EXAMPLES OF FOUR-DIMENSIONAL GEOMETRIC TRANSITION Joint with S. Riolo Fribourg, 8th May 2019
W HAT IS GEOMETRIC TRANSITION ? Roughly speaking, a geometric transition is a deformation of geometric structures on a manifold, which at some point “transitions” from one geometry to another. That is, one has a path P t of (G,X) -structures on a manifold M , for t in (- ε , ε ) , where G and X suddenly change for some t 0 . A classical example of geometric transition is obtained as a deformation from hyperbolic structures to spherical structures, “going through” Euclidean structures: X = ℍ n , G = Isom( ℍ n ) for t < 0 for t = 0 X = 𝔽 n , G = Isom( 𝔽 n ) for t > 0 X = 𝕋 n , G = Isom( 𝕋 n )
H YPERBOLIC -E UCLIDEAN - SPHERICAL TRANSITION AN EXAMPLE A simple example of this transition is obtained by considering a hyperbolic sphere with three cone points, i.e. the double of a (regular, say) hyperbolic triangle. One can then “zoom" around this limit point, and obtain a transition of the geometric structure to a Euclidean sphere with cone points.
H YPERBOLIC -E UCLIDEAN - SPHERICAL TRANSITION ANOTHER EXAMPLE Similarly, starting from a hyperbolic quadrilateral, we can construct a hyperbolic torus with one cone point. When you let the cone angle go to 2 π , the torus collapses to a point. Again, by “zooming", one finds a Euclidean torus as a limit.
In this talk, we will be interested in hyperbolic structures which “collapse" to a co-dimension one totally geodesic hyperplane. An example (from Cooper-Hodgson-Kerckhoff's book): By glueing opposite faces, and then doubling, one constructs a hyperbolic structure on T 3 , singular along a link, which collapses to a hyperbolic torus with one cone point. We want to study "rescaled limits” for this type of degenerations.
W HY DO WE CARE ? HALF-PIPE GEOMETRY In dimension 3, Danciger in 2011 introduced the “right” limit geometry ( half-pipe ) and understood a fairly general condition which ensures the existence of transition on a compact three-manifold M , with a very simple (conical) singularities along a knot K . He proves that collapsing hyperbolic structures on M, with cone singularities along a knot K, admits a geometric transition to half- pipe and Anti-de Sitter structures under the following condition on the character variety: H 1 Ad ρ ( π 1 ( M ∖ K ), 𝔱𝔭 (2,1)) ≅ ℝ . for a representation ρ : π 1 ( M ∖ K ) → SO(2,1) .
W HY DO WE CARE ? HALF-PIPE GEOMETRY In general, it is a difficult problem to determine which manifolds admit a transition of geometric structures. In this talk, we are interested in higher-dimensional examples. It is already pretty hard to produce higher-dimensional examples of hyperbolic manifolds (or other geometric structures) and study their deformations. We will produce here a class of examples of geometric transition from a hyperbolic structure to half-pipe and Anti-de Sitter structures on a closed finite-volume four-manifold with a simple not too complicated singular locus.
W HY DO WE CARE ? THE KERCKHOFF-STORM POLYTOPES In 2010, Kerckhoff and Storm described an interesting deformation of hyperbolic polytopes in dimension 4, depending on a real parameter. Starting from the 24-cell and removing two walls, they construct a polytope which can now be deformed to a family P t , collapsing to a three-dimensional polytope (cuboctahedron) when t=0 . This family of polytopes has been used by several authors (Martelli- Riolo, Saratchandran, Kolpakov-Riolo, Riolo-Slavich). In their paper, Kerckhoff and Storm mention that when t=0 the family of polytopes P t is expected to have interesting geometric limits. It turns out that half-pipe geometry is well-suited to describe this infinitesimal behaviour.
R EAL PROJECTIVE STRUCTURES A REMINDER A real projective structure on a manifold M is given by an atlas with values in n- dimensional projective space, where transition functions are the restrictions of projective transformations. This is the definition of U i U j (G,X)- structure, for: X = ℝ P n ' i : U i ! X ' j : U j ! X ' j ◦ ' − 1 2 G i G = PGL( n + 1, ℝ ) ' j ( U j ) ⊆ X ' i ( U i ) ⊆ X
R EAL PROJECTIVE STRUCTURES SPECIAL CASES Here we are interested in three special types of real projective structures: • Hyperbolic structures X 1 = ℍ n = P { − x 2 n < 0 } , 0 + x 2 1 + … + x 2 n − 1 + x 2 G 1 = PO( n ,1) • Anti-de Sitter structures X − 1 = 𝔹 d 𝕋 n = P { − x 2 0 + x 2 1 + … + x 2 n − 1 − x 2 n < 0 } , G − 1 = PO( n − 1,2) • Half-pipe structures X 0 = ℍℙ n = P { − x 2 n < 0 } , 0 + x 2 1 + … + x 2 n − 1 +0 ⋅ x 2 G 0 = { [ } < PGL( n + 1, ℝ ) v T ±1 ] : A ∈ O( n − 1,1), v ∈ ℝ n A 0
M OTIVATION FOR HALF - PIPE GEOMETRY 1. LIMIT OF HYP&ADS GEOMETRY Consider the projective transformations r t = [ diag ( 1,…,1,1 t ) ] . { X t = r t ( ℍ n ) t > 0 Then define X t = r | t | ( 𝔹 d 𝕋 n ) t < 0 Namely, X t is defined by the inequality X t = P { − x 2 0 + x 2 1 + … + x 2 n − 1 + t | t | x 2 n < 0 } . Then X t converges to: X 0 = ℍℙ n = P { − x 2 n − 1 < 0 } . 0 + x 2 1 + … + x 2
M OTIVATION FOR HALF - PIPE GEOMETRY 1. LIMIT OF HYP&ADS GEOMETRY Moreover, G t converges to: G 0 = { [ } v T ±1 ] : A ∈ O( n − 1,1), v ∈ ℝ n 0 A which is what we define as the “isometry" group of half-pipe space. Half-pipe space X 0 is topologically the X 0 ≅ ℍ n − 1 × ℝ . product However, its geometry is substantially different. For instance, the second factor represents a degenerate direction.
M OTIVATION FOR HALF - PIPE GEOMETRY 2. DUALITY WITH MINKOWSKI SPACE There is another reason why the "isometry group" G 0 of half-pipe geometry is naturally defined as before. In fact, n- dimensional half-pipe space naturally parameterizes ℝ n − 1,1 . spacelike hyperplanes in Minkowski space Given ( x 0,…, x n ), we associate the hyperplane defined by {( y 0 , … y n − 1 ) ∈ ℝ n − 1,1 : − x 0 y 0 + x 1 y 1 + … + x n − 1 y n − 1 = x n } . This is a spacelike hyperplane as a consequence of the condition − x 2 0 + x 2 1 + … + x 2 n − 1 < 0 , ℍℙ n . and, by homogeneity, the correspondence is well-defined on
M OTIVATION FOR HALF - PIPE GEOMETRY 2. DUALITY WITH MINKOWSKI SPACE The "isometry group" G 0 of half-pipe space is then the group Isom( ℝ n − 1,1 ) ≅ O( n − 1,1) ⋉ ℝ n − 1,1 . induced by the action of In fact, it is an exercise to check that the action on spacelike hyperplanes defines an isomorphism Isom( ℝ n − 1,1 ) ≅ G 0 = { [ } . v T ±1 ] : A ∈ O( n − 1,1), v ∈ ℝ n A 0 There is more additional structure in half-pipe geometry determined by the structure of the group G 0 . In fact, it makes sense to talk about a (non-degenerate) totally geodesic hyperplane , which corresponds to the hyperplanes in Minkowski space passing through a given point.
H OW TO PRODUCE EXAMPLES OF TRANSITION Here’s my favourite recipe: • Take a continuous family of finite-volume hyperbolic polytopes P t , for t in [0, ε ), such that P 0 collapses to a polytope in a co- dimension one totally geodesic subspace. • Glue a finite number of copies of P t , so as to obtain a family of hyperbolic structures on a fixed manifold (with some singularities!) • Apply the “rescaling” transformations r t =[diag(1,…,1,1/t)]. • Show that the rescaled polytopes glue to a half-pipe structure when t=0, and even continue towards Anti-de Sitter structures for t in (- ε ,0].
E XAMPLES OF TRANSITION A FIRST EXAMPLE IN 3D Let us now go back to our first 3d example of geometric transition. For simplicity, we send some of the vertices at infinity. Here is an affine picture of the hyperbolic polyhedron we obtain (in the projective model).
E XAMPLES OF TRANSITION A FIRST EXAMPLE IN 3D We can now deform this hyperbolic (ideal) polyhedron, by “flattening” towards a hyperbolic quadrilateral. By glueing faces/doubling, one obtains a deformation of geometric structures (with cone singularities!).
E XAMPLES OF TRANSITION A FIRST EXAMPLE IN 3D Let us now apply the projective rescaling r t =[diag(1,…,1,1/t)]. One shows that the polytope limits to a polytope in half-pipe space, and continues to Anti-de Sitter space!
T HE STRUCTURE OF THE SINGULAR LOCUS When we glue opposite faces, and double, we get finite-volume “cusped” structures, with singularities along a two-component link. • When t>0, this is a cone singularity, for which the cone angle approaches 2 π . • When t<0, this is a tachyon singularity. • When t=0, the singularity is the infinitesimal version of cone angles and tachyons.
T HE STRUCTURE OF THE SINGULAR LOCUS Why does there exist such an element of G 0 , the analogue of a rotation along a line, in half-pipe geometry? Recalling that two totally geodesic hyperplanes P 1 and P 2 in half-pipe space correspond to two points q 1 and q 2 in Minkowski space, P 1 and P 2 intersect if and only if q 1 and q 2 are separated by a spacelike segment. q 2 P 1 q 1 P 2 In this case, the element of G 0 that we are looking for corresponds to the translation sending q 1 to q 2 .
E XAMPLES OF TRANSITION A SECOND EXAMPLE IN 3D We shall now see another Let us now send these three simple example which dihedral angles to π . will be useful later on. All the angles of this polytopes are π /2, except at the three top edges.
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