arXiv research

Closed finite gap curves are dense in Sobolev spaces of curves in hyperbolic and Euclidean 3-spaces.

The study classifies spaces with specific conformal vector fields.

Study describes limits of surfaces in a mathematical space.

Mapping class group dynamics tracked through Teichmüller space.

Study constructs closed curves with constant curvature on cylinders and tori.

We obtain a topological and weakly equivariant classification of closed three-dimensional Alexandrov spaces with an effective isometric circle action. As an application of the classification we prove a version of the Borel conjecture for closed three-dimensional Alexandrov spaces with circle symmetry.

The paper proves Simon's conjecture for closed Einstein spaces, setting eigenvalue bounds.

In this paper we extend our correlation functions to the open/closed case. This gives rise to actions of an open/closed version of the Sullivan PROP as well as an action of the relevant moduli space. There are several unexpected structures and conditions that arise in this extension which are forced upon us by consider…

Higher KdV flows on spaces of closed equicentroaffine plane curves are studied and it is shown that the flows are described as certain multi-Hamiltonian systems on the spaces. Multi-Hamiltonian systems describing higher mKdV flows are also given on spaces of closed Euclidean plane curves via the geometric Miura transfo…

Study of geodesics on tetrahedra in hyperbolic space.

Let $X$ be a Banach space and $Conv_H(X)$ be the space of non-empty closed convex subsets of $X$, endowed with the Hausdorff metric $d_H$. We prove that each connected component of the space $Conv_H(X)$ is homeomorphic to one of the spaces: a singleton, the real line, a closed half-plane, the Hilbert cube multiplied by…

In this paper, we show a local energy convexity of $W^{1,2}$ maps into $CAT(K)$ spaces. This energy convexity allows us to extend Colding and Minicozzi's width-sweepout construction to produce closed geodesics in any closed Alexandrov space of curvature bounded from above, which also provides a generalized version of t…

We find the minimal value of the length in de Sitter space of closed space-like curves with non-vanishing non-space-like geodesic curvature vector. These curves are in correspondence with closed almost-regular canal surfaces, and their length is a natural magnitude in conformal geometry. As an application, we get a low…

Study on Finsler spaces with unique prime geodesics.

Closed curves in 3D and 4D spheres are dense in Sobolev norms.

Totally nonnegative Grassmannian and related spaces are shown to be like closed balls.

We calculate the rational equivariant cohomology of the spaces of non-contractible loops in compact space forms and show how to apply these calculations for proving the existence of closed geodesics.

We study closed three-dimensional Alexandrov spaces with a lower Ricci curvature bound in the $\mathsf{CD}^*(K,N)$ sense, focusing our attention on those with positive or nonnegative Ricci curvature. First, we show that a closed three-dimensional $\mathsf{CD}^*(2,3)$-Alexandrov space must be homeomorphic to a spherical…

CAT(0) spaces close to Euclidean spheres are homeomorphic to Euclidean spaces.

Estimates the number of closed curves on surfaces with power-saving error terms.

Paper determines Assouad-Nagata dimension for all minor-closed metrics.

In the present paper, we determine the topologies of three-dimensional closed Alexandrov spaces which converge to lower dimensional spaces in the Gromov-Hausdorff topology.

In this paper, we use Chas-Sullivan theory on loop homology and Leray-Serre spectral sequence to investigate the topological structure of the non-contractible component of the free loop space on the real projective spaces with odd dimensions. Then we apply the result to get the resonance identity of non-contractible ho…

The paper finds at least four prime closed characteristics on star-shaped hypersurfaces in 8D space.

We study three-dimensional Alexandrov spaces with a lower curvature bound, focusing on extending three classical results on three-dimensional manifolds: First, we show that a closed three-dimensional Alexandrov space of positive curvature, with at least one topological singularity, must be homeomorphic to the suspensio…

Study bounds the length of shortest periodic geodesics on certain curved spaces.

First example of a hyperbolic 4-orbifold underlying $\mathbb{P}^2$.

Clarifies definitions of global hyperbolicity in various spaces.

We consider the quantifier-free languages, Bc and Bc0, obtained by augmenting the signature of Boolean algebras with a unary predicate representing, respectively, the property of being connected, and the property of having a connected interior. These languages are interpreted over the regular closed sets of n-dimension…

Survey on geodesics on tetrahedra in curved spaces.

This is a survey paper on the space of symplectic structures on closed 4-manifolds, for the Proceedings ICCM 2004

The analog of the Schauder inequality for closed surfaces in Euclidean spaces is obtained in this article.

Unified framework for Alexandrov 3-spaces, extending manifold results.

The paper finds the optimal number of closed geodesics on Finsler compact spaces.

Found a 6D Lie group with a closed geodesic.

Study on flat metrics on 3D and 4D manifolds, focusing on topology and algebra.

We obtain a structure theorem for closed, cohomogeneity one Alexandrov spaces and we classify closed, cohomogeneity one Alexandrov spaces in dimensions 3 and 4. As a corollary, we obtain the classification of closed, $n$-dimensional, cohomogeneity one Alexandrov spaces admitting an isometric $T^{n-1}$ action. In contra…

New invariants help find closed geodesics on curved spaces.

The kth finite subset space of a topological space X is the space exp_k X of non-empty finite subsets of X of size at most k, topologised as a quotient of X^k. The construction is a homotopy functor and may be regarded as a union of configuration spaces of distinct unordered points in X. We show that the finite subset …

Paper proves almost Schur Lemma on smooth metric measure spaces.

It is well known that the category of Frolicher spaces and smooth mappings is Cartesian closed. The principal objective in this paper is to show that the full subcategory of Frolicher spaces that believe in fantasy that every Weil functor is really an exponentiation by the corresponding infinitesimal object is also Car…

We provide an algebraic description of the Teichmüller space and moduli space of flat metrics on a closed manifold or orbifold and study its boundary, which consists of (isometry classes of) flat orbifolds to which the original object may collapse. It is also shown that every closed flat orbifold can be obtained by col…

The goal of the chapter is to present certain aspects of the relationship between the study of simple closed geodesics and Teichmüller spaces.

In this paper, we give the generalization of the criterion for a 3-space curve to be closed given by [3] to an n-space curve in Minkowski space-time E_v^n. Furthermore, we apply this criterion for a curve lying on an oriented surface in the Minkowski space E_v^n as an application.

Frolicher spaces and smooth mappings form a cartesian closed category. It was shown in our previous paper [Far East Journal of Mathematical Sciences, 35 (2009), 211-233] that its full subcategory of Weil exponentiable Frolicher spaces is cartesian closed. By emancipating microlinearity from within a well-adapted model …

In this paper we examine different aspects of the geometry of closed conformal vector fields on Riemannian manifolds. We begin by getting obstructions to the existence of closed conformal and nonparallel vector fields on complete manifolds with nonpositive Ricci curvature, thus generalizing a theorem of T. K. Pan. Then…

Model learns latent dynamics of complex systems with closed transformation paths.

Defines Vassiliev complexity measures for open and closed curves in 3D space.