Related papers: Entropy Rigidity and Hilbert Volume

Hyperideal tetrahedra are the fundamental building blocks of hyperbolic 3-manifolds with geodesic boundary. The study of their geometric properties (in particular, of their volume) has applications also in other areas of low-dimensional…

In the space of closed $G_2$-structures equipped with Bryant's Dirichlet-type metric, we continue to utilise the geodesic, constructed in our previous article, to show that, under a normalisation condition Hitchin's volume functional is…

This paper shows that many hyperbolic manifolds obtained by glueing arithmetic pieces embed into higher-dimensional hyperbolic manifolds as codimension-one totally geodesic submanifolds. As a consequence, many Gromov--Pyatetski-Shapiro and…

For closed hyperbolic $3$-manifolds $M$ with volume less than a constant $V$, we prove an inequality regarding the geometric $L^2$-norm and the topological Thurston norm, which is qualitatively sharp and verifies a conjecture of Brock and…

In this paper we obtain new upper bounds on volumes of right-angled polyhedra in hyperbolic space $\mathbb{H}^3$ in three different cases: for ideal polyhedra with all vertices on the ideal hyperbolic boundary, for compact polytopes with…

We provide a detailed proof of the following folklore theorem: Let mu > 0 be a Margulis constant for 3-dimensional hyperbolic space. Then for any d>0 there exists a constant K>0, depending on mu and d, so that for any complete finite volume…

We study a notion of relative entropy for certain hypersurfaces in hyperbolic space. We relate this quantity to the renormalized area introduced by Graham-Witten[RW99]. We also obtain a monotonicity formula for relative entropy applied to…

Following the work of Li-Shi-Qing, we propose the definition of the relative volume function for an AH manifold. It is not a constant function in general and we study the egularity of this function. We use this function to give an accurate…

We prove that a closed surface with a CAT($\kappa$) metric has Hausdorff dimension = 2, and that there are uniform upper and lower bounds on the two-dimensional Hausdorff measure of small metric balls. We also discuss a connection between…

A question about Ricci flow is when the diameters of the manifold under the evolving metrics stay finite and bounded away from 0. Topping \cite{T:1} addresses the question with an upper bound that depends on the $L^{(n-1)/2}$ bound of the…

For a single cusped hyperbolic 3-manifold, Hodgson proved that there are only finitely many Dehn fillings of it whose trace fields have bounded degree. In this paper, we conjecture the same for manifolds with more cusps, and give the first…

In this paper we study existence and lack thereof of closed embedded orientable co-dimension one totally geodesic submanifolds of minimal volume cusped orientable hyperbolic manifolds.

A periodic geodesic on a surface has a natural lift to the unit tangent bundle; when the complement of this lift is hyperbolic, its volume typically grows as the geodesic gets longer. We give an upper bound for this volume which is linear…

In his paper "On the Schlafli differential equality", J. Milnor conjectured that the volume of n-dimensional hyperbolic and spherical simplices, as a function of the dihedral angles, extends continuously to the closure of the space of…

In this paper, we prove that profinitely rigid finite-volume hyperbolic manifolds form a closed set under geometric topology. This observation implies the profinite rigidity of a large family of cusped hyperbolic manifolds via…

This paper contains a purely topological theorem and a geometric application. The topological theorem states that if M is a simple closed orientable 3-manifold such that \pi_1(M) contains a genus g surface group and H_1(M;Z/2Z) has rank at…

We study a volume/area preserving curvature flow of hypersurfaces that are convex by horospheres in the hyperbolic space, with velocity given by a generic positive, increasing function of the mean curvature, not necessarly homogeneous. For…

We obtain a bound for the area of a capillary $H-$surface in a three-manifold with umbilic boundary and controlled sectional curvature. We then analyze the geometry when this area bound is realized, and obtain rigidity theorems. As a side…

In a variety of settings we provide a method for decomposing a 3-manifold $M$ into pieces. When the pieces have the appropriate type of hyperbolicity, then the manifold $M$ is hyperbolic and its volume is bounded below by the sum of the…

Let ${\mathfrak M}$ be a closed, orientable, hyperbolic 3-orbifold whose singular set is a link, and such that $\pi_1({\mathfrak M})$ contains no hyperbolic triangle group. We show that if the underlying manifold $|{\mathfrak M}|$ is…