arXiv research

Study on ordering geodesics on hyperbolic surfaces.

For hyperbolic 3-manifolds, Ruelle zeta function vanishing order is 4 minus Betti number.

Study on uniqueness of hypersurfaces in hyperbolic space with constant mean curvature.

We investigate the orderability properties of fundamental groups of 3-dimensional manifolds. Many 3-manifold groups support left-invariant orderings, including all compact P^2-irreducible manifolds with positive first Betti number. For seven of the eight geometries (excluding hyperbolic) we are able to characterize whi…

This paper shows hyperbolic knots can have arbitrarily large torsion in knot Floer homology.

We give a complete list of the cobounded actions of solvable Baumslag-Solitar groups on hyperbolic metric spaces up to a natural equivalence relation. The set of equivalence classes carries a natural partial order first introduced by Abbott-Balasubramanya-Osin, and we describe the resulting poset completely. There are …

The paper studies rigidity and stability of minimal submanifolds in hyperbolic space.

Minimal Delaunay triangulations on hyperbolic surfaces have linear number of vertices.

New positive mass theorem for hyperbolic 3-manifolds using Green functions.

Of all real Lagrangian--Grassmannians $LG(n,2n)$, only $LG(2,4)$ admits a distinguished (Lorentzian) conformal structure and hence is identified with the indefinite M\"obius space $S^{1,2}$. Using Cartan's method of moving frames, we study hyperbolic (timelike) surfaces in $LG(2,4)$ modulo the conformal symplectic gro…

We prove that a smooth Riemannian manifold admitting an imaginary generalized Killing spinor whose Dirac current satisfies an additional algebraic constraint condition can be embedded as spacelike Cauchy hypersurface in a smooth Lorentzian manifold on which the given spinor extends to a null parallel spinor. This is in…

The paper consists of two parts. In the first part, by using the Gauss-Bonnet curvature, which is a natural generalization of the scalar curvature, we introduce a higher order mass, the Gauss-Bonnet-Chern mass $m^{\H}_k$, for asymptotically hyperbolic manifolds and show that it is a geometric invariant. Moreover, we pr…

We calculate the Lefschetz number of a Galois automorphism in the cohomology of certain arithmetic congruence groups arising from orders in quaternion algebras over number fields. As an application we give a lower bound for the first Betti number of a class of arithmetically defined hyperbolic 3-manifolds and we deduce…

We propose an extension of the recently-proposed volume conjecture for closed hyperbolic 3-manifolds, to all orders in perturbative expansion. We first derive formulas for the perturbative expansion of the partition function of complex Chern-Simons theory around a hyperbolic flat connection, which produces infinitely-m…

Researchers recover Riemannian manifolds and lower order terms from travel time data.

Study stabilizes second-order systems to first-order dynamics.

New examples of hyperbolic links with generalized torsion elements found.

Unified framework for hyperbolic embeddings from mixed data types.

There are various results that frame left-orderability of a group as a geometric property. Indeed, the fundamental group of a 3-manifold is left-orderable whenever the first Betti number is positive; in the case that the first Betti number is zero this property is closely tied to the existence of certain nice foliation…

We obtain the exact values of the systoles of these hyperbolic surfaces of genus $g$ with cyclic symmetries of the maximum order and the next maximum order. Precisely: for genus $g$ hyperbolic surface with order $4g+2$ cyclic symmetry, the systole is $2\mathrm{arccosh} (1+\cos \fracπ{2g+1}+\cos \frac{2π}{2g+1})$ when $…

The paper studies hyperbolic equations in a spacetime foliation, proving existence and uniqueness.

An important problem in quaternionic hyperbolic geometry is to classify ordered $m$-tuples of pairwise distinct points in the closure of quaternionic hyperbolic n-space, $\overline{{\bf H}_\bh^n}$, up to congruence in the holomorphic isometry group ${\rm PSp}(n,1)$ of ${\bf H}_\bh^n$. In this paper we concentrate on tw…

Three counterexamples show higher eigenvalue multiplicities than conjectured.

Localized Kasner-like singularities constructed in spacetime.

This paper is the first in a series of two articles whose aim is to extend a recent result of Guillarmou-Lefeuvre on the local rigidity of the marked length spectrum from the case of compact negatively-curved Riemannian manifolds to the case of manifolds with hyperbolic cusps. In this first paper, we deal with the line…

New quantum invariant is asymptotically multiplicative under cyclic covers.

The paper connects hyperbolic spinors to non-null framed curves in Minkowski 3-space.

Prove first-band large-diameter asymptotics for Dirichlet spectrum on horoconvex domains in real hyperbolic space.

Learning graph representations via low-dimensional embeddings that preserve relevant network properties is an important class of problems in machine learning. We here present a novel method to embed directed acyclic graphs. Following prior work, we first advocate for using hyperbolic spaces which provably model tree-li…

Study determines left-orderable properties of knot covers.

We prove that the first order theory of nonabelian free groups eliminates the "there exists infinitely many" quantifier (in eq). Equivalently, since the theory of nonabelian free groups is stable, it does not have the finite cover property. We also extend our results to torsion-free hyperbolic groups under some conditi…

We define a stochastic model of a two-sided limit order book in terms of its key quantities \textit{best bid [ask] price} and the \textit{standing buy [sell] volume density}. For a simple scaling of the discreteness parameters, that keeps the expected volume rate over the considered price interval invariant, we prove a…

The paper deals with a formally self-adjoint first order linear differential operator acting on m-columns of complex-valued half-densities over an n-manifold without boundary. We study the distribution of eigenvalues in the elliptic setting and the propagator in the hyperbolic setting, deriving two-term asymptotic form…

We describe the first-order variations of the angles of Euclidean, spherical or hyperbolic polygons under infinitesimal deformations such that the lengths of the edges do not change. Using this description, we introduce a vector-valued quadratic invariant $b$ on the space of those isometric deformations which, for conv…

New insights into a complex hyperbolic braid group quotient.

Central limit theorem for Green metrics on hyperbolic groups.

The paper shows how different geodesic flows on surfaces can be mapped to each other.

We are concerned with orderable groups and particularly those with orderings invariant not only under multiplication, but also under a given automorphism or family of automorphisms. Several applications to topology are given: we prove that the fundamental groups of hyperbolic nonorientable surfaces, and the groups of c…

Study shows surgeries on certain knots yield left-orderable 3-manifolds.

New method detects left-orderable surgeries on knot 6_2.

Paper presents a new method for learning hyperbolic representations using tree structures.

Sharp inequalities for radial functions on hyperbolic spaces without boundary conditions.

We study complex hyperbolic disc bundles over closed orientable surfaces that arise from discrete and faithful representations H_n->PU(2,1), where H_n is the fundamental group of the orbifold S^2(2,...,2) and thus contains a surface group as a subgroup of index 2 or 4. The results obtained provide the first complex hyp…

This paper is a further extension of the method proposed in Itkin, 2014 as applied to another set of jump-diffusion models: Inverse Normal Gaussian, Hyperbolic and Meixner. To solve the corresponding PIDEs we accomplish few steps. First, a second-order operator splitting on financial processes (diffusion and jumps) is …

Extensions (entropies) play a central role in the theory of hyperbolic conservation laws by providing intrinsic selection criteria for weak solutions. For a given hyperbolic system u_t+f(u)_x=0, a standard approach is to analyze directly the second order PDE system for the extensions. Instead we find it advantageous to…

A Coxeter $n$-orbifold is an $n$-dimensional orbifold based on a polytope with silvered boundary facets. Each pair of adjacent facets meet on a ridge of some order $m$, whose neighborhood is locally modeled on ${\mathbb R}^n$ modulo the dihedral group of order $2m$ generated by two reflections. For $n \geq 3$, we study…

The paper defines two types of hyperbolicity for complex manifolds and proves related results.

The gluing equations of a cusped hyperbolic 3-manifold $M$ are a system of polynomial equations in the shapes of an ideal triangulation $\calT$ of $M$ that describe the complete hyperbolic structure of $M$ and its deformations. Given a Neumann-Zagier datum (comprising the shapes together with the gluing equations in a …