arXiv research

New manifolds with small systoles not quasi-arithmetic.

Small sets of systoles fill hyperbolic surfaces of large genus.

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

Löbell polyhedra have small systoles and are quasi-arithmetic.

Positive scalar curvature implies small 2-systoles in Kähler manifolds

The study of systoles in arithmetic hyperbolic manifolds.

Optimal curves minimize crossings on surfaces.

We prove that manifolds of Lusternik-Schnirelmann category 2 necessarily have free fundamental group. We thus settle a 1992 conjecture of Gomez-Larranaga and Gonzalez-Acuna, by generalizing their result in dimension 3, to all higher dimensions. We examine its ramifications in systolic topology, and provide a sufficient…

The paper calculates critical points of systole function on Teichmüller space.

Study growth of systoles in arithmetic manifolds, focusing on $k$-dimensional cases.

The systole size affects the smallest eigenvalue of hyperbolic surfaces.

Automorphisms of curve graphs and systolic complexes are isomorphic to mapping class groups for small k.

The paper explores the shape of filling-systole subspace in surface moduli space and critical points of systole function.

We outline the current state of knowledge regarding geometric inequalities of systolic type, and prove new results, including systolic freedom in dimension 4. Namely, every compact, orientable, smooth 4-manifold X admits metrics of arbitrarily small volume such that every orientable, immersed surface of smaller than un…

New bounds on knot distortion and Seifert surface properties.

We are interested in the maximum value achieved by the systole function over all complete finite area hyperbolic surfaces of a given signature $(g,n)$. This maximum is shown to be strictly increasing in terms of the number of cusps for small values of $n$. We also show that this function is greater than a function that…

We prove an inequality between the $L^{\infty}$-norm of the contact Hamiltonian of a positive loop of contactomorphims and the minimal Reeb period. This implies that there are no small positive loops on hypertight or Liouville fillable contact manifolds. Non-existence of small positive loops for overtwisted 3-manifolds…

New bounds on hyperbolic surfaces' properties using linear programming.

Our main result is that for all sufficiently large $x_0>0$, the set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed invariant trace field $k$ and systole bounded below by $x_0$ has density one within the set of all commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds wit…

Given a pair of integers m and n such that 1 < m < n, we show that every n-dimensional manifold admits metrics of arbitrarily small total volume, and possessing the following property: every m-dimensional submanifold of less than unit m-volume is necessarily torsion in homology. This result is different from the case o…

We show that the asymptotic growth rate for the minimal cardinality of a set of simple closed curves on a closed surface of genus $g$ which fill and pairwise intersect at most $K\ge 1$ times is $2\sqrt{g}/\sqrt{K}$ as $g \to \infty$ . We then bound from below the cardinality of a filling set of systoles by $g/\log(g)$.…

In this article we show that for every finite area hyperbolic surface $X$ of type $(g,n)$ and any harmonic Beltrami differential $μ$ on $X$, then the magnitude of $μ$ at any point of small injectivity radius is uniform bounded from above by the ratio of the Weil-Petersson norm of $μ$ over the square root of the systole…

The paper proves optimal systolic inequalities for Möbius strip and Klein bottle.

A new systolic inequality for mod 2 systoles is established.

Minimal displacement set in weakly systolic complexes is systolic and embeds isometrically.

Let X be a closed manifold of dimension 2m >= 6 with torsion-free middle-dimensional homology. We construct metrics on X of arbitrarily small volume, such that every middle-dimensional submanifold of less than unit volume necessarily bounds. Thus, Loewner's theorem has no higher-dimensional analogue.

A new systolic inequality with a remainder for the real projective plane.

We prove a new systolic volume lower bound for non-orientable n-manifolds, involving the stable 1-systole and the codimension 1 systole with coefficients in Z_2. As an application, we prove that Lusternik-Schnirelmann category and systolic category agree for non-orientable closed manifolds of dimension 3, extending our…

In this paper, we prove that the systolic volume of a closed aspherical 3-manifold is bounded below in terms of complexity. Systolic volume is defined as the optimal constant in a systolic inequality. Babenko showed that the systolic volume is a homotopy invariant. Moreover, Gromov proved that the systolic volume depen…

In this paper we investigate the systolic landscape of translation surfaces for fixed genus and fixed angles of their cone points. We furthermore study how the systoles of a translation surface relate to the systoles of its graph of saddle connections. This allows us to develop an algorithm to compute the systolic rati…

Max systoles on spheres with punctures are counted.

Study on complexity of systolic geodesics on Bolza surface.

New finding links hyperbolic manifold systolic volume to triangulation complexity.

Upper bound found for systolic geometry on manifolds with positive scalar curvature.

Sharp inequalities for Kähler manifolds' systolic invariants are established.

Systoles of hyperbolic manifolds are dense and related to Salem numbers.

Random surfaces with long systoles created from graph theory ideas.

Upper bounds found for systole function critical points on surface moduli space.

Extremal length systole is maximized at the Bolza surface.

For all systolic groups we construct boundaries which are EZ--structures. This implies the Novikov conjecture for torsion--free systolic groups. The boundary is constructed via a system of distinguished geodesics in a systolic complex, which we prove to have coarsely similar properties to geodesics in CAT(0) spaces.

Sharp inequalities link manifold's systole to curvature of boundary.

New algorithms compute points on generalized Bolza surfaces.

New systolic inequality for 3D contact forms on Seifert bundles.

The aim of this text is to present the concept of systole of a compact riemannian manifold and to give an overview of systolic geometry. I will also present the "regularization technique", which leads to major results in systolic geometry. I will detail how this technique allows to link the systolic volume of some clos…

The paper analyzes systoles of complex projective spaces under various metrics.

New Finsler metric on sphere disproves systolic ratio conjecture.

The study finds a unique systole maximum in non-hyperelliptic surfaces.

Extends Gromov's optimal systolic inequality to manifolds with specific cohomology properties.