Related papers: Volume maximization and the extended hyperbolic sp…
Let $M$ be a non-compact hyperbolic $3$-manifold with finite volume and totally geodesic boundary components. By subdividing mixed ideal polyhedral decompositions of $M$, under some certain topological conditions, we prove that $M$ has an…
Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle $\mathrm{PSL}_2(\mathbb{Z})\backslash\mathrm{PSL}_2(\mathbb{R})$. The complement of any finite number of orbits is a…
Let $S$ be a closed oriented surface of genus at least $2$. Using the parameterisation of the deformation space of globally hyperbolic maximal anti-de Sitter structures on $S \times \mathbb{R}$ by the cotangent bundle over the Teichm\"uller…
The symmedian point of a triangle enjoys several geometric and optimality properties, which also serve to define it. We develop a new dynamical coordinatization of the symmedian, which naturally generalizes to other ideal hyperbolic…
We give an upper bound for the number of compact essential orientable non-isotopic surfaces, with Euler characteristic at least some constant $\chi$, properly embedded in a finite-volume hyperbolic 3-manifold $M$, closed or cusped. This…
We consider contracting flows in $(n+1)$-dimensional hyperbolic space and expanding flows in $(n+1)$-dimensional de Sitter space. When the flow hypersurfaces are strictly convex we relate the contracting hypersurfaces and the expanding…
A three-dimensional orthoscheme is defined as a tetrahedron whose base is a right-angled triangle and an edge joining the apex and a non-right-angled vertex is perpendicular to the base. A generalization, called complete orthoschemes, of…
We investigate several topics of triangle geometry in the elliptic and in the extended hyperbolic plane, such as: centers based on orthogonality, centers related to circumcircles and incircles, radical centers and centers of similitude,…
In this paper we give lower and upper bounds for the volume growth of a regular hyperbolic simplex, namely for the ratio of the $n$-dimensional volume of a regular simplex and the $(n-1)$-dimensional volume of its facets. In addition to the…
It is proved that the volume of spherical or hyperbolic simplices, when considered as a function of the dihedral angles, can be extended continuously to degenerated simplices.
In this paper we study the systoles of arithmetic hyperbolic 2- and 3-manifolds. Our first result is the construction of infinitely many arithmetic hyperbolic 2- and 3-manifolds which are pairwise noncommensurable, all have the same…
We propose an approach to find constant curvature metrics on triangulated closed 3-manifolds using a finite dimensional variational method whose energy function is the volume. The concept of an angle structure on a tetrahedron and on a…
In this paper, we study the rigidity of hyperbolic polyhedral 3-manifolds and the volume optimization program of angle structures. We first study the rigidity of decorated 1-3 type hyperbolic polyhedral metrics on 3-manifolds which are…
We study the problem of bounding the number of cusps of a complex hyperbolic manifold in terms of its volume. Applying algebro-geometric methods using Mumford's work on toroidal compactifications and its generalization due to N. Mok and…
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…
We study the symplectic geometry of the moduli space of closed n-gons with fixed side-lengths in hyperbolic 3-space. We prove that these moduli spaces have a symplectic structure coming from Poisson Lie theory. We construct completely…
We construct for every connected surface $S$ of finite negative Euler characteristic and every $H \in [0,1)$, a hyperbolic 3-manifold $N(S,H)$ of finite volume and a proper, two-sided, totally umbilic embedding $f\colon S\to N(S,H)$ with…
We study the supremum of the volume of hyperbolic polyhedra with some fixed combinatorics and with vertices of any kind (real, ideal or hyperideal). We find that the supremum is always equal to the volume of the rectification of the…
We compare the volume of a hyperbolic 3-manifold $M$ of finite volume and the complexity of its fundamental group.
We produce a large class of hyperbolic homology 3-spheres admitting arbitrarily many distinct tight contact structures. We also produce a sub-class admitting arbitrarily many distinct tight contact structures within the same homotopy class…