arXiv research

Central limit theorem for Green metrics on hyperbolic groups.

Network metrics form a fundamental part of the network analysis toolbox. Used to quantitatively measure different aspects of the network, these metrics can give insights into the underlying network structure and function. In this work, we connect network metrics to modern probabilistic machine learning. We focus on the…

New centrality-based graph shift operators improve graph neural networks.

In this paper we prove that the Kähler-Einstein metrics for a degeneration family of Kähler manifolds with ample canonical bundles Gromov-Hausdorff converge to the complete Kähler-Einstein metric on the smooth part of the central fiber when the central fiber has only normal crossing singularities inside smooth total sp…

Study of centralizer elements preserving geodesic flow foliations on covers.

Solutions to the $n$-dimensional Laplace equation which are constant on a central quadric are found. The associated twistor description of the case $n=3$ is used to characterise Gibbons-Hawking metrics with tri-holomorphic $SL(2, \C)$ symmetry.

Study on 4-manifolds for special Kähler metrics with constant Ricci determinant.

We prove central limit theorems for the random walks on either the mapping class group of a closed, connected, orientable, hyperbolic surface, or on $\text{Out}(F_N)$, each time under a finite second moment condition on the measure (either with respect to the Teichmüller metric, or with respect to the Lipschitz metric …

Generalizes k-means to graphs using PageRank.

The study proves a central limit theorem for Gaussian holomorphic sections on Kähler manifolds.

Study shows mass distribution of random holomorphic sections follows a central limit theorem.

Given a (smooth) complex analytic family of compact complex manifolds, we prove that the central fibre must be Moishezon if the other fibres are Moishezon. Using a "strongly Gauduchon metric" on the central fibre whose existence was proved in our previous work on limits of projective manifolds, we show that the irreduc…

This paper explores the impact of metric choice on Fréchet regression.

Study finds significant price declines and capital reallocation from centralized to decentralized exchanges after FTX collapse.

In this paper we characterize planar central configurations in terms of a sectional curvature value of the Jacobi-Maupertuis metric. This characterization works for the $N$-body problem with general masses and any $1/r^α$ potential with $α> 0$. We also observe dynamical consequences of these curvature values for relati…

Analyzes Kähler-Einstein metrics on families of Fano varieties.

For a Riemannian foliation on a closed manifold, the first secondary invariant of Molino's central sheaf is an obstruction to tautness. Another obstruction is the class defined by the basic component of the mean curvature with respect to some metric. Both obstructions are proved to be the same up to a constant, and oth…

Conditions for the existence of Kähler-Einstein metrics and central Kähler metrics [MS] along with examples, both old and new, are given on classes of Lorentzian $4$-manifolds with two distinguished vector fields. The results utilize the general construction [AM] of Kähler metrics on such manifolds. The examples includ…

Zamolodchikov's c-theorem type argument (and also string theory effective action constructions) imply that the RG flow in 2d sigma model should be gradient one to all loop orders. However, the monotonicity of the flow of the target-space metric is not obvious since the metric on the space of metric-dilaton couplings is…

Survey on Thurston metric on Teichmüller space, focusing on extremal maps.

In this paper we prove that the Kähler-Einstein metrics for a toroidal canonical degeneration family of Kähler manifolds with ample canonical bundles Gromov-Hausdorff converge to the complete Kähler-Einstein metric on the smooth part of the central fiber when the base locus of the degeneration family is empty. We also …

The goals of this article are twofold : 1) to compute the conjugate locus of a geodesic that lies in the center of a simply connected, 2-step nilpotent Lie group with a left invariant metric 2) compare the isometry types of two such nilpotent Lie groups whose conjugate loci for central geodesics are "the same" in a sui…

The study of curvature properties of homogeneous Finsler spaces with $(α, β)$-metrics is one of the central problems in Riemann-Finsler geometry. In the present paper, the existence of invariant vector fields on homogeneous Finsler spaces with square $(α, β)$-metric and Randers changed square $(α, β)$-metric is proved.…

The paper studies a special sphere in ALE spaces.

A generalized flag manifold is a homogeneous space of the form $G/K$, where $K$ is the centralizer of a torus in a compact connected semisimple Lie group $G$. We classify all flag manifolds with four isotropy summands and we study their geometry. We present new $G$-invariant Einstein metrics by solving explicity the Ei…

We explain how the formal aspects of the theory of Kahler-Einstein metrics can be developed in the framework of moment maps. The central result we use is the Berndtsson convexity theorem, which is interpreted as defining a metric on the space of complex structures. We discuss some applications of these ideas to the Kah…

The paper studies Einstein-Hilbert functional and its relation to K-semistability.

We partially confirm an old conjecture of Donaldson that if there exists a cscK metrics in a given Kähler class, then there is no degenerated geodesic ray which is tamed by a bounded ambient geometry unless it parallels to a holomorphic line consists of cscK metrics only. We also prove that for simple test configuratio…

The paper studies metrics on Lie groups SO(2) and SO(3) and their integrability.

We consider two types of $p$-centro affine flows on smooth, centrally symmetric, closed convex planar curves, $p$-contracting, respectively, $p$-expanding. Here $p$ is an arbitrary real number greater than 1. We show that, under any $p$-contracting flow, the evolving curves shrink to a point in finite time and the only…

Complex networks are ubiquitous to several Computer Science domains. Centrality measures are an important analysis mechanism to uncover vital elements of complex networks. However, these metrics have high computational costs and requirements that hinder their applications in large real-world networks. In this tutorial,…

This study compares price discovery in ETH and BTC markets between centralized and decentralized exchanges.

This paper deals with relative normalizations of skew ruled surfaces in the Euclidean space $\mathbb{E}^{3}$. In section 2 we investigate some new formulae concerning the Pick invariant, the relative curvature, the relative mean curvature and the curvature of the relative metric of a relatively normalized ruled surface…

We present a countably infinite number of new explicit co-homogeneity one Sasaki-Einstein metrics on S^2 x S^3, in both the quasi-regular and irregular classes. These give rise to new solutions of type IIB supergravity which are expected to be dual to N=1 superconformal field theories in four-dimensions with compact or…

Study degenerations of Kähler-Einstein metrics on surfaces.

In this short note, we show that given a special Kähler-Einstein degeneration with bounded geometry, for any noncentral fiber, there exists a Kähler-Ricci flow which converges to the Kähler-Einstein metric of the central fiber. As an example, Tian's deformations \cite{Tian97} of the Mukai 3-fold admit Kähler-Ricci flow…

Kähler-Einstein metrics on toric submanifolds cannot be induced by projective immersions.

Vertex centrality measures are a multi-purpose analysis tool, commonly used in many application environments to retrieve information and unveil knowledge from the graphs and network structural properties. However, the algorithms of such metrics are expensive in terms of computational resources when running real-time ap…

Deviation inequalities and limit laws for random walks on metric spaces.

Motivated by considerations of euclidean quantum gravity, we investigate a central question of spectral geometry, namely the question of reconstructability of compact Riemannian manifolds from the spectra of their Laplace operators. To this end, we study analytic paths of metrics that induce isospectral Laplace-Beltram…

We determine the 6-dimensional solvmanifolds admitting an invariant complex structure with holomorphically trivial canonical bundle. Such complex structures are classified up to isomorphism, and the existence of strong Kähler with torsion (SKT), generalized Gauduchon, balanced and strongly Gauduchon metrics is studied.…

Investigates metric degeneracies on symplectic leaves using a generalized gradient flow.

Study calculates curvature for fluid dynamics group, proving positivity.

Proves local isometric embedding of low-differentiability metrics in 3D space.

We refine the recent local rigidity result for the marked length spectrum obtained by the first and third author in \cite{Guillarmou-Lefeuvre-18} and give an alternative proof using the geodesic stretch between two Anosov flows and some uniform estimate on the variance appearing in the central limit theorem for Anosov …

Study Ricci curvature of homogeneous Finsler spaces with specific metrics.

Statistical analysis of Diffusion Tensor Imaging (DTI) data requires a computational framework that is both numerically tractable (to account for the high dimensional nature of the data) and geometric (to account for the nonlinear nature of diffusion tensors). Building upon earlier studies that have shown that a Rieman…

Existence of Ricci flat metric on Kummer K3 surface proven.