arXiv research

New Finsler metric on sphere disproves systolic ratio conjecture.

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…

Proves upper bound on systolic ratio for circle fillings.

The systolic ratio of a contact form on a closed three-manifold is the quotient of the square of the shortest period of closed Reeb orbits by the contact volume. We show that every co-orientable contact structure on any closed three-manifold is defined by a contact form with arbitrarily large systolic ratio. This shows…

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

Let $(M,g)$ be a closed, oriented, Riemannian manifold of dimension $m$. We call a systole a shortest non-contractible loop in $(M,g)$ and denote by $sys(M,g)$ its length. Let $SR(M,g)=\frac{{sys(M,g)}^m}{vol(M,g)}$ be the systolic ratio of $(M,g)$. Denote by $SR(k)$ the supremum of $SR(S,g)$ among the surfaces of fixe…

The study proves surfaces with high genus have a specific inequality.

3D contact manifolds have optimal higher systolic ratios.

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 …

We find an upper bound for the entropy of a systolically extremal surface, in terms of its systole. We combine the upper bound with A. Katok's lower bound in terms of the volume, to obtain a simpler alternative proof of M. Gromov's asymptotic estimate for the optimal systolic ratio of surfaces of large genus. Furthermo…

A compact manifold is called Bieberbach if it carries a flat Riemannian metric. Bieberbach manifolds are aspherical, therefore the supremum of their systolic ratio, over the set of Riemannian metrics, is finite by a fundamental result of M. Gromov. We study the optimal systolic ratio of compact of $3$-dimensional orien…

We prove that the systolic ratio of a sphere of revolution $S$ does not exceed $π$ and equals $π$ if and only if $S$ is Zoll. More generally, we consider the rotationally symmetric Finsler metrics on a sphere of revolution which are defined by shifting the tangent unit circles by a Killing vector field. We prove that i…

We study optimal curvature-free inequalities of the type discovered by C. Loewner and M. Gromov, using a generalisation of the Wirtinger inequality for the comass. Using a model for the classifying space BS^3 built inductively out of BS^1, we prove that the symmetric metrics of certain two-point homogeneous manifolds t…

Sharp inequalities found for orbifold metrics.

Study improves bounds on p-covectors and proves stable systolic inequalities.

We study the systolic area (defined as the ratio of the area over the square of the systole) of the 2-sphere endowed with a smooth riemannian metric as a function of this metric. This function, bounded from below by a positive constant over the space of metrics, have the standard metric $g\_0$ for critic point, althoug…

We prove an optimal systolic inequality for CAT(0) metrics on a genus~2 surface. We use a Voronoi cell technique, introduced by C.~Bavard in the hyperbolic context. The equality is saturated by a flat singular metric in the conformal class defined by the smooth completion of the curve y^2=x^5-x. Thus, among all CAT(0) …

Let $α$ be a contact form on a connected closed three-manifold $Σ$. The systolic ratio of $α$ is defined as $ρ_{\mathrm{sys}}(α):=\tfrac{1}{\mathrm{Vol}(α)}T_{\min}(α)^2$, where $T_{\min}(α)$ and $\mathrm{Vol}(α)$ denote the minimal period of periodic Reeb orbits and the contact volume. The form $α$ is said to be Zoll …

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…

Let S be a closed orientable surface of genus at least 2 and let G be a semisimple real algebraic group of non-compact type. We consider a class of representations from the fundamental group of S to G called positively ratioed representations. These are Anosov representations with the additional condition that certain …

If a contact form on a (2n+1)-dimensional closed contact manifold admits closed Reeb orbits, then its systolic ration is defined to be the quotient of (n+1)th power of the shortest period of Reeb orbits by the contact volume. We prove that every co-orientable contact structure on any closed contact manifold admits a co…

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…

We give asymptotic bounds for the optimal Lipschitz constants for the systole map from the Teichmuller space to the curve complex. We give similar results to those known for closed surfaces in the cases when the genus is fixed or the ratio of genus and punctures is a rational number.

The study connects contact forms and Ruelle invariant in convex domains.

The systolic ratio of a contact form $α$ on the three-sphere is the quantity \[ ρ_{\mathrm{sys}}(α) = \frac{T_{\min}(α)^2}{\mathrm{vol}(S^3,α\wedge dα)}, \] where $T_{\min}(α)$ is the minimal period of closed Reeb orbits on $(S^3,α)$. A Zoll contact form is a contact form such that all the orbits of the corresponding R…

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.

We investigate the geometry of $π_1$-injective surfaces in closed hyperbolic 3-manifolds. First we prove that for any $e>0$, if the manifold $M$ has sufficiently large systole $\sys_1(M)$, the genus of any such surface in $M$ is bounded below by $\exp((1/2-e)\sys_1(M))$. Using this result we show, in particular, that f…

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…

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

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.

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.

Following an idea of Dadok, Harvey and Lawson, we apply the triality property of SO(8) to study the comass of certain self-dual 4-forms on R^8. In particular, we prove that the Cayley 4-form has comass 1 and that any self-dual 4-form realizing the maximal Wirtinger ratio is SO(8)-conjugate to the Cayley 4-form. We also…

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.

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

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