arXiv research

The paper shows dense and residual sets of continuous maps with positive metric mean dimension.

We prove that the upper metric mean dimension of $C^0$-generic homeomorphisms, acting on a compact smooth boundaryless manifold with dimension greater than one, coincides with the dimension of the manifold. In the case of continuous interval maps we also show that each level set for the metric mean dimension is $C^0$-d…

We consider the notion of dimension in four categories: the category of (unbounded) separable metric spaces and (metrically proper) Lipschitz maps, and the category of (unbounded) separable metric spaces and (metrically proper) uniform maps. A unified treatment is given to the large scale dimension and the small scale …

This paper studies the Yang--Mills ASD equation over the cylinder as a non-linear evolution equation. We consider a dynamical system consisting of bounded orbits of this evolution equation. This system contains many chaotic orbits, and moreover it becomes an infinite dimensional and infinite entropy system. We study th…

The abstract applies waist inequality to dynamical systems and entropy.

Constructs uniformly positive scalar curvature metrics on open manifolds

Paper extends foliation results in higher dimensions for Schwarzschild spaces.

Compact metrics found with specific curvature properties on 3D surfaces.

We study a conformal flow for compact Riemannian manifolds of dimension greater than two with boundary. Convergence to a scalar-flat metric with constant mean curvature on the boundary is established in dimensions up to seven, and in any dimensions if the manifold is spin or if it satisfies a generic condition.

By a Cantor group we mean a topological group homeomorphic to the Cantor set. We show that a compact metric space of rational cohomological dimension $n$ can be obtained as the orbit space of a Cantor group action on a metric compact space of covering dimension $n$. Moreover, the action can be assumed to be free if $n=…

Study negative scalar curvature metrics with positive boundary mean curvature.

We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension $\as_{\Z} X$ of metric spaces. We show that it agrees with the asymptotic dimension $\as X$ when the later is finite. Then we use this fact to construct an example of a metric space $X$ of boun…

The study finds either many or few constant mean curvature hypersurfaces on a manifold.

We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension $n\geq 3$. We prove the existence of such conformal metrics in the cases of $n=6,7$ or the manifold is spin and some other remai…

Study local structure of Einstein metrics with boundary conditions.

The paper examines the stability of a specific flow on complex manifolds.

In this paper we prove that every Riemannian metric on a locally conformally flat manifold with umbilic boundary can be conformally deformed to a scalar flat metric having constant mean curvature. This result can be seen as a generalization to higher dimensions of the well known Riemann mapping Theorem in the plane.

Study shows curvature rigidity of specific metric types.

The paper studies T-tensor of spherically symmetric Finsler metrics and characterizes metrics satisfying the T-condition.

Study finds infinitely many non-radial solutions for negative scalar curvature in higher dimensions.

In 1992, motivated by Riemann mapping theorem, Escobar considered a version of Yamabe problem on manifolds of dimension n greater than 2 with boundary. The problem consists in finding a conformal metric such that the scalar curvature is zero and the mean curvature is constant on the boundary. By using a local test func…

The paper provides statistical guarantees for generative models using dimension reduction.

Special Liouville metrics with Ricci-like conditions are determined by elliptic functions.

Sharp upper bounds on inscribed radius for metric spaces with convex boundary.

The paper presents new metrics to quantify and test for (i) the equality of distributions and (ii) the independence between two high-dimensional random vectors. We show that the energy distance based on the usual Euclidean distance cannot completely characterize the homogeneity of two high-dimensional distributions in …

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

Study on subgroups' evolution in 3D Lie groups using mean curvature flow.

This is an exposition of some recent results on ECS manifolds, by which we mean pseudo-Riemannian manifolds of dimensions greater than 3 that are neither conformally flat nor locally symmetric, and have parallel Weyl tensor. All ECS metrics are indefinite. We state two classification theorems, describing the local stru…

By a Cantor group we mean a topological group homeomorphic to the Cantor set. The author earlier proved that every compact metric space of rational cohomological dimension n can be obtained as the orbit space of a Cantor group action on a metric compact space of covering dimension n. In this paper, we consider actions …

Study finds solutions to curvature equation with boundary conditions.

The study proves a new positive energy theorem for manifolds with specific curvature properties.

A Riemannian or Finsler metric on a compact manifold M gives rise to a length function on the free loop space ΛM, whose critical points are the closed geodesics in the given metric. If X is a homology class on ΛM, the minimax critical level cr(X) is a critical value. Let M be a sphere of dimension >2, and fix a metric …

Paper introduces new bounds linking data compressibility to generalization error.

Generative neural network simulates characteristic functions.

Defines a geometric dimension for Lorentzian spaces, distinguishing spacelike and null subspaces.

Proves multiplicity one for boundary minimal hypersurfaces in compact manifolds.

The article classifies curvature functions on compact manifolds with boundaries.

New metrics avoid high-dimensional analysis challenges, proving convergence without 'curse of dimensionality'.

Let (M,g) be a compact n-dimensional Riemannian manifold with boundary. This article is concerned with the set of scalar-flat metrics on M which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. We construct examples of metrics on the unit ball, in dimensions n>=25, for wh…

The paper finds infinitely many half-volume CMC hypersurfaces on generic or Ricci-positive manifolds.

The paper studies curvature properties of direct image bundles.

Method identifies mixed Nash equilibria in high dimensions for training mixtures of GANs.

We consider two cases of the asymptotically flat scalar-flat Yamabe problem on a non-compact manifold with boundary, in dimension $n\geq3$. First, following arguments of Cantor and Brill in the compact case, we show that given an asymptotically flat metric $g$, there is a conformally equivalent asymptotically flat scal…

The purpose of this paper is to revisit the Bianchi identities existing for the Riemann and Weyl tensors in the combined framework of the formal theory of systems of partial differential equations (Spencer cohomology, differential systems, formal integrability) and Algebraic Analysis (homological algebra, differential …

New algorithms for clustering and dimension reduction using relative von Neumann entropy.

Let (M,g) be a compact Riemannian manifold with boundary. This paper is concerned with the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. We prove that this set is compact for dimensions greater than or equal to 7 under the generic condi…

Consider a sequence of closed, orientable surfaces of fixed genus $g$ in a Riemannian manifold $M$ with uniform upper bounds on mean curvature and area. We show that on passing to a subsequence and choosing appropriate parametrisations, the inclusion maps converge in $C^0$ to a map from a surface of genus $g$ to $M$. W…

Deep networks can approximate high-dimensional distributions from low-dimensional ones.