Related papers: Alexandrov embedded closed magnetic geodesics on S…

We prove that a proper geodesic metric space has non-positive curvature in the sense of Alexandrov if and only if it satisfies the Euclidean isoperimetric inequality for curves. Our result extends to non-geodesic spaces and non-zero…

We use the weighted Hsiung-Minkowski integral formulas and Brendle's inequality to show new rigidity results. First, we prove Alexandrov type results for closed embedded hypersurfaces with radially symmetric higher order mean curvature in a…

We consider geodesic nets (critical points of a length functional on the space of embedded graphs) on doubled polygons (topological 2-spheres endowed with a flat metric away from finitely many cone singularities). We use the theorem of…

In this paper, we prove that for every Finsler $n$-sphere $(S^n, F)$ for $n\ge 3$ with reversibility $\lambda$ and flag curvature $K$ satisfying $(\frac{\lambda}{\lambda+1})^2<K\le 1$, either there exist infinitely many prime closed…

In the present paper, we consider several valid notions of orientability of Alexandov spaces and prove that all such conditions are equivalent. Further, we give topological and geometric applications of the orientability. In particular, a…

This paper aims to give an elementary proof for Toponogov's theorem in Alexandrov geometry with lower curvature bound. The idea of the proof comes from the fact that, in Riemannian geometry, sectional curvature can be embodied in the second…

Let $M=S^n/ \Gamma$ and $h \in \pi_1(M)$ be a non-trivial element of finite order $p$, where the integers $n, p\geq2$ and $\Gamma$ is a finite abelian group which acts on the sphere freely and isometrically, therefore $M$ is diffeomorphic…

In this paper we study half-geodesics, those closed geodesics that minimize on any subinterval of length $l(\gamma)/2$. For each nonnegative integer $n$, we construct Riemannian manifolds diffeomorphic to $S^2$ admitting exactly $n$…

For non-reversible Finsler metrics of positive flag curvature on spheres and projective spaces we present results about the number and the length of closed geodesics and about their stability properties.

We show that the shortest closed geodesic on a 2-sphere with non-negative curvature has length bounded above by three times the diameter. We prove a new isoperimetric inequality for 2-spheres with pinched curvature; this allows us to…

We construct a concrete example of constant Gauss curvature $K = 1$ on the 2-sphere having all geodesics closed and of same length.

We study arrangements of geodesic arcs on a sphere, where all arcs are internally disjoint and each arc has its endpoints located within the interior of other arcs. We establish fundamental results concerning the minimum number of arcs in…

We study non-resonant circles for strong magnetic fields on a closed, connected, oriented surface and show how these can be used to prove the existence of trapping regions and of periodic magnetic geodesics with prescribed low speed. As a…

This paper is devoted to prove that if an Alexandrov space of curvature not less than $\kappa$ with a codimension one extremal subset which admits an isometric involution with respect to the induced length metric, then the metric space…

In this paper we prove that in the class of metric measure spaces with Alexandrov curvature bounded from below the Riemannian curvature-dimension condition $RCD(K,N)$ with $K\in \mathbb{R}$ and $N\in [1,\infty)$ is preserved under doubling…

We prove that for any reversible Finsler metric on S2, the number of prime closed geodesics grows quadratically with respect to length. The main tools are an improvement on Franks' theorem about the number of periodic points of…

In this paper, we study a complete noncompact nonnegatively curved Alexandrov space $A$ with a soul $S$ of codimension two. We establish some structural results under additional regularity assumptions. As an application, we conclude that in…

In this paper, we prove that for every Finsler $n$-dimensional sphere $(S^n,F), n\ge 3$ with reversibility $\lambda$ and flag curvature $K$ satisfying $\left(\frac{\lambda}{1+\lambda}\right)^2<K\le 1$, there exist at least three distinct…

In this paper, we prove the existence of at least two distinct closed geodesics on every compact simply connected irreversible or reversible Finsler (including Riemannian) manifold of dimension not less than 2.

We study dual Abrikosov vortices in Abelian projected SU(2) gauge theory in the Polyakov gauge. We show that vortices are present in this gauge but they are suppressed with respect to the maximal Abelian gauge. We interpret this difference…