arXiv research

Study trace systoles on surfaces, finding optimal bounds and implications.

Method calculates systolic length of modular curves.

The study finds infinitely many semi-arithmetic Riemann surfaces with dense systoles and distinct invariant trace fields.

We apply a study of orders in quaternion algebras, to the differential geometry of Riemann surfaces. The least length of a closed geodesic on a hyperbolic surface is called its systole, and denoted syspi_1. P. Buser and P. Sarnak constructed Riemann surfaces X whose systole behaves logarithmically in the genus g(X). Th…

The systole of a hyperbolic surface is bounded by a logarithmic function of its genus. This bound is sharp, in that there exist sequences of surfaces with genera tending to infinity that attain logarithmically large systoles. These are constructed by taking congruence covers of arithmetic surfaces. In this article we p…

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…

We give a detailed description of the arithmetic Fuchsian group of the Bolza surface and the associated quaternion order. This description enables us to show that the corresponding principal congruence covers satisfy the bound sys(X) > 4/3 log g(X) on the systole, where g is the genus. We also exhibit the Bolza group a…

We prove an upper bound for the number of shortest closed geodesics in a closed hyperbolic manifold of any dimension in terms of its volume and systole, generalizing a theorem of Parlier for surfaces. We also obtain bounds on the number of primitive closed geodesics with length in a given interval that are uniform for …

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.

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.

New manifolds with small systoles not quasi-arithmetic.

Sharp inequalities for Kähler manifolds' systolic invariants are established.

Bounds on spectral gaps of hyperbolic 3-manifolds and orbifolds.

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.

The study of systoles in arithmetic hyperbolic manifolds.

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.

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.

Study finds bounds for systole length on arithmetic punctured spheres.

The study finds large systoles in translation surfaces and hyperelliptic ones.

Sharp systolic inequality for invariant tight contact forms on S1-bundles over S2.

Directly proves logarithmic systolic growth for all hyperbolic surfaces.

Study shows systole behavior changes significantly for large genus hyperbolic surfaces.

New links in 3-manifolds have large systole.

Paper proves optimal systolic inequality for manifolds with positive triRic curvature.

We give the first example of systolic freedom over torsion coefficients. The phenomenon is a bit unexpected (contrary to a conjecture of Gromov's) and more delicate than systolic freedom over the integers.

Proves upper bound on systolic ratio for circle fillings.

Study systoles and diameters on hyperbolic surfaces, finding an upper bound for their ratio.

We prove optimal systolic inequalities on Finsler Mobius bands relating the systole and the height of the Mobius band to its Holmes-Thompson volume. We also establish an optimal systolic in- equality for Finsler Klein bottles of revolution, which we conjecture to hold true for arbitrary Finsler metrics. Extremal metric…

New index characterizes non-smooth Zoll convex bodies.

For a translation surface, we define the systole to be the length of the shortest saddle connection. We give a characterization of the maxima of the systole function on a stratum, and give a family of examples providing local but nonglobal maxima on each stratum of genus at least three. We further study the relation be…

We show that for closed orientable manifolds the $k$-dimensional stable systole admits a metric-independent volume bound if and only if there are cohomology classes of degree $k$ that generate cohomology in top-degree. Moreover, it turns out that in the nonorientable case such a bound does not exist for stable systoles…