arXiv research

Inverts operator on hyperbolic surfaces, constructing invariant distributions.

The article constructs Feynman propagators for normally hyperbolic operators on curved spacetimes.

Study geometric isomorphisms between spacetime solutions using paracausal metrics.

Formula for Hadamard coefficients from Green's operators on spacetimes.

Study shows stability in X-ray transform on specific hyperbolic manifolds.

The paper derives expansions for Green's operators and resolvents using Hadamard methods.

The paper provides formulas for Hadamard coefficients using Green's operators.

The paper solves the Cauchy problem for Friedrichs systems on specific spacetime manifolds.

Differentiates Fréchet mean for hyperbolic space applications.

A new method reformulates Optimal Transport Conditional Flow Matching using proximal operators.

We derive upper eigenvalue bounds for the Dirac operator of a closed hypersurface in a manifold with Killing spinors such as Euclidean space, spheres or hyperbolic space. The bounds involve the Willmore functional. Relations with the Willmore inequality are briefly discussed. In higher codimension we obtain bounds on t…

In these lecture notes we discuss the solution theory of geometric wave equations as they arise in Lorentzian geometry: for a normally hyperbolic differential operator the existence and uniqueness properties of Green functions and Green operators is discussed including a detailed treatment of the Cauchy problem on a gl…

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 …

This paper considers asymptotically hyperbolic manifolds with a finite boundary intersecting the usual infinite boundary -- cornered asymptotically hyperbolic manifolds -- and proves a theorem of Cartan-Hadamard type near infinity for the normal exponential map on the finite boundary. As a main application, a normal fo…

We apply the ADM approach to obtain a Hamiltonian description of the Einstein-Hilbert action. In doing so we add four new ingredients: (i) We eliminate the diffeomorphism constraints. (ii) We replace the densities $\sqrt g$ by a function $\f(x,g_{ij})$ with the help of a fixed metric $χ$ such that the Lagrangian and he…

New normalizing flows in hyperbolic space improve posterior modeling for hierarchical data.

Let N be a complete Riemannian manifold of dimension n+1 whose Riemannian metric g is conformally equivalent to a metric with non-negative Ricci curvature. The normalized Steklov eigenvalues of a bounded domain in N are bounded above in terms of the isoperimetric ratio of the domain. Consequently, the normalized Steklo…

Mahan Mitra (Mj) proved Cannon--Thurston maps exist for normal hyperbolic subgroups of a hyperbolic group. We prove that Cannon--Thurston maps do not exist for infinite normal hyperbolic subgroups of non-hyperbolic CAT(0) groups with isolated flats with respect to the visual boundaries. We also show Cannon--Thurston ma…

We classify all of real hypersurfaces $M$ with Reeb invariant shape operator in complex hyperbolic two-plane Grassmannians $SU_{2,m}/S(U_2{\cdot}U_m)$, $m \geq 2$. Then it becomes a tube over a totally geodesic $SU_{2,m-1}/S(U_2{\cdot}U_{m-1})$ in $SU_{2,m}/S(U_2{\cdot}U_m)$ or a horosphere whose center at infinity is …

We prove that strictly hyperbolized smooth cube manifolds admit normal smooth structures.

The study examines the normal growth exponent of submanifolds in negatively curved manifolds.

CR Killing operator derived from tractor calculus for CR structures.

An automorphism $α$ of a group $G$ is normal if it fixes every normal subgroup of $G$ setwise. We give an algebraic description of normal automorphisms of relatively hyperbolic groups. In particular, we prove that for any relatively hyperbolic group $G$, $Inn(G)$ has finite index in the subgroup $Aut_n(G)$ of normal au…

Defines mass for non-smooth hyperbolic spaces using a modified flow.

In many problems of PDE involving the Laplace-Beltrami operator on manifolds with ends, it is often useful to introduce radial or geodesic normal coordinates near infinity. In this paper, we prove the existence of such coordinates for a general class of manifolds with ends, which contains asymptotically conical and hyp…

Unified framework for Riemannian deep learning across manifold-valued representations.

Constructs a solution operator for hyperbolic gluing in higher dimensions.

This study connects financial volatility to quantum mechanics on hyperbolic manifolds.

New Kähler manifolds found with nonpositive curvature operators.

Paper proves non-existence of certain hypersurfaces in complex quadric.

This note analyzes the normal form of gradient Ricci 4-solitons.

Quantizes Maxwell's theory on Lorentzian manifolds via a novel gauge-fixing method.

Geometric operators link solutions on different spacetimes.

This paper studies the normalized Ricci flow from a slight perturbation of the hyperbolic metric on $\mathbb H^n$. It's proved that if the perturbation is small and decays sufficiently fast at the infinity, then the flow will converge exponentially fast to the hyperbolic metric when the dimension $n>5$.

Green-hyperbolic operators are linear differential operators acting on sections of a vector bundle over a Lorentzian manifold which possess advanced and retarded Green's operators. The most prominent examples are wave operators and Dirac-type operators. This paper is devoted to a systematic study of this class of diffe…

Defines horocyclic evolutes, parallels, and involutes of spacelike frontals in hyperbolic 2-space.

Guillarmou extends X-ray transform to magnetic and thermostat flows.

Study on Dirac operator spectrum on hyperbolic surfaces with shrinking geodesics.

In this study, a numerical quadrature for the generalized inverse Gaussian distribution is derived from the Gauss-Hermite quadrature by exploiting its relationship with the normal distribution. The proposed quadrature is not Gaussian, but it exactly integrates the polynomials of both positive and negative orders. Using…

The study examines the normalized volumes of right-angled hyperbolic polyhedra and their spectra.

We prove a persistence result for noncompact normally hyperbolic invariant manifolds in Riemannian manifolds of bounded geometry. The bounded geometry of the ambient manifold is a crucial assumption in order to control the uniformity of all estimates throughout the proof.

The study classifies and normalizes 3D gl-regular Nijenhuis operators.

Unique inhomogeneous ruled hypersurface found in complex hyperbolic space.

Constructs spin hyperbolic surfaces with a spectral gap for Dirac operator.

We consider a hyperbolic Dirac-type operator with growing potential on a a spatially non-compact globally hyperbolic manifold. We show that the Atiyah-Patodi-Singer boundary value problem for such operator is Fredholm and obtain a formula for this index in terms of the local integrals and the relative eta-invariant int…

Study finds both existence and non-existence of maps in Morse boundaries.

The study finds the minimum average area ratio on hyperbolic manifolds and its relation to scalar curvature.

Study geodesic X-ray transforms on hyperbolic surfaces, proposing new reconstruction methods.