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Register a new userThe Space of Strictly-convex Real-projective structures on a closed manifold
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This is an expository proof that, if $M$ is a compact $n$-manifold with no boundary, then the set of holonomies of strictly-convex real-projective structures on $M$ is a subset of $operatorname{Hom}(pi_1M,operatorname{PGL}(n+1,mathbb RR))$ that is both open and closed.
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Let $N$ be a compact manifold with a foliation $mathscr{F}_N$ whose leaves are compact strictly convex projective manifolds. Let $M$ be a compact manifold with a foliation $mathscr{F}_M$ whose leaves are compact hyperbolic manifolds of dimension bigger than or equal to $3$. Suppose to have a foliation-preserving homeomorphism $f:(N,mathscr{F}_N) rightarrow (M,mathscr{F}_M)$ which is $C^1$-regular when restricted to leaves. In the previous situation there exists a well-defined notion of foliated volume entropies $h(N,mathscr{F}_N)$ and $h(M,mathscr{F}_M)$ and it holds $h(M,mathscr{F}_M) leq h(N,mathscr{F}_N)$. Additionally, if equality holds, then the leaves must be homothetic.
In this paper, a generalized cusp is a properly convex manifold with strictly convex boundary that is diffeomorphic to $M times [0, infty)$ where $M$ is a closed Euclidean manifold. These are classified in [2]. The marked moduli space is homeomorphic to a subspace of the space of conjugacy classes of representations of $pi_1(M)$. It has one description as a generalization of a trace-variety, and another description involving weight data that is similar to that used to describe semi-simple Lie groups. It is also a bundle over the space of Euclidean similarity (conformally flat) structures on $M$, and the fiber is a closed cone in the space of cubic differentials. For 3-dimensional orientable generalized cusps, the fiber is homeomorphic to a cone on a solid torus.
These notes grew out of our learning and applying the methods of Fock and Goncharov concerning moduli spaces of real projective structures on surfaces with ideal triangulations. We give a self-contained treatment of Fock and Goncharovs description of the moduli space of framed marked properly convex projective structures with minimal or maximal ends, and deduce results of Marquis and Goldman as consequences. We also discuss the Poisson structure on moduli space and its relationship to Goldmans Poisson structure on the character variety.
We present an analytic study of conformal field theories on the real projective space $mathbb{RP}^d$, focusing on the two-point functions of scalar operators. Due to the partially broken conformal symmetry, these are non-trivial functions of a conformal cross ratio and are constrained to obey a crossing equation. After reviewing basic facts about the structure of correlators on $mathbb{RP}^d$, we study a simple holographic setup which captures the essential features of boundary correlators on $mathbb{RP}^d$. The analysis is based on calculations of Witten diagrams on the quotient space $AdS_{d+1}/mathbb{Z}_2$, and leads to an analytic approach to two-point functions. In particular, we argue that the structure of the conformal block decomposition of the exchange Witten diagrams suggests a natural basis of analytic functionals, whose action on the conformal blocks turns the crossing equation into certain sum rules. We test this approach in the canonical example of $phi^4$ theory in dimension $d=4-epsilon$, extracting the CFT data to order $epsilon^2$. We also check our results by standard field theory methods, both in the large $N$ and $epsilon$ expansions. Finally, we briefly discuss the relation of our analysis to the problem of construction of local bulk operators in AdS/CFT.
Let $C$ be a strictly convex domain in a $3$-dimensional Riemannian manifold with sectional curvature bounded above by a constant and let $Sigma$ be a constant mean curvature surface with free boundary in $C$. We provide a pinching condition on the length of the traceless second fundamental form on $Sigma$ which guarantees that the surface is homeomorphic to either a disk or an annulus. Furthermore, under the same pinching condition, we prove that if $C$ is a geodesic ball of $3$-dimensional space forms, then $Sigma$ is either a spherical cap or a Delaunay surface.
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