We compute the number of systoles, the shortest simple closed geodesics and 2-systoles, the second shortest simple closed geodesics on hyperbolic surfaces homeomorphic to once-punctured torus and four-punctured sphere.
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Complexity in Bolza surfacemath.DG
Study on complexity of systolic geodesics on Bolza surface.
problem Complexity of systolic geodesics on the Bolza surface.
method Analysis of intersections and triangulation of geodesics.
result There are 12 second systolic geodesics forming a triangulation of the surface.
The study of systoles in arithmetic hyperbolic manifolds.
problem Understanding the systoles of arithmetic hyperbolic manifolds.
method Construction and analysis of arithmetic hyperbolic manifolds.
result Explicit bounds on volumes and systoles of arithmetic hyperbolic manifolds.
The article disproves a local systolic inequality and shows a lower bound on filling area.
problem Proving a local systolic inequality and a lower bound on filling area.
method Analyzing Gromov's filling area conjecture and showing a computational mistake.
result The local systolic inequality was disproved and a lower bound on filling area was shown.
The paper analyzes systoles of complex projective spaces under various metrics.
problem Behavior of systoles in complex projective spaces for different metrics.
method Integral geometric techniques and careful analysis of systole functional.
result Balanced metrics locally minimize the systole on volume-normalized metrics.
Shape of filling-systole subspace in surface moduli space and critical points of systole functionmath.GT
The paper explores the shape of filling-systole subspace in surface moduli space and critical points of systole function.
problem Understanding the structure and critical points of the filling-systole subspace in surface moduli space.
method Analyzing Teichmüller and Weil-Petersson distances to determine the proximity of points to the subspace.
result Most points in are within a specific Teichmüller distance from and have a certain distance from the thick part of .
A new topological gap theorem improves the systole of 3-manifolds with positive scalar curvature.
problem Improving the systole of 3-manifolds with positive scalar curvature.
method Weak inverse mean curvature flow.
result The systole of 3-manifolds is no greater than an improved constant c ≈ 5.44π.
Optimal inequalities for systole, inradius, and volume in hyperbolic 3-manifolds
problem Establishing optimal inequalities relating systole, inradius, and volume in hyperbolic 3-manifolds
method Using systole-volume inequalities for extremal manifolds
result Extremal manifolds for systole, inradius, and volume are identified
Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so-called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surface (we call these admissible). There is a natural necessary condition on such grap…
Constructs minimal laminations with controlled leaf topologies.
problem Creating minimal laminations with specific surface topologies.
method Using towers of finite coverings and developing a relative version of residual finiteness.
result Established finite covers with control on the second systole.
Kissing numbers for surfacesmath.GT
The so-called {\it kissing number} for hyperbolic surfaces is the maximum number of homotopically distinct systoles a surface of given genus can have. These numbers, first studied (and named) by Schmutz Schaller by analogy with lattice sphere packings, are known to grow, as a function of genus, at least like $g^{\s…
Algorithms find second and third shortest non-trivial closed walks on surfaces.
problem Finding non-trivial closed walks on surfaces efficiently.
method Algorithms based on careful analysis of shortest curves and configurations.
result Second shortest walk found in time, third in time.
The paper proves optimal systolic inequalities for Möbius strip and Klein bottle.
problem Optimal systolic inequalities for Möbius strip and Klein bottle.
method Alternative proof using -distance of conformal factor.
result Estimates on systolic defect for Möbius strip and Klein bottle.
A new systolic inequality for mod 2 systoles is established.
problem Bounding the product of mod 2 systoles in Riemannian manifolds.
method Analyzing the product of systoles of dimensions 1 and n-1 in closed manifolds with bounded local geometry.
result A systolic inequality is derived for the product of mod 2 systoles, showing a power-law relationship with volume.
Minimal displacement set in weakly systolic complexes is systolic and embeds isometrically.
problem Structure of minimal displacement set in weakly systolic complexes.
method Investigation of minimal displacement set properties and embeddings.
result Minimal displacement set is systolic and embeds isometrically into the complex.
Study systolic geometry of translation surfaces and origamis.
problem Investigate systolic ratios of translation surfaces and origamis.
method Analyze systoles and saddle connections, develop algorithm, compute ratios.
result Compute maximal systolic ratio of origamis in up to 67 squares.
Authors find no positive spun triangulations for certain hyperbolic 3-manifolds.
problem Finding positive spun triangulations for hyperbolic 3-manifolds.
method Using Choi's result, they provide examples of closed hyperbolic 3-manifolds and geodesics without positive spun ideal triangulations.
result They provide evidence for the conjecture that Vol3 has no positive spun ideal triangulation for any choice of geodesic.
A new systolic inequality with a remainder for the real projective plane.
problem Proving a stronger systolic inequality for the real projective plane.
method Developing a new systolic inequality with a remainder term.
result A stronger systolic inequality with a remainder for the real projective plane.
The systolic ratio of a contact form on the three-sphere is the quantity \[ ρ_{\mathrm{sys}}(α) = \frac{T_{\min}(α)^2}{\mathrm{vol}(S^3,α\wedge dα)}, \] where is the minimal period of closed Reeb orbits on . A Zoll contact form is a contact form such that all the orbits of the corresponding R…
Extended systolic inequality for 2-complexes to improve group systolic area bounds.
problem Improving bounds on systolic area of various groups.
method Extended systolic inequality for piecewise Riemannian 2-complexes.
result Improved universal lower bound for systolic area of many groups.
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…
Max systoles on spheres with punctures are counted.
problem Finding the maximum number of systoles on spheres with punctures.
method Analyzing complete Riemannian metrics on spheres with punctures.
result Determined the maximal number of systoles.
New finding links hyperbolic manifold systolic volume to triangulation complexity.
problem Understanding the relationship between systolic volume and triangulation complexity in hyperbolic manifolds.
method Proof based on Jørgensen and Thurston's theorem of hyperbolic volume.
result Systolic volume of hyperbolic manifolds is related to triangulation complexity.
New surfaces with few filling systoles found.
problem Finding surfaces with a small number of systoles that still fill.
method Constructing a hyperbolic surface with a specified number of systoles that fill.
result Disproved the lower bound on systole count for filling surfaces.
Upper bound found for systolic geometry on manifolds with positive scalar curvature.
problem Relationship between systolic geometry and positive scalar curvature.
method Spinorial methods combined with geometric measure theory and curvature estimates.
result Upper bound for the two-dimensional stable systole on certain manifolds.
New manifolds with small systoles not quasi-arithmetic.
problem Finding manifolds with small systoles not quasi-arithmetic.
method Hybrid construction of known manifolds.
result Exhibited manifolds with arbitrarily small systoles.
Sharp inequalities for Kähler manifolds' systolic invariants are established.
problem Understanding systolic invariants of Kähler manifolds.
method Analyzing metrics with positive scalar curvature on Kähler manifolds and their products.
result Bounds for systolic invariants attain equality for specific manifolds.
Systoles of hyperbolic manifolds are dense and related to Salem numbers.
problem Density of systoles in hyperbolic manifolds.
method Analyzing systoles and arithmetic hyperbolic manifolds.
result Systoles of closed arithmetic hyperbolic manifolds are dense in .
Random surfaces with long systoles created from graph theory ideas.
problem Finding surfaces with long systoles.
method Two constructions inspired by graph theory.
result Proved a new lower bound on systole length.
An upperbound for the number of critical points of the systole function on the surface moduli spacemath.GT
Upper bounds found for systole function critical points on surface moduli space.
problem Finding upper bounds for systole function critical points.
method Analyzing the systole function on the surface moduli space.
result Upper bounds for critical points of systole function and their systole values.
Extremal length systole is maximized at the Bolza surface.
problem Finding the surface with the maximum extremal length systole.
method Analyzing the Bolza surface and comparing its extremal length systole to others.
result The extremal length systole of the Bolza surface is and is a strict local maximum.
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.
problem Finding relationships between manifold's systole and curvature.
method Proved inequalities relating homological systoles to scalar and mean curvature.
result Equality case implies universal cover is a cylinder.
New systolic inequality for 3D contact forms on Seifert bundles.
problem Bounding the shortest Reeb orbit period in terms of contact volume.
method Proved a general systolic inequality for S1-invariant contact forms on Seifert bundles.
result Validated systolic inequality on Seifert bundles with non-zero Euler number.
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…
Exact systole values found for hyperbolic surfaces with specific cyclic symmetries.
problem Finding exact systole lengths for hyperbolic surfaces with maximal cyclic symmetries.
method Analyzing hyperbolic surfaces of different genera with specified cyclic symmetries and calculating exact systole lengths.
result Exact formulas for systole lengths of hyperbolic surfaces with maximal cyclic symmetries.
Survey on systole growth in congruence symmetric spaces.
problem Understanding systole growth in congruence symmetric spaces.
method Survey and simple proofs for best constants.
result Best possible constants for systole growth in symmetric spaces.
Study systole of large genus minimal surfaces in positive Ricci curvature.
problem Understanding the systole of minimal surfaces in manifolds with positive Ricci curvature.
method Colding--Minicozzi lamination theory.
result Results on the systole of large genus minimal surfaces.
New Finsler metric on sphere disproves systolic ratio conjecture.
problem Proving the maximal systolic ratio on 2-sphere.
method Inspired by Cossarini-Sabourau, constructs a Finsler metric.
result Systolic ratio of new Finsler metric is .
The study finds a unique systole maximum in non-hyperelliptic surfaces.
problem Understanding systole functions on translation surfaces.
method Analyzing local and global maxima of systole functions.
result Local maxima are not global in non-hyperelliptic components.
Study spherical surfaces with conical points, proving systole inequality.
problem Characterize moduli spaces of spherical metrics with conical singularities.
method Analyze systole inequality and properness of forgetful map.
result Explicit systole inequality linking metric and conformal invariants.
Extends Gromov's optimal systolic inequality to manifolds with specific cohomology properties.
problem Finding optimal systolic inequalities for manifolds with complex cohomology structures.
method Extends Gromov's inequality to manifolds with fundamental cohomology classes as cup products of 2-dimensional classes.
result Provides an optimal systolic inequality for a new class of manifolds.
Study finds bounds for systole length on arithmetic punctured spheres.
problem Finding the shortest essential curve on arithmetic punctured spheres.
method Correspondence between surfaces and planar triangulations to bound systole length.
result Arithmetic surfaces do not achieve maximal systole length for .
The study finds large systoles in translation surfaces and hyperelliptic ones.
problem Investigating large systoles in translation surfaces.
method Analyzing surfaces of increasing genus and hyperelliptic surfaces to find large systoles.
result An infinite sequence of translation surfaces with large systoles.
Sharp systolic inequalities for invariant tight contact forms on principal S1-bundles over S2math.SG
Sharp systolic inequality for invariant tight contact forms on S1-bundles over S2.
problem Finding the shortest period of closed Reeb orbits on invariant tight contact forms.
method Proving a sharp systolic inequality based on the Euler class of the bundle.
result A behavior of the systolic inequality depends on the Euler class of the bundle.
Directly proves logarithmic systolic growth for all hyperbolic surfaces.
problem Proving logarithmic systolic growth for all hyperbolic surfaces.
method Using original Brooks/Buser-Sarnak surfaces through a direct approach.
result Directly proves logarithmic systolic growth for all hyperbolic surfaces.
Study shows systole behavior changes significantly for large genus hyperbolic surfaces.
problem Understanding systole behavior in large genus hyperbolic surfaces.
method Analysis of random surfaces with respect to Weil-Petersson volume.
result Expected value of separating systole behaves like for large genus.