering map π : Ω → S , and (cid:101) ι i : = ι i ◦ π ∈ C ∞ ( Ω )denote the lift of ι i to Ω . In order to construct the required function v , we shall construct v i ∈ C ∞ ( (cid:101) U i ) with Λ (cid:48) -invariant graph for each i = ··· , N and sum up them using the partition of unity ( (cid:101) ι i ). To this end, wetreat i = i ≥ i ≥
1, since U i is simply connected, we can write (cid:101) U i as a disjoint union (cid:101) U i = (cid:91) A ∈ Λ A ( W )where W is a connected component of (cid:101) U i . We can then take an arbitrary w ∈ C ∞ ( W ) and obtain therequired v i ∈ C ∞ ( (cid:101) U i ) using the Λ (cid:48) -action on the graph of w . More precisely, v i is given bygr( v i ) = (cid:91) Φ ∈ Λ (cid:48) Φ (cid:161) gr( w ) (cid:162) .For instance, if w ≡
0, then u i is an affine function on each component on (cid:101) U i .For i =
0, we may assume U = (cid:83) nj = V j , where we label the punctures of S by 1, ··· , n , and V j isa neighborhood of the j th puncture. To construct v , we only need to construct v ( j )0 ∈ C ∞ ( (cid:101) V j ) with Λ (cid:48) -invariant graph on (cid:101) V j : = π − ( V j ) for each j and put v : = (cid:80) nj = v ( j )0 . So we fix j and a connected component Z of (cid:101) V j . The subgroup of Λ preserving Z is generated by some parabolic element A ∈ Λ . Since the element Φ A of Λ (cid:48) projecting to A has a fixed point by assumption, Φ A preserves some non-vertical affine plane P by Cor. 4.7. Therefore, we can define v ( j )0 by first letting v ( j )0 (cid:175)(cid:175) Z be the affine function whose graph is P ,then using the Λ (cid:48) -action to define v ( j )0 on the other components of (cid:101) V j . Namely, v ( j )0 is given bygr( v ( j )0 ) = (cid:91) Φ ∈ Λ (cid:48) / 〈 Φ A 〉 Φ (cid:161) P ∩ ( Z × (cid:82) ) (cid:162) .This finishes the construction of the v i ’s.We can now construct the required v as v : = N (cid:88) i = (cid:101) ι i v i ∈ C ∞ ( Ω ).In order to check that the graph gr( v ) is preserved by any Φ ∈ Λ (cid:48) , we pick x ∈ Ω and let x (cid:48) ∈ Ω be its imageby π ( Φ ) ∈ Aut ( Ω ). Since ι i is the lift of a function on S and gr( v i ) is preserved by Λ (cid:48) , we have (cid:101) ι i ( x ) = (cid:101) ι i ( x (cid:48) )and Φ ( x , v i ( x )) = ( x (cid:48) , v i ( x (cid:48) )). Therefore, by Lemma 2.8 (1) (whose statement is only about two points butcan be generalized to N points by applying repeatedly), we have Φ ( x , v ( x )) = Φ (cid:179) x , (cid:88) i (cid:101) ι i ( x ) v i ( x ) (cid:180) = (cid:179) x (cid:48) , (cid:88) i (cid:101) ι i ( x (cid:48) ) v i ( x (cid:48) ) (cid:180) = ( x (cid:48) , v ( x (cid:48) )).This shows that gr( v ) is preserved by Φ . Finally, since (cid:101) ι = (cid:101) U ⊂ (cid:101) U , we have v = v in (cid:101) U , which restrictsto an affine function on each component of (cid:101) U by construction. Therefore, v satisfies the requirements andthe proof is complete. (cid:3) This lemma enables us to show the implication “(a) ⇒ (b)” in Thm. F (1), namely: Proposition 5.2.
Let Ω , Λ and Λ (cid:48) be as in Lemma 5.1. Then there exists ϕ ∈ C ( ∂ Ω ) with graph preservedby Λ (cid:48) . FFINE DEFORMATIONS OF QUASI-DIVISIBLE CONVEX CONES 31
Proof.
Let U and (cid:101) U be as in Lemma 5.1, K ⊂ Ω \ (cid:101) U be a compact set with (cid:83) A ∈ Λ A ( K ) = Ω \ (cid:101) U , and v ∈ C ∞ ( Ω )be a function produced by the lemma. Since the Cheng-Yau support function w Ω from Thm. 3.10 issmooth and strongly convex ( i.e. has positive definite Hessian) in Ω , by compactness of K , we can pick asufficiently large constant M > u − : = v + Mw Ω , u + : = v − Mw Ω are strongly convex and strongly concave in K , respectively. By Lemma 2.8 (2), the graph of u ± is pre-served by Λ (cid:48) , hence u ± is strongly concave/convex on Ω \ (cid:101) U . Since v restricts to an affine functions on eachcomponent of (cid:101) U by construction, it follows that u ± is strongly concave/convex throughout Ω . Moreover,the boundary values of u + and u − (in the sense of §3.1) are the same function ϕ on ∂ Ω because w Ω iscontinuous on Ω with vanishing boundary value. Since the boundary value of a convex (resp. concave)function is lower (resp. upper) semicontinuous by construction (see §3.1), we conclude that ϕ is continuouswith Λ (cid:48) -invariant graph, as required. (cid:3) Lower semicontinuous boundary functions with Λ -invariant graph. Recall from §2.6 thatthe group
Aut ( Ω ) of orientation-preserving projective automorphisms of Ω can be identified with the groupof those automorphisms of the convex tube domain Ω × (cid:82) which preserve the slice Ω × { } . As an ingredientin the proof of Thm. F (2), we now identify all the lower semicontinuous functions on ∂ Ω with graphpreserved by a subgroup Λ < Aut ( Ω ) which quasi-divides Ω : Proposition 5.3.
Let Ω ⊂ (cid:82) be a bounded convex domain quasi-divisible by a torsion-free group Λ < Aut ( Ω ) , F ⊂ ∂ Ω be the set of fixed points of parabolic elements in Λ , and pick p , ··· , p n ∈ F such that F isthe disjoint union of the orbits Λ . p j , j = ··· , n. Given µ = ( µ , ··· , µ n ) ∈ (cid:82) n ≥ , let ψ µ be the function on ∂ Ω ,with graph preserved by Λ (as a group of automorphisms of Ω × (cid:82) , which also acts on ∂ Ω × (cid:82) ), such that ψ µ ( p j ) = − µ j for all j; ψ µ = on ∂ Ω \ F . Then ψ µ is lower semicontinuous. Moreover, these are the only lower semicontinuous functions ∂ Ω → (cid:82) ∪ { +∞ } with graphs preserved by Λ that are not constantly +∞ .Proof. We claim that for any convex function u : Ω → (cid:82) with Λ -invariant graph, the boundary value u | ∂ Ω (in the sense of §3.1) vanishes on ∂ Ω \ F .In order to show this, let us compare u with the Cheng-Yau support function w Ω , which is strictlynegative on Ω and has Λ -invariant graph. Letting (cid:101) U ⊂ Ω be the open set lifted from some neighborhoodsof punctures of Ω / Λ as in Lemma 5.1, we can take a compact subset K ⊂ Ω such that (cid:83) A ∈ Λ A ( K ) = Ω \ (cid:101) U and take a sufficiently large constant M > − Mw Ω ≥ u ≥ Mw Ω in K . Then the Λ -invariance ofthe graphs of both u and w Ω imply the same inequalities on Ω \ (cid:101) U . It follows that u | ∂ Ω vanishes on ∂ Ω \ F because for any x ∈ ∂ Ω \ F , u ( x ) is the limit of u ( x ) as x tends to x along a line segment joining x withsome x ∈ Ω , and this segment contains a sequence of points in Ω \ (cid:101) U tending to x . We have thus proventhe claim.To show that the specific function ψ µ is lower semicontinuous, we first pick a generator A j for thestabilizer of p j in Λ (which is an infinite cyclic group generated by a parabolic element), and apply Lemma4.8 to find an A j -invariant smooth convex function f j : Ω → (cid:82) whose boundary value at p j is − µ j . Thenwe can construct a smooth function v µ ∈ C ∞ ( Ω ) with Λ -invariant graph satisfyinglim s → v µ ((1 − s ) p j + sx ) = − µ j for any x ∈ Ω , j = ··· , n using a partition of unity similarly as in the proof of Lemma 5.1, only replacing the A -invariant affineplane P used in that proof by the graph of f j . Next, by taking a large enough M > u µ = v µ + Mw Ω ∈ C ∞ ( Ω ) with Λ -invariant graph, which hasthe same limit property as v µ above because w Ω is continuous on Ω and has vanishing boundary value.This means that the boundary value u µ | ∂ Ω of u µ , which is a lower semicontinuous function on ∂ Ω with Λ -invariant graph, coincides with ψ µ at every p j . The Λ -invariance then implies that u µ | ∂ Ω = ψ µ holds on F . It follows that the equality actually holds on the whole boundary ∂ Ω , because ψ µ vanishes on ∂ Ω \ F by definition and so does u µ | ∂ Ω by the above claim. As a consequence, ψ µ is lower semicontinuous, asrequired.To show the “Moreover” statement, let ψ : ∂ Ω → (cid:82) ∪ { +∞ } be an arbitrary lower semicontinuous functionwith graph preserved by Λ , such that ψ ( x ) < +∞ for some x ∈ ∂ Ω . As explained in §3.1, ψ is the boundaryvalue of its convex envelope ψ ∈ LC ( (cid:82) ). The effective domain dom ( ψ ) : = { x ∈ (cid:82) | ψ ( x ) < +∞ } is a convexset containing the Λ -orbit of x , which is dense in ∂ Ω by Prop. 4.3 (2), hence it contains Ω . This means ψ only takes finite values in Ω , so we can invoke the above claim and conclude that ψ = ∂ Ω \ F . Thedensity of ∂ Ω \ F and the lower semicontinuity of ψ then imply ψ ≤ F . Therefore, ψ must equalsome ψ µ , as required. (cid:3) We can now prove the first two parts of Theorem F.
Proof of Thm. F (1) and (2).
Suppose ϕ ∈ C ( ∂ Ω ) has graph preserved by Λ (cid:48) . Let (cid:98) ϕ : ∂ Ω → (cid:82) ∪ { +∞ } be alower semicontinuous function also with graph preserved by Λ (cid:48) such that (cid:98) ϕ (cid:54)≡ +∞ . By Lemma 2.8, wecan write (cid:98) ϕ = ϕ + ψ for a lower semicontinuous ψ : ∂ Ω → (cid:82) ∪ { +∞ } with graph preserved by Λ . By Prop.5.3, ψ must equal some ψ µ described in that proposition. It follows that (cid:98) ϕ equals some ϕ µ described inthe required statement (2). Also, since every ψ µ vanishes on the dense subset ∂ Ω \ F of ∂ Ω by definition,the only continuous one among the ψ µ ’s is the zero function ψ . It follows that the only continuous oneamong the ϕ µ ’s is ϕ itself. This proves Part (2).As for Part (1), the implication “(b) ⇒ (c)” is trivial, and we have already shown “(a) ⇒ (b)” in Prop. 5.2.To show the implication “(c) ⇒ (a)”, let (cid:98) ϕ be as in (c). We just proved that (cid:98) ϕ equals some ϕ µ , hence inparticular only takes values in (cid:82) . Then, for each Φ ∈ Λ (cid:48) whose projection π ( Φ ) ∈ Λ is parabolic, if we let p ∈ ∂ Ω be the fixed point of π ( Φ ), then Φ fixes ( p , (cid:98) ϕ ( p )) ∈ ∂ Ω × (cid:82) because of the Λ (cid:48) -invariance of gr( ϕ ), hence(a) holds. This completes the proof of Part (1). (cid:3) Solving the Monge-Ampère problem.
We shall deduce Part (3) of Theorem F from the followingresult in [NS19].
Theorem 5.4 ([NS19, Thm. A (cid:48) in §8.1]) . Let Ω ⊂ (cid:82) be a bounded convex domain and ϕ : ∂ Ω → (cid:82) be alower semicontinuous function. Then(1) For each t ∈ (cid:82) , there exists a unique convex solution u t ∈ C ∞ ( Ω ) to the Dirichlet problem of Monge-Ampère equation (cid:40) det D u = e − t w − Ω in Ω , u | ∂ Ω = ϕ . Here w Ω is the Cheng-Yau support function of Ω (see Thm. 3.10).(2) For any x ∈ ∂ Ω , if Ω satisfies the exterior circle condition at x , then u t has infinite inner deriva-tives at x (see §3.2 for the definition).(3) For any fixed x ∈ Ω , t (cid:55)→ u t ( x ) is a strictly increasing concave function tending to −∞ and ϕ ( x ) as ttends to −∞ and +∞ , respectively.Remark . The original statement of [NS19, Thm. A (cid:48) ] is more general in that ϕ is only assumed totake values in (cid:82) ∪ { +∞ } . It asserts the unique-existence of a lower semicontinuous u : Ω → (cid:82) ∪ { +∞ } with u | ∂ Ω = ϕ , satisfying the same equation det D u = e − t w − Ω in the convex domain U : = intdom ( u ) (ses§3.1 for the notation), under the extra constraint that u has infinite inner derivatives at every point of ∂ U ∩ Ω = ∂ U \ ∂ Ω . Proof of Thm. F (3).
The unique-existence of the required u t and the last bullet point in the statement ofThm. F (3) are given immediately by Thm. 5.4.We henceforth fix t ∈ (cid:82) and denote u : = u t for simplicity. To see that the graph gr( u ) ⊂ Ω × (cid:82) is preservedby any Φ ∈ Λ (cid:48) , we let (cid:101) u denote the convex function on Ω such that gr( (cid:101) u ) = Φ (gr( u )), which has the sameboundary value ϕ as u because gr( ϕ ) ⊂ ∂ Ω × (cid:82) is preserved by Λ (cid:48) . By Prop. 3.12, the graph Σ of theLegendre transform u ∗ over D u ( Ω ) is an affine ( C , e t )-surface (at this stage, we do not know whether FFINE DEFORMATIONS OF QUASI-DIVISIBLE CONVEX CONES 33 the graph is entire, i.e. whether D u ( Ω ) = (cid:82) ), whereas it follows from Lemma 3.4 that the graph (cid:101) Σ of (cid:101) u ∗ over D (cid:101) u ( Ω ) is the image of Σ by the affine transformation in Aut ( C ) (cid:110) (cid:82) corresponding to Φ . Since theproperty of being an affine ( C , k )-surface is preserved by Aut ( C ) (cid:110) (cid:82) , (cid:101) Σ is an affine ( C , e t )-surface as well,which implies, again by Prop. 3.12, that (cid:101) u is also a solution to the same Dirichlet problem. Thus, we have u = (cid:101) u by the uniqueness. This shows that gr( u ) is preserved by Λ (cid:48) .Now it only remains to be shown that (cid:107) D u (cid:107) tends to +∞ on ∂ Ω , which is equivalent, by Lemma 3.3 (3),to the condition that u has infinite inner derivatives at every point of ∂ Ω .To this end, we first claim that u is strictly smaller than the convex envelope ϕ in Ω (we already have u ≤ ϕ by construction, see §3.1). Suppose by contradiction that u ( x ) = ϕ ( x ) for some x ∈ Ω and let a ( x ) : = u ( x ) + ( x − x ) · D u ( x ) be the supporting affine function of u at x . Then we have a ≤ u ≤ ϕ in Ω ,with both equalities achieved at x . By [NS19, Lemma 4.9], the set (cid:169) x ∈ Ω | ϕ ( x ) = a ( x ) (cid:170) is the convex hullof some subset of ∂ Ω . Since this convex hull contains x , it also contains some line segment I passingthrough x . It follows that u = a on I , contradicting the strict convexity of u , hence the claim is proved.If Ω is divisible by Λ , we take a compact set K ⊂ Ω with (cid:83) A ∈ Λ A ( K ) = Ω . Since ϕ − u is strictly positiveby the claim, we can pick sufficiently small (cid:178) > u ≤ ϕ + (cid:178) w Ω in K . By Lemma 2.8 (2), the graphs of the functions on both sides are preserved by Λ (cid:48) , so this inequalityactually holds on the whole Ω . Also, the equality is achieved on ∂ Ω . Therefore, for any x ∈ ∂ Ω , x ∈ Ω and s ∈ (0,1], we have u ( x + s ( x − x )) − u ( x ) s (5.2) ≤ ϕ ( x + s ( x − x )) − ϕ ( x ) s + (cid:178) w Ω ( x + s ( x − x )) − w Ω ( x ) s .Each of the three fractions in (5.2) is increasing in s by convexity, hence has a limit in [ −∞ , +∞ ) as s → −∞ because w Ω has infinite inner derivatives at x by Theorem 3.10and Lemma 3.3 (3). As a result, the left-hand side tends to −∞ , hence u has infinite inner derivatives at x , as required.If Ω is quasi-divisible but not divisible by Λ , we adapt the argument as follows. Let U be an opensubset of the punctured surface Ω / Λ as in Lemma 5.1, consisting of neighborhoods of punctures, (cid:101) U ⊂ Ω be the lift of U , and F be the subset of ∂ Ω consisting of fixed points of parabolic elements in Λ . Then thesame reasoning as above shows that (5.1) holds on Ω \ (cid:101) U . It follows that (5.2) holds for those s ∈ (0,1] suchthat x + s ( x − x ) ∈ Ω \ (cid:101) U . Now, if x ∈ ∂ Ω is not in F , then there exists a sequence of such s ’s convergingto 0, and it follows that u has infinite inner derivatives at x in the same way as before. Otherwise, x ∈ F is the fixed point of some parabolic element in Λ , hence Ω satisfies the exterior circle condition at x byLemma 4.1. In this case, we can apply Thm. 5.4 (1) and conclude that u has infinite inner derivatives at x as well. This completes the proof. (cid:3) Proofs of Thm. A, B and Cor. C.
To prove the results in the introduction about affine deformations,we now take a proper convex cone C ⊂ (cid:82) quasi-divisible by a torsion-free group Γ < Aut ( C ). By Prop. 4.3(4), the dual cone C ∗ ⊂ (cid:82) ∗ is quasi-divisible by the image Γ ∗ : = (cid:169) t A − | A ∈ Γ (cid:170) of Γ in Aut ( C ∗ ) ( c.f. §2.2).The dictionary between the two geometries explained in the introduction and §2 then translates eachaffine deformation Γ τ of Γ to a group Γ ∗ τ < Aut ( Ω × (cid:82) ) of automorphisms of the convex tube domain Ω × (cid:82) (where Ω ∼= (cid:80) ( C ∗ ), see §2.2) which projects bijectively to Γ ∗ through the projection π : Aut ( Ω × (cid:82) ) → Aut ( Ω ) = Aut ( C ∗ ) ( c.f. §2.4). Furthermore, by Prop. 4.4 and the identification N C ∼= ∂ Ω × (cid:82) ( c.f. Prop. 2.2), the cocycle τ is admissible if and only if every Φ ∈ Γ ∗ τ , whose projection π ( Φ ) ∈ Γ ∗ is parabolic, has a fixed point in ∂ Ω × (cid:82) .The above discussion and the framework set up in the previous sections allow us to deduce most ofthe statements in Theorems A, B and Corollary C of the introduction immediately from Theorem F. Weproceed to give a formal proof. Let us reformulate the statements as follows: Theorem 5.6 (Thm. A, B and Cor. C) . In the above setting, the followings hold.(1) There exists a C-regular domain in (cid:65) preserved by Γ τ if and only if τ is admissible. In this case,there is a unique equivariant continuous map f from ∂ (cid:80) ( C ) to the space of C-null planes in (cid:65) . Thecomplement of (cid:83) x ∈ ∂ (cid:80) ( C ) f ( x ) in (cid:65) has two connected components D + and D − , which are C-regularand ( − C ) -regular domains preserved by Γ τ , respectively.(2) If C is divisible by Γ , then D + is the unique C-regular domain preserved by Γ τ . Otherwise, assumethe surface S : = (cid:80) ( C )/ Γ has n ≥ punctures and τ is admissible, then all the C-regular domainspreserved by Γ τ form a family ( D µ ) parameterized by µ ∈ (cid:82) n ≥ , such that D (0, ··· ,0) = D + and we haveD µ ⊂ D µ (cid:48) if and only if µ is coordinate-wise larger than or equal to µ (cid:48) .(3) For any C-regular domain D preserved by Γ τ , the Γ τ -action on D is free and properly discontinuous,and there exists a homeomorphism between the quotient D / Γ τ and S × (cid:82) satisfying the followingconditions: • Let K : D → (cid:82) be the function given by composing the quotient map D → D / Γ τ with the projec-tion D / Γ τ ∼= S × (cid:82) → (cid:82) to the (cid:82) -factor. Then K is convex. • For each t ∈ (cid:82) , the level surface K − ( t ) of this function is a complete affine ( C , e t ) -surface in (cid:65) generating D, which is unique.Moreover, the surface K − ( t ) is asymptotic to the boundary of D (c.f. §3.3). Parts (1) and (2) here are exactly the first two parts of Thm. A, and it is easy to see that Part (3)contains Thm. A (3), Thm. B and Cor. C at the same time.
Proof. (1) We apply Thm. F (1) to the convex tube domain Ω × (cid:82) in the aforementioned setting, taking Λ (cid:48) and Λ in the assumption of the theorem to be Γ ∗ τ and Γ ∗ , respectively. As explained, Condition (a) in Thm.F (1) is equivalent to the admissibility of τ , whereas Condition (c) is equivalent to the existence of a Γ τ -invariant C -regular domain by the correspondence between C -regular domains and lower semicontinuousfunctions on ∂ Ω (see Thm. 3.6 (1)). Therefore, the required “if and only if” statement is a consequence ofThm. F (1).To show the statements about the map f , we first note that the domain ∂ (cid:80) ( C ) of the map can bereplaced by ∂ Ω = ∂ (cid:80) ( C ∗ ). In fact, since ∂ (cid:80) ( C ∗ ) can be identified with the set of projective lines in (cid:82)(cid:80) tangent to ∂ (cid:80) ( C ), by the strict convexity and C property of ∂ (cid:80) ( C ) (see Prop. 4.3 (1)), we have a equivarianthomeomorphism ∂ (cid:80) ( C ) ∼= ∂ (cid:80) ( C ∗ ) given by assigning to each x ∈ ∂ (cid:80) ( C ) the line tangent to ∂ (cid:80) ( C ) at x .In view of the identification between the space N C of C -null planes and ∂ Ω × (cid:82) (see §2.1), one can checkthat a map f : ∂ Ω → N C is equivariant if and only if it has the form f ( x ) = ( f ( x ), ϕ ( x )) ∈ ∂ Ω × (cid:82) ∼= N C , ∀ x ∈ ∂ Ω .for some Γ ∗ -equivariant map f : ∂ Ω → ∂ Ω and some function ϕ : ∂ Ω → (cid:82) whose graph is preserved by Γ ∗ τ . If f is further assumed to be continuous, then f can only be the identity because the fixed points ofelements in Γ ∗ form a dense subset of ∂ Ω (see Prop. 4.3 (1)), and ϕ can only be the one provided by Thm.F (1). This shows the unique-existence of f .Now let D + : = ep ◦ ( ϕ ∗ ) be the C -regular domain corresponding to this ϕ (see Thm. 3.6 (1)), and D − ⊂ (cid:65) be the ( − C )-regular domain obtained in a symmetric way. In other words, as explained in the lastparagraph of §3.3, a point in D + (resp. D − ) corresponds to an affine plane in (cid:65) ∗ which cross the convextube domain Ω × (cid:82) ⊂ (cid:65) ∗ from below (resp. above) of gr( ϕ ). So D + and D − are disjoint convex domains.Moreover, we have (cid:65) \ ( D + ∪ D − ) = (cid:91) x ∈ ∂ (cid:80) ( C ) f ( x )because on one hand, a point in the set on the left-hand side corresponds to a non-vertical affine planein (cid:65) ∗ which intersects gr( ϕ ); on the other hand, for each x ∈ ∂ (cid:80) ( C ) ∼= ∂ Ω , points in the C -null plane f ( x )correspond to non-vertical plans in (cid:65) ∗ passing through the point ( x , ϕ ( x )) ∈ gr( ϕ ). Therefore, D ± areexactly the two connected components of (cid:65) \ (cid:83) x ∈ ∂ (cid:80) ( C ) f ( x ). Also, the Γ ∗ τ -invariance of gr( ϕ ) implies that D ± are preserved by Γ τ . This completes the proof of Part (1). FFINE DEFORMATIONS OF QUASI-DIVISIBLE CONVEX CONES 35 (2) The C -regular domains in (cid:65) preserved by Γ τ are exactly the domains of the form ep ◦ ( (cid:98) ϕ ∗ ) for some (cid:98) ϕ described by Condition (c) in Thm. F (1). Therefore, by Thm. F (2), D + is the unique such domain in thedivisible case, whereas in the quasi-divisible but not divisible case, D µ : = ep ◦ ( ϕ ∗ µ ), µ ∈ (cid:82) n ≥ are exactly all such domains. Moreover, the condition D µ ⊂ D µ (cid:48) is equivalent to ϕ ∗ µ ≥ ϕ ∗ µ (cid:48) , which is inturn equivalent to ϕ µ ≤ ϕ µ (cid:48) and hence ϕ µ ≤ ϕ µ (cid:48) by basic properties of Legendre transforms and convexenvelopes. By construction of ϕ µ , the last inequality means µ is coordinate-wise larger than or equal to µ (cid:48) , as required.(3) Fix a C -regular domain D = D µ as above, with µ ∈ (cid:82) n ≥ , and let u t ∈ C ∞ ( Ω ) denote the unique convexsolution to the Dirichlet problem (cid:40) det D u = e − t w − Ω u | ∂ Ω = ϕ µ produced by Thm. F (3) for each t ∈ (cid:82) (although the equation here differs from the one in Thm. F (3) in thatthe parameter t is multiplied by , the conclusions are clearly not affected). Then u t has Γ ∗ τ -invariantgraph and the one-parameter family ( u t ) fulfills the first condition in Prop. 3.13. Moreover, u t satisfiesthe inequalities(5.3) ϕ µ + e − t w Ω ≤ u t ≤ ϕ µ .The second inequality is just because u t | ∂ Ω = ϕ µ (see §3.1), while the first one is given by [NS19, Lemma8.5] and follows easily from the Comparison Principle for Monge-Ampère equations and basic propertiesof Monge-Ampère measures.We shall deduce the required homeomorphism S × (cid:82) ∼ → D / Γ τ from the map F : T − : = (cid:169) ( x , ξ ) ∈ Ω × (cid:82) (cid:175)(cid:175) ξ < ϕ ( x )) (cid:170) −→ D = ep ◦ ( ϕ ∗ µ )( x , u t ( x )) (cid:55)→ (cid:161) D u t ( x ), x · D u t ( x ) − u t ( x ) (cid:162) studied in §3.6, which is a homeomorphism by Prop. 3.14.While the group Γ τ acts on the target D of F by affine transformation, it also acts on the domain T − byprojective transformations via Γ ∗ τ ( i.e. via the isomorphism Aut ( C ) (cid:110) (cid:82) ∼= Aut ( Ω × (cid:82) )), and the geometricdefinition of F in §3.6 implies that F is equivariant with respect to the two actions. But T − is foliatedby the graphs (gr( u t )) ( c.f. Figure 3.2), and the quotient T − / Γ ∗ τ is homeomorphic to S × (cid:82) = ( Ω / Γ ∗ ) × (cid:82) insuch a way that each leaf gr( u t ) corresponds to the slice S × { t } (more precisely, this homeomorphism isgiven by the bijection Ω × (cid:82) ∼ → T − , ( x , t ) (cid:55)→ ( x , u t ( x )), which is shown to be a homeomorphism in the proof ofProp. 3.14). Therefore, both actions are free and properly discontinuous, and F induces a homeomorphismbetween the quotients S × (cid:82) ∼= T − / Γ ∗ τ ∼ −→ D / Γ τ ,sending each slice S × { t } to the quotient of the surface gr( u ∗ t ).This homeomorphism has the required properties in the two bullet points because K is exactly thefunction studied in Prop. 3.13 and proved to be convex therein, whereas K − ( t ) = gr( u ∗ t ) is the uniqueaffine ( C , e t )-surface generating D by Thm. 3.6, Prop. 3.12 and Thm. 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