arXiv research

We propose a new two-component geodesic equation with the unusual property that the underlying space has constant positive curvature. In the special case of one space dimension, the equation reduces to the two-component Hunter-Saxton equation.

Score-based methods fail with isolated components and incorrect mixing proportions.

Explicit computation of symplectic form for $\mathrm{PGL}_n(\mathbb{R})$-Hitchin component.

Let D a divisor with simple normal crossings in a Kahler manifold X. The purpose of this short note is to show that the existence of a Poincare type metric with constant scalar curvature in on the complement of D implies for any component of the divisor that the scalar curvature of Poincare type metric outside of D is …

Computes constants for cyclic covers of translation surfaces.

We show that in any spacetime dimension $D\ge 4$, degenerate components of the event horizon do not exist in static vacuum configurations with positive cosmological constant. We also show that without a cosmological constant asymptotically flat solutions cannot possess a degenerate horizon component. Several independen…

The computation of the sparse principal component of a matrix is equivalent to the identification of its principal submatrix with the largest maximum eigenvalue. Finding this optimal submatrix is what renders the problem ${\mathcal{NP}}$-hard. In this work, we prove that, if the matrix is positive semidefinite and its …

We show that there are Montesinos knots with $n+1\geq 4$ tangles whose character varieties contain arbitrarily many irreducible components of dimension $d$ for any $1\le d\le n-2$. Moreover, these irreducible components can be chosen so that the trace of the meridian is non-constant.

Paper finds first examples of unlinked knots that can't be separated.

Study refines Siegel-Veech constants for abelian differentials.

The paper studies curves of constant-ratio in pseudo-Galilean space.

Algorithm learns Gaussian mixtures robust to outliers.

In this article, we study local holomorphic isometric embeddings from ${\BB}^n$ into ${\BB}^{N_1}\times... \times{\BB}^{N_m}$ with respect to the normalized Bergman metrics up to conformal factors. Assume that each conformal factor is smooth Nash algebraic. Then each component of the map is a multi-valued holomorphic m…

The connected components of the zero set of any conformal vector field $v$, in a pseudo-Riemannian manifold $(M,g)$ of arbitrary signature, are of two types, which may be called `essential' and `nonessential'. The former consist of points at which $v$ is essential, that is, cannot be turned into a Killing field by a lo…

In this paper, we consider compact free boundary constant mean curvature surfaces immersed in a mean convex body of the Euclidean space or in the unit sphere. We prove that the Morse index is bounded from below by a linear function of the genus and number of boundary components.

The study examines parallel forms on manifolds, focusing on specific dimensions and forms.

Researchers classify 3D self-shrinkers in 4D space.

Every biharmonic Wintgen ideal submanifold in a Riemannian manifold of constant sectional curvature is either minimal or has constant mean curvature.

Gradient almost para-Ricci-like solitons have constant coefficients and scalar curvatures.

A 3D metric conformally related to Arnold cat fast dynamo metric: ${ds_{A}}^{2}=e^{-λz}dp^{2}+e^{λz}dq^{2}+dz^{2}$ is shown to present a behaviour of non-dynamos where the magnetic field exponentially decay in time. The Riemann-Christoffel connection and Riemann curvature tensor for the Arnold and its conformal counter…

Let $M$ be a smooth connected orientable compact surface. Denote by $F(M,S^1)$ the space of all Morse functions $f:M\to S^1$ having no critical points on the boundary of $M$ and such that for every boundary component $V$ of $M$ the restriction $f|_{V}:V\to S^1$ is either a constant map or a covering map. Endow $F(M,S^1…

In this paper we show that the topological closure of the holonomy group of a certain class of projectively flat Finsler 2-manifolds of constant curvature is maximal, that is isomorphic to the connected component of the diffeomorphism group of the circle. This class of 2-manifolds contains the standard Funk plane of co…

Method constructs orthogonal curvilinear coordinates in constant curvature spaces.

Enhances mixture models with classifier-defined weights.

In this note we consider asymptotically flat manifolds with non-negative scalar curvature and an inner boundary which is an outermost minimal surface. We show that there exists an upper bound on the mean curvature of a constant mean curvature surface homologous to a subset of the interior boundary components. This boun…

Let $Γ$ be a nonuniform lattice acting on real hyperbolic n-space. We show that in dimension greater than or equal to 4, the volume of a representation is constant on each connected component of the representation variety of $Γ$ in SO(n,1). Furthermore, in dimensions 2 and 3, there is a semialgebraic subset of the repr…

Machine learning solves Einstein equations without symmetry assumptions.

For an un-oriented link $\mathcal{K}$, let $L(\mathcal{K})$ be the ropelength of $\mathcal{K}$. It is known that when $\mathcal{K}$ has more than one component, different orientations of the components of $\mathcal{K}$ may result in different braid index. We define the largest braid index among all braid indices corres…

We study stable constant mean curvature (CMC) hypersurfaces $Σ$ in slabs in a product space $M\times\r,$ where $M$ is an orientable Riemannian manifold. We obtain a characterization of stable cylinders and prove that if $Σ$ is not a cylinder then it is locally a vertical graph. Moreover, in case $M$ is $\h^n,\r^n$ or $…

We describe the connected components of the complement of a natural "diagonal" of real codimension 1 in a stratum of quadratic differentials on CP1. We establish a natural bijection between the set of these connected components and the set of generic configurations that appear on such "flat spheres". We also prove that…

We show that the Masur-Veech volumes and area Siegel-Veech constants can be obtained by intersection numbers on the strata of Abelian differentials with prescribed orders of zeros. As applications, we evaluate their large genus limits and compute the saddle connection Siegel-Veech constants for all strata. We also show…

The study classifies and constructs examples of surfaces with specific curvature and boundary conditions.

In this paper, we study submanifolds with constant $r$th mean curvature $S_r$. We investigate, the stability of such submanifolds in the case when they are leaves of a codimension one foliation. We also generalize recent results by Barros - Sousa and Alías - Colares, concerning conformal fields, to an arbitrary manifol…

In [20], Ros and Vergasta proved that an immersed orientable compact stable constant mean curvature surface $Σ$ with free boundary in a closed ball $B\subset\mathbb{R}^3$ must be a planar equator, a spherical cap or a surface of genus 1 with at most two boundary components. In this article, by using a modified Hersch t…

Natural metrics provide a way to induce a metric on the tangent bundle from the metric on its base manifold. The most studied type is the Sasaki metric, which applies the base metric separately to the vertical and horizontal components. We study a more general class of metrics which introduces interactions between the …

The paper defines and analyzes the adjoint Reidemeister torsion for connected sums of knots.

Consider a compact Kähler manifold X with a simple normal crossing divisor D, and define Poincaré type metrics on X\D as Kähler metrics on X\D with cusp singularities along D. We prove that the existence of a constant scalar curvature (resp. an extremal) Poincaré type Kähler metric on X\D implies the existence of a con…

Study on moduli spaces of negatively curved metrics on surfaces.

The paper classifies 3D self-expanders with specific properties.

Independent component analysis (ICA) is a statistical method for transforming an observable multi-dimensional random vector into components that are as statistically independent as possible from each other. Usually the ICA framework assumes a model according to which the observations are generated (such as a linear tra…

The paper calculates area Siegel--Veech constants for specific submanifolds of REL zero.

We prove the classical Yano-Obata conjecture by showing that the connected component of the group of holomorph-projective transformations of a closed, connected Riemannian Kähler manifold consists of isometries unless the metric has constant positive holomorphic curvature.

We investigate multi-dimensional Hamiltonian systems associated with constant Poisson brackets of hydrodynamic type. A complete list of two- and three-component integrable Hamiltonians is obtained. All our examples possess dispersionless Lax pairs and an infinity of hydrodynamic reductions.

We study Lagrangian submanifolds of the nearly Kähler $\mathbb{S}^3\times\mathbb{S}^3$ with respect to their, so called, angle functions. We show that if all angle functions are constant, then the submanifold is either totally geodesic or has constant sectional curvature and there is a classification theorem that follo…

The paper proves new curvature estimates in quaternionic contact geometry.

Euclidean geometry has historically been the typical "workhorse" for machine learning applications due to its power and simplicity. However, it has recently been shown that geometric spaces with constant non-zero curvature improve representations and performance on a variety of data types and downstream tasks. Conseque…

Let $F$ be a closed essential surface in a hyperbolic 3-manifold $M$ with a toroidal cusp $N$. The depth of $F$ in $N$ is the maximal distance from points of $F$ in $N$ to the boundary of $N$. It will be shown that if $F$ is an essential pleated surface which is not coannular to the boundary torus of $N$ then the depth…

Study on Ricci-like solitons and gradient solitons on specific manifolds.