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Register a new userGeneralized Cusps in Real Projective Manifolds: Classification
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A generalized cusp $C$ is diffeomorphic to $[0,infty)$ times a closed Euclidean manifold. Geometrically $C$ is the quotient of a properly convex domain by a lattice, $Gamma$, in one of a family of affine groups $G(psi)$, parameterized by a point $psi$ in the (dual closed) Weyl chamber for $SL(n+1,mathbb{R})$, and $Gamma$ determines the cusp up to equivalence. These affine groups correspond to certain fibered geometries, each of which is a bundle over an open simplex with fiber a horoball in hyperbolic space, and the lattices are classified by certain Bieberbach groups plus some auxiliary data. The cusp has finite Busemann measure if and only if $G(psi)$ contains unipotent elements. There is a natural underlying Euclidean structure on $C$ unrelated to the Hilbert metric.
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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.
This paper uses work of Haettel to classify all subgroups of PGL(4,R) isomorphic to (R^3 , +), up to conjugacy. We use this to show there are 4 families of generalized cusps up to projective equivalence in dimension 3.
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, we exploit a subtle indeterminacy in the definition of the spherical Kervaire-Milnor invariant which was discovered by R. Stong to construct non-spin 4-manifolds with even intersection form and prescribed signature.
We study the limit geometry of complete projective special real manifolds. By limit geometry we mean the limit of the evolution of the defining polynomial and the centro-affine fundamental form along certain curves that leave every compact subset of the initial complete projective special real manifold. We obtain a list of possible limit geometries, which are themselves complete projective special real manifolds, and find a lower limit for the dimension of their respective symmetry groups. We further show that if the initial manifold has regular boundary behaviour, every possible limit geometry is isomorphic to $mathbb{R}_{>0}ltimesmathbb{R}^{n-1}$.
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