arXiv research

We prove the existence of infinitely many periodic points of symplectomorphisms isotopic to the identity if they admit at least one (non-contractible) hyperbolic periodic orbit and satisfy some condition on its flux. The obtained periodic points correspond to periodic orbits whose free homotopy classes are formed by it…

The study proves conditions for symplectic torus actions on manifolds with non-contractible orbits.

We show that the existence of noncontractible periodic orbits for compactly supported time-dependent Hamiltonian on the disk cotangent bundle of a Finsler manifold provided that the Hamiltonian is sufficiently large over the zero section. We generalize the BPS capacities and earlier constructions of Weber (2006 Duke Ma…

Extending work of Chen, we prove the Weinstein conjecture in dimension three for strongly fillable contact structures with either non-vanishing first Chern class or with strong and exact filling having non-trivial canonical bundle. This implies the Weinstein conjecture for certain Stein fillable contact structures obta…

In this paper, we establish first the resonance identity for non-contractible homologically visible prime closed geodesics on Finsler $n$-dimensional real projective space $(\mathbb{R}P^n,F)$ when there exist only finitely many distinct non-contractible closed geodesics on $(\mathbb{R}P^n,F)$, where the integer $n\geq2…

The study finds non-contractible geodesics on compact Finsler space forms without intersections.

The paper finds non-contractible loops of Legendrian tori from knot families.

In this paper, we prove that for every irreversible Finsler $n$-dimensional real projective space $(\mathbb{R}P^n,F)$ with reversibility $λ$ and flag curvature $K$ satisfying $\frac{16}{9}\left(\fracλ{1+λ}\right)^2<K\le 1$ with $λ<3$, there exist at least $n-1$ non-contractible closed geodesics. In addition, if the met…

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…

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.

In this paper we establish the existence of periodic orbits belonging to any $σ$-atoroidal free homotopy class for Hamiltonian systems in the twisted disc bundle, provided that the compactly supported time-dependent Hamiltonian function is sufficiently large over the zero section and the magnitude of the weakly exact $…

The paper proves conditions for the existence of multiple non-contractible closed geodesics on Finsler compact space forms.

The first two authors showed in~\cite{AM1} how the Conley-Zehnder index of any contractible periodic Reeb orbit of a non-degenerate toric contact form on a good toric contact manifold with zero first Chern class, i.e. a Gorenstein toric contact manifold, can be explicitly computed using moment map data. In this paper w…

Let $M=S^n/ Γ$ and $h$ be a nontrivial element of finite order $p$ in $π_1(M)$, where the integer $n\geq2$, $Γ$ is a finite group which acts freely and isometrically on the $n$-sphere and therefore $M$ is diffeomorphic to a compact space form. In this paper, we establish first the resonance identity for non-contractibl…

We prove that for compact, non-contractible, one dimensional geodesic spaces, a version of the marked length spectrum conjecture holds. For a compact one dimensional geodesic space X, we define a subspace Conv(X). When X is non-contractible, we show that X deformation retracts to Conv(X). If two such spaces X, Y have t…

Using a new method we give elementary estimates for the capacity of non-contractible annuli on cylinders and provide examples, where these inequalities are sharp. Here the lower bound depends only on the area of the annulus. In the case of constant curvature this lower bound is obtained with the help of a symmetrizatio…

In this article we calculate the n-string braid groups of certain non-contractible graphs. We use techniques from the work of A. Abrams, F. Connolly and M. Doig combined with Van Kampen's Theorem to prove these results.

We prove that the existence of a positively defined, invariant Einstein metric $m$ on a connected homogeneous space $G/H$ of a compact Lie group $G$ is the consequence of non-contractibility of some compact set $C=X_{G,H}^Σ$ (Böhm polyhedron) introduced by C.Böhm. There is a natural continuous map of $C$ onto the flag …

This note proves that any locally extremal non-self-conjugate geodesic loop in a Riemannian manifold is a closed geodesic. As a consequence, any complete and non-contractible Riemannian manifold with diverging injectivity radii along diverging sequences and without points conjugate to themselves, possesses a minimizing…

In this paper, we establish that, for statistically convex-cocompact actions, contracting elements are exponentially generic in counting measure. Among others, the following exponential genericity results are obtained as corollaries for the set of hyperbolic elements in relatively hyperbolic groups, the set of rank-1 e…

Let B be a thick spherical building equipped with its natural CAT(1) metric and let M be a proper, convex subset of B. If M is open or if M is a closed ball of radius pi/2, then the maximal subcomplex supported by the complement of M is spherical and non contractible.

New structure found in loops on quasi-surfaces.

Tight embeddings of 2-tori in 3D space contain short loops.

Constructing translating solitons from Lagrangian Grim Reapers.

We apply the techniques of totally twisted Khovanov homology to the constructions by M. Asaeda, J. Przytycki, and A. Sikora of Khovanov type homologies for links and tangles in I-bundles over (orientable) surfaces. As a result we describe an invariant chain complex built out of resolutions with only non-contractible ci…

Generalizes moment-angle manifolds to arbitrary nice manifolds with corners.

We prove that the least area of the non-contractible immersed spheres is no more than $4π$ in any oriented compact manifold with dimension $n+2\leq 7$ which satisfies $R\geq 2$ and admits a map to $\mathbf S^2\times T^n$ with nonzero degree. We also prove a rigidity result for the equality case. This can be viewed as a…

We study contractivity properties of gradient flows for functions on normed spaces or, more generally, on Finsler manifolds. Contractivity of the flows turns out to be equivalent to a new notion of convexity for the functions. This is different from the usual convexity along geodesics in non-Riemannian Finsler manifold…

It is a classical theorem of Loewner that the systole of a Riemannian torus can be bounded in terms of its area. We answer a question of a similar flavor of Robert Young showing that if $T$ is a Riemannian 2-torus with boundary in $\mathbb R ^n$, such that the boundary curve is a standard unit circle, then the length o…

Graev's nerve implies invariant Einstein metrics on homogeneous spaces.

The systole of a compact non simply connected Riemannian manifold is the smallest length of a non-contractible closed curve ; the systolic ratio is the quotient $(\mathrm{systole})^n/\mathrm{volume}$. Its supremum on the set of all the riemannian metrics, is known to be finite for a large class of manifolds, including …

For a closed symplectic manifold $(M,ω)$, a compatible almost complex structure $J$, a 1-periodic time dependent symplectic vector field $Z$ and a homotopy class of closed curves $γ$ we define a Floer complex based on 1-periodic trajectories of $Z$ in the homotopy class $γ$. We suppose that the closed 1-form $i_{Z_t}ω$…

The study finds infinite geodesics on manifolds with specific homotopy group properties.

The paper proves existence of solutions for mean field equations on compact Riemann surfaces.

Uniform systole bounds for arithmetic orbifolds and number fields.

Suppose that $X,X'$ are simply-connected closed exotic 4-manifolds. It is well-known that $X'$ is obtained by an order 2 cork twist of $X$. We give an infinite exotic family of 4-manifolds not generated by any infinite order cork. This is the first example admitting such a condition. We prove a necessary condition of 4…

Reformulates binary classification on manifolds using Yang-Mills-Higgs theory.

Study limits of adjoint orbits for Lie groups, describing nilpotent orbits.

In a previous article, analytic 1-submanifolds had been classified w.r.t. their symmetry under a given regular and separately analytic Lie group action on an analytic manifold. It was shown that such an analytic 1-submanifold is either free or (via the exponential map) analytically diffeomorphic to the unit circle or a…

We define the notion of the orbit group of a quandle via its connectivity and compute the orbit groups for some basic quandles. We also show that the orbit group counts the number of orbits of certain quandles.

The paper finds linked periodic orbits in disc homeomorphisms using braids.

The paper finds symplectic compactifications of coadjoint orbits.

New insights into pseudo-Anosov flows with special periodic orbits.

The study counts periodic orbits on smooth manifolds, adding ghost orbits for completeness.

The paper proves conditions for the existence of small Urysohn width hypersurfaces in manifolds with positive scalar curvature.

A lamination of a graph embedded on a surface is a collection of pairwise disjoint non-contractible simple closed curves drawn on the graph. In the case when the surface is a sphere with three punctures (a.k.a. a pair of pants), we first identify the lamination space of a graph embedded on that surface as a lattice pol…

Study properties of orbits of Hermann actions without commutability assumptions.

Smooth approximations for continuous functions on orbit spaces.