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.
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.
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…
Proves upper bound on systolic ratio for circle fillings.
problem Bounding systolic ratio for circle fillings.
method Proved upper bound on systolic ratio depending on genus.
result Filling Area Conjecture holds for large genus.
The study finds bounds on systoles for genus two Riemann surfaces with abelian differentials.
problem Bounding systoles on genus two Riemann surfaces with abelian differentials.
method Analyzing holomorphic 1-forms on Riemann surfaces, providing upper bounds on systoles.
result For genus two, there are at most 10 systolic loops, with a unique realization.
Study systoles and diameters on hyperbolic surfaces, finding an upper bound for their ratio.
problem Understanding the relationship between systoles and diameters on hyperbolic surfaces.
method Exploring the inequality between systoles and diameters, deducing an upper bound for their ratio.
result The ratio of systoles and diameters has a genus-dependent upper bound.
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.
Lower bounds for systole growth in quaternionic hyperbolic manifolds.
problem Finding lower bounds for systole growth in quaternionic hyperbolic manifolds.
method Explicit lower bound calculation for systole in congruence covers.
result Optimal lower bound for systole growth proved.
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.
Uniform systole bounds for arithmetic orbifolds and number fields.
problem Bounding systole lengths in arithmetic orbifolds.
method Geometric methods and Mahler measure.
result Uniform lower bounds for systole lengths.
New bounds set for stable 2-systole in specific geometric spaces.
problem Bounding stable 2-systole in metrics with positive scalar curvature.
method Proving uniform bounds on specific manifolds.
result Stable 2-systole is uniformly bounded for certain manifolds.
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 construction for surfaces with logarithmically large systoles.
problem Bounding the systole of hyperbolic surfaces.
method Combining graph constructions and matrix counting.
result Constructs surfaces with logarithmically large systoles.
A new method using mod n covering improves systolic inequalities.
problem Stable systolic inequalities in Riemannian geometry.
method Mod n covering approach to force nonzero cup products or indices.
result Improved stable two systolic bounds for various 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 n=7,10,11. 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…
We prove a finiteness result for the systolic area of groups, answering a question of M. Gromov. Namely, we show that there are only finitely many possible unfree factors of fundamental groups of~2-complexes whose systolic area is uniformly bounded. Furthermore, we prove a uniform systolic inequality for all 2-complexe…
Method calculates systolic length of modular curves.
problem Computing upper bounds on systolic length of Riemann surfaces.
method Using congruence subgroups of hyperbolic triangle groups and traces of generators.
result Systolic length grows logarithmically with genus.
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.
Study bounds magnetic geodesics on surfaces using systolic inequalities.
problem Bounding magnetic geodesics on surfaces with prescribed geodesic curvature.
method Applied local systolic-diastolic inequality to contact forms and odd-symplectic forms on three-manifolds.
result Results hold for curves with prescribed curvature close to Zoll or large enough.
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…
Study improves bounds on p-covectors and proves stable systolic inequalities.
problem Improving bounds on p-covectors and proving stable systolic inequalities.
method Analyzing Euclidean norms and fundamental cohomology classes.
result Improved upper bounds and stable systolic inequalities proved.
The paper fills hyperbolic surfaces with a minimal number of systoles.
problem Finding the minimum number of systoles to fill a hyperbolic surface.
method The approach is simpler than previous methods, using O(lngg) systoles. result The number of systoles needed is O(lngg). In my masters thesis I prove a square root bound on the distance of homological codes that come from two dimensional surfaces, as a result of the systolic inequality. I also give a detailed version of M.H. Freedman's proof that due to systolic freedom, this bound does not hold in higher dimensions.
Study trace systoles on surfaces, finding optimal bounds and implications.
problem Optimal systolic inequalities on hyperbolic manifolds and non-Fuchsian representations.
method Defined trace systole, used Markoff maps correspondence, computed bounds.
result Explicit optimal bounds for one-holed torus, four-holed sphere, and non-orientable surface of genus 3.
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.
The paper bounds Pachner moves and systoles in hyperbolic 3-manifolds.
problem Bounding Pachner moves and systoles in cusped hyperbolic 3-manifolds.
method Using geometric ideal triangulations and dihedral angles, the paper gives bounds on Pachner moves and systoles.
result Lower bounds on systole length and Pachner move sequence length.
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.
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 L2-distance of conformal factor. result Estimates on systolic defect for Möbius strip and Klein bottle.
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…
Sharp inequalities found for orbifold metrics.
problem Bounding systolic ratios on rotationally symmetric orbifolds.
method Analyzing spindle orbifolds and Besse metrics.
result Upper bounds on systolic ratios are attained at Besse metrics.
Upper bound found for 2-systole in stretched S² x S² metrics.
problem Bounding the 2-systole in stretched S² x S² metrics.
method Using Gromov and Zhu's developments, derived an upper bound.
result Upper bound for 2-systole derived.
This paper shows a relationship between 3-manifold complexity and systolic volume.
problem Understanding the complexity of 3-manifolds.
method Using systolic volume and triangulation complexity, the paper establishes a relationship between these two concepts.
result The systolic volume of a 3-manifold is related to its complexity.
Calculates the systolic growth of nilpotent Lie groups, providing new insights.
problem Understanding the growth of systolic volume in nilpotent Lie groups.
method Expressed systolic growth in terms of discrete subrings, developed methods for lower bounds.
result First computations of systolic growth for non-equivalent volume growth cases.
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.
We study the number and the length of systoles on complete finite area orientable hyperbolic surfaces. In particular, we prove upper bounds on the number of systoles that a surface can have (the so-called kissing number for hyperbolic surfaces). Our main result is a bound which only depends on the topology of the surfa…
Given a closed manifold M, we prove the upper bound of (n+d)/2 for the length of a product of systoles that can form a curvature-free lower bound for the total volume of M, in the spirit of M. Gromov's systolic inequalities. Here n is the dimension of M, while d is the is the cohomological dimension of its fundamental …
The study bounds and constructs surfaces in arithmetic hyperbolic 3-manifolds.
problem Understanding the growth of Heegard genus across commensurability classes.
method Examined minimal surfaces, gave bounds, constructed incommensurable manifolds, analyzed genera growth.
result Infinitely many incommensurable manifolds with controlled volume and 1-systole.
In 1972, Marcel Berger defined a metric invariant that captures the `size' of k-dimensional homology of a Riemannian manifold. This invariant came to be called the k-dimensional SYSTOLE. He asked if the systoles can be constrained by the volume, in the spirit of the 1949 theorem of C. Loewner. We construct metrics, ins…
Sharp inequality for complex projective space's systoles under scalar curvature bounds.
problem Proving a sharp stable 2-systolic inequality for complex projective space.
method Uses Spin^c Dirac operators, comass estimate, and stable norm-comass duality.
result Equality holds only for the Fubini-Study metric, up to biholomorphism.
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.
We prove the simultaneous (k,n-k)-systolic freedom, for a pair of adjacent integers k smaller than n/2, of a simply connected n-manifold X. Our construction, related to recent results of I. Babenko, is concentrated in a neighborhood of suitable k-dimensional submanifolds of X. We employ calibration by differential form…
The study uses symplectic capacities to bound the systole on the sphere.
problem Bounding the systole on the sphere using symplectic capacities.
method Using symplectic capacities and properties of fiberwise balanced hypersurfaces.
result Upper bounds on the systole in terms of geometric data and β. Density result for arithmetic hyperbolic orbifolds with systole bound.
problem Understanding the density of arithmetic hyperbolic orbifolds with a given systole bound.
method Using bounds for the absolute logarithmic Weil height of algebraic integers and precise estimates for quaternion algebras.
result The set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed invariant trace field and systole bounded below by x0 has density one. The systole length of hyperbolic n-manifolds is bounded by a function of n and t.
problem Bounding the systole length of hyperbolic n-manifolds.
method Relating the number of simplices to the diameter, and using bounds on Margulis tubes.
result The systole length is bounded by a function of n and t.
Given a closed hyperbolic 3-manifold M of volume V, and a link L in M such that the complement M \ L is hyperbolic, we establish a bound for the systole length of M \ L in terms of V. This extends a result of Adams and Reid, who showed that in the case that M is not hyperbolic, there is a universal bound of 7.35534... …
Uniform bounds on harmonic Beltrami differentials and Weil-Petersson curvatures established.
problem Bounding the magnitude of harmonic Beltrami differentials and Weil-Petersson curvatures.
method Using the systole of a hyperbolic surface, the authors derive uniform bounds for the magnitude of harmonic Beltrami differentials and the Weil-Petersson Ricci curvature.
result Uniform bounds on Weil-Petersson curvatures and magnitudes of harmonic Beltrami differentials are established.
New bounds on hyperbolic surfaces' properties using linear programming.
problem Finding bounds on various geometric and spectral properties of hyperbolic surfaces.
method Adapted linear programming methods from sphere packings to hyperbolic surfaces.
result Obtained new upper and lower bounds on multiple properties of hyperbolic surfaces.