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The Area of Convex Projective Surfaces  [PDF]
Ilesanmi Adeboye,Daryl Cooper
Mathematics , 2015,
Abstract: The area in the Hilbert metric of a compact, properly convex, projective surface $F$ is at least $\pi^2|\chi (F)|$. The area of an ideal triangle with Fock-Goncharov parameter $t$ is at least $(\pi^2+(\log t)^2)/2$.
Moduli spaces of convex projective structures on surfaces  [PDF]
V. V. Fock,A. B. Goncharov
Mathematics , 2004,
Abstract: We define convex projective structures on 2D surfaces with holes and investigate their moduli space. We prove that this moduli space is canonically identified with the higher Teichmuller space for the group PSL_3 defined in our paper math/0311149. We define the quantum version of the moduli space of convex projective structures on surfaces with holes. The present paper can serve as an introduction to math/0311149. In the Appendix we show that the space of configurations of 5 flags in the projective plane is of cluster type E_7.
The degeneration of convex RP^2 structures on surfaces  [PDF]
Tengren Zhang
Mathematics , 2013,
Abstract: Let M be a compact surface of negative Euler characteristic and let C(M) be the deformation space of convex real projective structures on M. For every choice of pants decomposition for M, there is a well known parameterization of C(M) known as the Goldman parameterization. In this paper, we study how some geometric properties of the real projective structure on M degenerates as we deform it so that the internal parameters of the Goldman parameterization leave every compact set while the boundary invariants remain bounded away from zero and infinity.
Convex real projective structures and Hilbert metrics  [PDF]
Inkang Kim,Athanase Papadopoulos
Mathematics , 2014,
Abstract: We review some basic concepts related to convex real projective structures from the differential geometry point of view. We start by recalling a Riemannian metric which originates in the study of affine spheres using the Blaschke connection (work of Calabi and of Cheng-Yau) mentioning its relation with the Hilbert metric. We then survey some of the deformation theory of convex real projective structures on surfaces. We describe in particular how the set of (Hilbert) lengths of simple closed curves is used in a parametrization of the deformation space in analogy with the classical Fenchel-Nielsen parameters of Teichm\"uller space (work of Goldman). We then mention parameters of this deformation space that arise in the work of Hitchin on the character variety of representations of the fundamental group of the surface in $\mathrm{SL}(3,\mathbb{R})$. In this character variety, the component of the character variety that corresponds to projective structures is identified with the vector space of pairs of holomorphic quadratic and cubic differentials over a fixed Riemann surface. Labourie and Loftin (independently) obtained parameter spaces that use the cubic differentials and affine spheres. We then display some similarities and differences between Hilbert geometry and hyperbolic geometry using geodesic currents and topological entropy. Finally, we discuss geodesic flows associated to Hilbert metrics and compactifications of spaces of convex real projective structures on surfaces. This makes another analogy with works done on the Teichm\"uller space of the surface.
Automorphisms of surfaces: Kummer rigidity and measure of maximal entropy  [PDF]
Serge Cantat,Christophe Dupont
Mathematics , 2014,
Abstract: We classify complex projective surfaces with an automorphism of positive entropy for which the unique invariant measure of maximal entropy is absolutely continuous with respect to Lebesgue measure.
An algorithm for the Euclidean cell decomposition of a cusped strictly convex projective surface  [PDF]
Stephan Tillmann,Sampson Wong
Mathematics , 2015,
Abstract: Cooper and Long generalised Epstein and Penner's Euclidean cell decomposition of cusped hyperbolic manifolds of finite volume to non-compact strictly convex projective manifolds of finite volume. We show that Weeks' algorithm to compute this decomposition for a hyperbolic surface generalises to strictly convex projective surfaces.
Entropies of compact strictly convex projective manifolds  [PDF]
Micka?l Crampon
Mathematics , 2009,
Abstract: Let M be a compact manifold of dimension n with a strictly convex projective structure. We consider the geodesic flow of the Hilbert metric on it, which is known to be Anosov. We prove that its topological entropy is less than n-1, with equality if and only if the structure is Riemannian, that is hyperbolic. As a corollary, we get that the volume entropy of a divisible strictly convex set is less than n-1, with equality if and only if it is an ellipsoid.
Complex projective surfaces and infinite groups  [PDF]
Fedor Bogomolov,Ludmil Katzarkov
Mathematics , 1997,
Abstract: The paper contains a general construction which produces new examples of non simply-connected smooth projective surfaces. We analyze the resulting surfaces and their fundamental groups. Many of these fundamental groups are expected to be non-residually finite. Using the construction we also suggest a series of potential counterexamples to the Shafarevich conjecture which claims that the universal covering of smooth projective variety is holomorphically convex. The examples are only potential since they depend on group theoretic questions, which we formulate, but we do not know how to answer. At the end we formulate an arithmetic version of the Shafarevich conjecture.
Theorems on tangencies in projective and convex geometry  [PDF]
Roland Abuaf
Mathematics , 2011,
Abstract: We discuss various phenomena of tangency in projective and convex geometry.
On Convex Projective Manifolds and Cusps  [PDF]
Daryl Cooper,Darren Long,Stephan Tillmann
Mathematics , 2011,
Abstract: This study of properly or strictly convex real projective manifolds introduces notions of parabolic, horosphere and cusp. Results include a Margulis lemma and in the strictly convex case a thick-thin decomposition. Finite volume cusps are shown to be projectively equivalent to cusps of hyperbolic manifolds. This is proved using a characterization of ellipsoids in projective space. Except in dimension 3, there are only finitely many topological types of strictly convex manifolds with bounded volume. In dimension 4 and higher, the diameter of a closed strictly convex manifold is at most 9 times the diameter of the thick part. There is an algebraic characterization of strict convexity in terms of relative hyperbolicity.
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