arXiv research

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…

Study on complexity of systolic geodesics on Bolza surface.

The article constructs Bolza-like surfaces for infinitely many genera and studies their properties.

Max systoles on spheres with punctures are counted.

Study finds bounds for systole length on arithmetic punctured spheres.

We extend a systolic inequality of Guth for Riemannian manifolds of maximal $\mathbb{Z}_2$ cup-length to piecewise Riemannian complexes of dimension 2. As a consequence we improve the previous best universal lower bound for the systolic area of groups for a large class of groups, including free abelian and surface grou…

Formula found for maximal systole of hyperbolic surfaces with largest $S^3$ symmetry.

Extremal length systole is maximized at the Bolza surface.

Random surfaces with long systoles created from graph theory ideas.

Let $S$ be a compact hyperbolic Riemann surface of genus $g \geq 2$. We call a systole a shortest simple closed geodesic in $S$ and denote by $\mathop{sys}(S)$ its length. Let $\mathop{msys(g)}$ be the maximal value that $\mathop{sys}(\cdot)$ can attain among the compact Riemann surfaces of genus $g$. We call a (global…

New Finsler metric on sphere disproves systolic ratio conjecture.

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…

3D contact manifolds have optimal higher systolic ratios.

We prove an optimal systolic inequality for nonpositively curved Dyck's surfaces. The extremal surface is flat with eight conical singularities, six of angle theta and two of angle 9pi - theta, for a suitable theta with cos(theta) in Q(sqrt{19}). Relying on some delicate capacity estimates, we also show that the extrem…

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

We bound two global invariants of cusped hyperbolic manifolds: the length of the shortest closed geodesic (the systole), and the radius of the biggest embedded ball (the inradius). We give an upper bound for the systole, expressed in terms of the dimension and simplicial volume. We find a positive lower bound on the in…

Study on combinatorial $k$-systoles on surfaces, showing growth in intersection numbers.

We find a canonical decomposition of a geodesic current on a surface of finite type arising from a topological decomposition of the surface along special geodesics. We show that each component either is associated to a measured lamination or has positive systole. For a current with positive systole, we show that the in…

Geometrically, spherical 3-manifolds emerge from flat SU(2)-bundles over hyperbolic surfaces.

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 study the Weyl chamber length boundary both of the Hitchin and of the maximal character varieties and determine therein an open set of discontinuity for the action of the mapping class group. This result is obtained as consequence of a canonical decomposition of a geodesic current on a surface of finite type arising…

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…

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.

We apply recently developed convex programs to find the minimal-area Riemannian metric on $2n$-sided polygons ($n\geq 3$) with length conditions on curves joining opposite sides. We argue that the Riemannian extremal metric coincides with the conformal extremal metric on the regular $2n$-gon. The hexagon was considered…

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

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

The study of systoles in arithmetic hyperbolic manifolds.

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

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

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.

Proves upper bound on systolic ratio for circle fillings.

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.