arXiv research

Extremal length systole is maximized at the Bolza surface.

Extremal length generalized for higher dimensions and applied to complex systolic inequalities.

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…

Cube edges curves minimize systole length.

Flat surfaces with finitely many singularities solve systolic extremal problem.

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…

Optimal inequalities for systole, inradius, and volume in hyperbolic 3-manifolds

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…

Optimal curves minimize crossings on surfaces.

In this article we address a number of features of the moduli space of spherical metrics on connected, compact, orientable surfaces with conical singularities of assigned angles, such as its non-emptiness and connectedness. We also consider some features of the forgetful map from the above moduli space of spherical sur…

A compact manifold is called Bieberbach if it carries a flat Riemannian metric. Bieberbach manifolds satisfy an isosystolic inequality by a general and fundamental result of M. Gromov. In dimension 3, there exist four classes of non-orientable Bieberbach manifolds up to an affine diffeomorphism. In this paper, We prove…

The paper finds minimal-area metrics on polygons with multiple crossing geodesics.

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…

Extends curve functions to geodesic currents with a simple criterion.

The study of random surfaces reveals asymptotic lengths of separating geodesics.

We present a new optimal systolic inequality for a closed Riemannian manifold X, which generalizes a number of earlier inequalities, including that of C. Loewner. We characterize the boundary case of equality in terms of the geometry of the Abel-Jacobi map, A_X, of X. For an extremal metric, the map A_X turns out to be…

In this paper we raise the question whether every closed Riemannian manifold has a spine of minimal area, and we answer it affirmatively in the surface case. On constant curvature surfaces we introduce the spine systole, a continuous real function on moduli space that measures the minimal length of a spine in each surf…

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.

Extended systolic inequality for 2-complexes to improve group systolic area bounds.

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.

New surfaces with few filling systoles found.

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.

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…

Survey on systole growth in congruence symmetric spaces.

Exact systole values found for hyperbolic surfaces with specific cyclic symmetries.

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.