Related papers: Projective Deformations of Hyperbolic Coxeter 3-Or…

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $\mathrm{SL}(n+1, \mathbb{R})$ or $\mathrm{PGL}(n+1, \mathbb{R})$. A recent work shows that many hyperbolic…

We study geometry, topology and deformation spaces of noncompact complex hyperbolic manifolds (geometrically finite, with variable negative curvature), whose properties make them surprisingly different from real hyperbolic manifolds with…

In this paper, we demonstrate that the complete hyperbolic structure of various two-bridge knots and links cannot be deformed to an inequivalent strictly convex projective structure. We also prove a complementary result showing that under…

Let X be a space of constant curvature and P be a convex polyhedron in X. A Coxeter decomposition of the polyhedron P is a decomposition of P into finitely many Coxeter polyhedra, such that any two polyhedra having a common facet are…

Motivated by classical theorems on minimal surface theory in compact hyperbolic three-manifolds, we investigate the questions of existence and deformations for least area minimal surfaces in complete noncompact hyperbolic three-manifold of…

This is the third in a series of papers constructing hyperbolic structures on all Haken three-manifolds. This portion deals with the mixed case of the deformation space for manifolds with incompressible boundary that are not acylindrical,…

We construct two-dimensional families of complex hyperbolic structures on disc orbibundles over the sphere with three cone points. This contrasts with the previously known examples of the same type, which are locally rigid. In particular,…

Let $(M, \partial M)$ be a compact 3-manifold with boundary which admits a complete, convex co-compact hyperbolic metric. For each hyperbolic metric $g$ on $M$ such that $\dr M$ is smooth and strictly convex, the induced metric on $\dr M$…

Since the end of the 19th century, and after the works of F. Klein and H. Poincar\'e, it is well known that models of elliptic geometry and hyperbolic geometry can be given using projective geometry, and that Euclidean geometry can be seen…

We establish Marstrand-type projection theorems for orthogonal projections along geodesics onto m-dimensional subspaces of hyperbolic $n$-space by a geometric argument. Moreover, we obtain a Besicovitch-Federer type characterization of…

The space of marked n distinct points on the complex projective line up to projective transformations will be called a configuration space in this paper. There are two families of complex hyperbolic structures on the configuration space…

We study deformations of complex hyperbolic surfaces which furnish the simplest examples of: (i) negatively curved K\"ahler manifolds and (ii) negatively curved Riemannian manifolds not having {\it constant} curvature. Although such complex…

We prove that a closed 3-orbifold that fibers over a hyperbolic polygonal 2-orbifold admits a family of hyperbolic cone structures that are viewed as regeneration of the polygon, provided that the perimeter is minimal.

We compute Coxeter diagrams of several ``large'' reflective even 2-elementary hyperbolic lattices and their maximal parabolic subdiagrams, and give some applications of these results to the theory of K3 surfaces and hyperkahler varieties.

We review several results related to the characterization of polyhedra in hyperbolic 3-space. In particular we present Rivin's theorem that gives a characterization of compact convex hyperbolic polyhedra, and Hodgson's proof of the Adreev's…

We prove 3-dimensional hyperbolic cone-manifolds are geometrically inflexible: a cone-deformation of a hyperbolic cone-manifold determines a bi-Lipschitz diffeomorphism between initial and terminal manifolds in the deformation in the…

Classical fully augmented links have explicit hyperbolic geometry, and have diagrams on the 2-sphere in the 3-sphere. We generalise to construct fully augmented links projected to the reflection surface of any 3-manifold obtained by…

In $\mathbb{R}^3$, a hyperbolic paraboloid is a classical saddle-shaped quadric surface. Recently, Elser has modeled problems arising in Deep Learning using rectangular hyperbolic paraboloids in $\mathbb{R}^n$. Motivated by his work, we…

We show that any immersion, which is not a covering of an embedded 2-orbifold, of a totally geodesic hyperbolic turnover in a complete orientable hyperbolic 3-orbifold is contained in a hyperbolic 3-suborbifold with totally geodesic…

We show that the moduli space M of marked cubic surfaces is biholomorphic to the quotient by a discrete group generated by complex reflections of the complex four-ball minus the reflection hyperplanes of the group. Thus M carries a complex…