On convex projective manifolds and cusps

Cooper, Daryl; Long, D. D.; Tillmann, Stephan

doi:10.1016/j.aim.2015.02.009

Cited by 63 publications

(156 citation statements)

References 47 publications

Supporting

Mentioning

143

Contrasting

Unclassified

Order By: Relevance

“…This function is convex so the sublevel set B(φ, t) = h −1 φ (0, t] is convex and is called a horoball associated to φ. It is in general different from the algebraic horoballs defined in [3]. The boundary…”

mentioning

confidence: 99%

“…No similar result holds in the strictly convex setting: there are examples of one cusped 3-manifolds with families of finite volume strictly convex projective structure. This paper leads to a decomposition of the moduli space of such structures, but we do not know if the components of this decomposition are cells.Background for theory of cusped projective manifolds can be found in [3]. A subset Ω ⊂ RP n is properly convex if it is the interior of a compact convex set K that is disjoint from some codimension-1 projective hyperplane and strictly convex if in addition K contains no line segment of positive length in its boundary.…”

mentioning

confidence: 99%

“…Background for theory of cusped projective manifolds can be found in [3]. A subset Ω ⊂ RP n is properly convex if it is the interior of a compact convex set K that is disjoint from some codimension-1 projective hyperplane and strictly convex if in addition K contains no line segment of positive length in its boundary.…”

mentioning

confidence: 99%

“…It is proved in [3] that a strictly convex finite volume real projective manifold has finitely many ends and each is a maximal cusp.…”

mentioning

confidence: 99%

“…The projection, N = Ω − /Γ, of Ω − is a submanifold of M . Since M is finite volume and strictly convex, it is the union of a compact set and finitely many cusps, [3]. We replace Ω − by a K-neighborhood with K so large that the complement of N is now a subset of the cusps of M .…”

mentioning

confidence: 99%

See 4 more Smart Citations

A generalization of the Epstein-Penner construction to projective manifolds

Cooper

Long

2015

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. We extend the canonical cell decomposition due to Epstein and Penner of a hyperbolic manifold with cusps to the strictly convex setting. It follows that a sufficiently small deformation of the holonomy of a finite volume strictly convex real projective manifold is the holonomy of some nearby projective structure with radial ends, provided the holonomy of each cusp has a fixed point.One of the powerful constructions in theory of cusped hyperbolic n-manifolds is a cellulation constructed by Epstein & Penner in [5], which in the particular case that the manifold has one cusp, gives rise to a canonical cell decomposition. In this note we extend their results to the case of strictly convex real projective manifolds. One consequence is that a small deformation of the holonomy of a finite volume strictly convex structure on M is the holonomy of some (possibly not strictly convex) structure on M with radial ends provided the holonomy of each cusp has a fixed point in projective space, see Theorem 0.4.The proof of [5] employs Minkowski space and shows that if p is a point on the lightcone that corresponds to a parabolic fixed point then p has a discrete orbit. The convex hull of this orbit is an infinite sided polytope in Minkowski space that is preserved by the group. The boundary of the quotient of this polytope by the group gives the cell decomposition. This approach uses in an essential way the quadratic form β = x 2 1 + · · · + x 2 n − x 2 n+1 that defines O(n, 1) to identify Minkowski space with its dual. This gives a bijection between points in the orbit of p and horoballs that cover the cusp corresponding to p. The fact these horoballs are disjoint implies the orbit of p is discrete.In this paper we use a Vinberg hypersurface to give a bijection between the orbit of p and horoballs in the universal cover of the dual projective manifold that cover the dual cusp.In the hyperbolic case in dimension 2 one obtains a cell decomposition of moduli space from the result of Epstein and Penner, [8]. For finite volume hyperbolic structures Mostow-Prasad rigidity implies that in dimension at least 3 the moduli space is a point. No similar result holds in the strictly convex setting: there are examples of one cusped 3-manifolds with families of finite volume strictly convex projective structure. This paper leads to a decomposition of the moduli space of such structures, but we do not know if the components of this decomposition are cells.Background for theory of cusped projective manifolds can be found in [3]. A subset Ω ⊂ RP n is properly convex if it is the interior of a compact convex set K that is disjoint from some codimension-1 projective hyperplane and strictly convex if in addition K contains no line segment of positive length in its boundary.A strictly convex real projective n-manifold is M = Ω/Γ where Ω ⊂ RP n is strictly convex and Γ ∼ = π 1 M is a discrete group of projective transformations that preserves Ω and acts freely on it. We may, and will, lift Γ to a subgroup of SL(Ω) which is the group of matri...

show abstract

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…It is proved in [3] that a strictly convex finite volume real projective manifold has finitely many ends and each is a maximal cusp.…”

mentioning

confidence: 99%

mentioning

confidence: 99%