arXiv research

Extends graph factor system to quasi-median graphs.

New combinatorial structure for hierarchically hyperbolic spaces.

This paper deals with complex structures on Lie algebras $\ct_π \hh=\hh \ltimes_π V$, where $π$ is either the adjoint or the coadjoint representation. The main topic is the existence question of complex structures on $\ct_π \hh$ for $\hh$ a three dimensional real Lie algebra. First it was proposed the study of complex …

Let $X$ be a compact Kähler manifold and $\a \in H^{1,1}(X,\R)$ a Kähler class. We study the metric completion of the space $\HH_\a$ of Kähler metrics in $\a$, when endowed with the Mabuchi $L^2$-metric $d$. Using recent ideas of Darvas, we show that the metric completion $(\overline{\HH}_\a,d)$ of $(\HH_\a,d)$ is a CA…

Extends Alexandrov's result to unbounded convex domains in hyperbolic 3-space.

Assume that $f(s) = F'(s)$ where $F$ is a double-well potential. Under certain conditions on the Lipschitz constant of $f$ on $[-1,1]$, we prove that arbitrary bounded global solutions of the semilinear equation $Δu = f(u)$ on hyperbolic space $\HH^n$ must reduce to functions of one variable provided they admit asympto…

Geometric data uniquely determines convex subsets in hyperbolic manifolds.

HH-VAEM improves imputation and acquisition of missing data using hierarchical models and Hamiltonian Monte Carlo.

An explicit classification of homogeneous quaternionic Kaehler structures by real tensors is derived and we relate this to the representation-theoretic description found by Fino. We then show how the quaternionic hyperbolic space HH(n) is characterised by admitting homogeneous structures of a particularly simple type. …

The study classifies hypersurfaces in quaternionic space forms with constant principal curvatures.

Let $C$ be a differential graded coalgebra, $\barΩC$ the Adams cobar construction and $C^\vee$ the dual algebra. We prove that for a large class of coalgebras $C$ there is a natural isomorphism of Gerstenhaber algebras between the Hochschild cohomologies $HH^\ast (C^\vee, C ^\vee)$ and $HH^\ast (\barΩC ; \barΩC)$. Thi…

Recently, we have studied evolution of a family of Finsler metrics along Finsler Ricci flow and proved its convergence in short time. Here, evolution equation of the reduced $hh$-curvature and the Ricci scalar along the Finslerian Ricci flow is obtained and it is proved that the Ricci flow preserves positivity of reduc…

Let $\GG$ be a sub-Riemannian $k$-step Carnot group of homogeneous dimension $Q$. In this paper, we shall prove several geometric inequalities concerning smooth hypersurfaces (i.e. codimension one submanifolds) immersed in $\GG$, endowed with the $\HH$-perimeter measure.

Let $R$ be the $hh$-curvature associated with the Chern connection or the Cartan connection. Adopting the pulled-back tangent bundle approach to the Finslerian Geometry, an intrinsic characterization of $R$-Einstein metrics is given. Finslerian metrics which are locally conformally $R$-Einstein are classified.

Let $M$ be either the 2-sphere $\SS^2 \subset\RR^3$ or the hyperbolic plane $\HH^2 \subset \RR^3$. If $Δ(abc)$ is a geodesic triangle on $M$ with corners at $a,b,c\in M$, we denote by $α, β, γ\in M$ the midpoints of their sides. If $Ω$ denotes the oriented area of this triangle on $M$, it satisfies the relations: $$ \s…

Global stability proved for Navier-Stokes equations on hyperbolic space.

We answer a question of Durham, Hagen, and Sisto, proving that a Teichmüller geodesic ray does not necessarily converge to a unique point in the hierarchically hyperbolic space boundary of Teichmüller space. In fact, we prove that the limit set can be almost anything allowed by the topology.

We consider inverse curvature flows in $\Hh$ with star-shaped initial hypersurfaces and prove that the flows exist for all time, and that the leaves converge to infinity, become strongly convex exponentially fast and also more and more totally umbilic. After an appropriate rescaling the leaves converge in $C^\infty$ to…

Survey of tools for studying hierarchical hyperbolic spaces.

The rank of a hierarchically hyperbolic space is the maximal number of unbounded factors in a standard product region. For hierarchically hyperbolic groups, this coincides with the maximal dimension of a quasiflat. Examples for which the rank coincides with familiar quantities include: the dimension of maximal Dehn twi…

We prove a geometric model for HHS hierarchies as CAT(0) cube complexes.

The paper extends stability theorem for Navier-Stokes equations to negatively curved manifolds.

Under a pulled-back approach given in [1] and firstly presented in [2], we introduce, in this paper, the concepts of almost contact and normal almost contact Finsler structures on the pulled-back bundle. Properties of structures partly Sasakians are studied. Using the hh-curvature tensor of Chern connection given in [2…

Develops a new representation for constant mean curvature surfaces in hyperbolic 3-space.

The first part of this survey is a heuristic, non-technical discussion of what an HHS is, and the aim is to provide a good mental picture both to those actively doing research on HHSs and to those who only seek a basic understanding out of pure curiosity. It can be read independently of the second part, which is a deta…

We give a spinorial characterization of isometrically immersed surfaces into 3-dimensional homogeneous manifolds with 4-dimensional isometry group in terms of the existence of a particular spinor, called generalized Killing spinor. This generalizes results by T. Friedrich for $\R^3$ and B. Morel for $\Ss^3$ and $\HH^3$…

For almost any compact connected Lie group $G$ and any field $\mathbb{F}\_p$, we compute the Batalin-Vilkoviskyalgebra $H^{*+\text{dim }G}(LBG;\mathbb{F}\_p)$ on the loop cohomology of the classifying space introduced byChataur and the second author.In particular, if $p$ is odd or $p=0$, this Batalin-Vilkovisky algebra…

In this paper, we consider minimal hypersurfaces in the product space $\mathbb{H}^n \times \mathbb{R}$. We begin by studying examples of rotation hypersurfaces and hypersurfaces invariant under hyperbolic translations. We then consider minimal hypersurfaces with finite total curvature. This assumption implies that the …

Develops a new framework for large-scale geometry.

Let $D$ be a closed disk centered at the origin in the horizontal hyperplane $\{t=0\}$ of the sub-Riemannian Heisenberg group $\hh^n$, and $C$ the vertical cylinder over $D$. We prove that any finite perimeter set $E$ such that $D\subset E\subset C$ has perimeter larger than or equal to the one of the rotationally symm…

This paper studies geometric structures on manifolds with specific symplectic properties.

Given a symmetric nonnegative matrix $A$, symmetric nonnegative matrix factorization (symNMF) is the problem of finding a nonnegative matrix $H$, usually with much fewer columns than $A$, such that $A \approx HH^T$. SymNMF can be used for data analysis and in particular for various clustering tasks. In this paper, we p…

The Mahler measure of the polynomials $t(x^m-1) y - (x^n-1) \in \dC[x,y]$ is essentially the sum of volumes of a certain collection of ideal hyperbolic polyhedra in $\HH^3$, which can be determined a priori as a function on the parameter $t$. We obtain a formula that generalizes some previous formulas given by Cassaign…

Characterizes boundaries of HHGs and their properties.

Let $M$ be a compact oriented $d$-dimensional smooth manifold. Chas and Sullivan have defined a structure of Batalin-Vilkovisky algebra on $\mathbb{H}_*(LM)$. Extending work of Cohen, Jones and Yan, we compute this Batalin-Vilkovisky algebra structure when $M$ is a sphere $S^d$, $d\geq 1$. In particular, we show that $…

The Weyl problem is extended to hyperbolic and anti-de Sitter spaces, connecting geometry, analysis, and group theory.

A kinematic method selects the deformation Laplacian for fluid dynamics on Riemannian manifolds.

In this short note, using our geometric method introduced in a previous paper \cite{phl} and initiated by \cite{ave}, we derive an asymptotic swaption implied volatility at the first-order for a general stochastic volatility Libor Market Model. This formula is useful to quickly calibrate a model to a full swaption matr…

Novel theory combines combinatorial and topological elements.

The paper introduces combinatorial Calabi flows to find hyperbolic metrics on surfaces with boundary.

The paper studies critical points of horizontal energy functional in Riemannian foliations.

The paper develops algorithms for finding metrics with prescribed combinatorial curvature on polyhedral surfaces.

The paper introduces combinatorial curvature and flow for polyhedral surfaces, proving rigidity and solving the Yamabe problem.

For triangulated surfaces, we introduce the combinatorial Calabi flow which is an analogue of smooth Calabi flow. We prove that the solution of combinatorial Calabi flow exists for all time. Moreover, the solution converges if and only if Thurston's circle packing exists. As a consequence, combinatorial Calabi flow pro…

It is shown that, up to isometry, all but finitely many closed, orientable hyperbolic 3-manifolds with a given trace field $K$ admit 0.34 as a Margulis number. This is deduced from a more technical result giving a condition under which $\max(d(P,x\cdot P),d(P,y\cdot P))\ge0.34$ for every $P\in\HH^3$, where $x$ and $y$ …

Fractional combinatorial flow improves surface conformal structures.

We show meromorphic extension and analyze the divisors of a Selberg zeta function of odd type $Z_{Γ,Σ}^{\rm o}(λ)$ associated to the spinor bundle $Σ$ on odd dimensional convex co-compact hyperbolic manifolds $X:=Γ\backslash\hh^{2n+1}$. We define a natural eta invariant $η(D)$ associated to the Dirac operator $D$ on $X…

Combinatorial method computes Legendrian knot invariant.