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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Introduces systolic inequalities in Riemannian and symplectic geometry.
problem Exploring systolic inequalities in different geometric settings.
method Comparing classical Riemannian metrics to recent symplectic measurements.
result Illustrates connections between Riemannian and symplectic geometry.
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.
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.
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.
Extremal length generalized for higher dimensions and applied to complex systolic inequalities.
problem Estimating and calculating extremal length in higher dimensions.
method Generalization of extremal length to complex manifolds and formulation of geometric inequalities.
result Formulated complex systolic inequalities in terms of extremal length.
Paper proves optimal systolic inequality for manifolds with positive triRic curvature.
problem Optimal systolic inequality for manifolds with positive triRic curvature.
method Stable weighted k-slicing, volume comparison theorem, and metric deformation. result Proves an optimal systolic inequality and characterizes the equality case.
The study proves surfaces with high genus have a specific inequality.
problem Proving surfaces with high genus satisfy a specific inequality.
method Using volume entropy and systolic ratio inequality.
result Every closed surface of genus at least 18 satisfies Loewner's systolic ratio inequality.
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.
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.
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…
Two lectures on metric geometry of manifolds.
problem Understanding metric geometry properties of manifolds.
method Discussion of specific inequalities and concepts.
result Exploration of metric geometry properties of manifolds.
Optimal systolic inequality proved for manifolds with positive bi-Ricci curvature.
problem Proving optimal systolic inequalities on manifolds with positive bi-Ricci curvature.
method Minimal surfaces method under the Generic Regularity Hypothesis.
result Optimal systolic inequality proved in all dimensions.
We establish combinatorial versions of various classical systolic inequalities. For a smooth triangulation of a closed smooth manifold, the minimal number of edges in a homotopically non-trivial loop contained in the 1-skeleton gives an integer called the combinatorial systole. The number of top-dimensional simplices…
Proves a new inequality for certain complex surfaces.
problem Analyzes compact Kähler surfaces with positive scalar curvature.
method Uses properties of holomorphic maps and ruled surfaces.
result Proves a 2-systolic inequality on these surfaces.
We show that the systolic constant, the minimal entropy, and the spherical volume of a manifold depend only on the image of the fundamental class under the classifying map of the universal covering. Moreover, we compute the systolic constant of manifolds with fundamental group of order two (modulo the value on the real…
3D contact manifolds have optimal higher systolic ratios.
problem Optimizing higher systolic ratios in 3D contact manifolds.
method Proving Besse contact forms maximize certain ratios.
result Besse contact forms are local maximizers of higher systolic ratios.
The paper establishes a local systolic inequality for odd-symplectic forms.
problem Formulating and proving a systolic inequality for odd-symplectic forms.
method Defining volume and action of periodic orbits, proving polynomial relationship, and applying to specific cases.
result Established a polynomial relationship between volume and action for Zoll odd-symplectic forms, recovering known inequalities.
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
We investigate the filling area conjecture, optimal systolic inequalities, and the related problem of the nonvanishing of certain linking numbers in 3-manifolds.
Tight embeddings of 2-tori in 3D space contain short loops.
problem Finding the shortest non-contractible loops in twisted 2-tori.
method Proving systolic inequalities for T2 embeddings in R3. result Highly twisted 2-tori contain non-contractible loops of small diameter.
We show that the geometry of a Riemannian manifold (M,g) is sensitive to the apparently purely homotopy-theoretic invariant of M known as the Lusternik-Schnirelmann category, denoted cat_{LS}(M). Here we introduce a Riemannian analogue of cat_{LS}(M), called the systolic category of M. It is denoted cat_{sys}(M), and d…
Sharp systolic bounds for spheres and Finsler spheres are derived.
problem Bounding the systolic ratio of spheres and Finsler spheres.
method Rotationally symmetric Finsler metrics on spheres are analyzed using Killing vector fields.
result The systolic ratio of spheres and Finsler spheres does not exceed π and equals π if and only if the metric is Riemannian and Zoll.
We give a short proof of the systolic inequality for the n-dimensional torus. The proof uses minimal hypersurfaces. It is based on the Schoen-Yau proof that an n-dimensional torus admits no metric of positive scalar curvature.
We introduce the notions of categorical systoles and categorical volumes of Bridgeland stability conditions on triangulated categories. We prove that for any projective K3 surface, there exists a constant C depending only on the rank and discriminant of its Picard group, such that $$\mathrm{sys}(σ)^2\leq C\cdot\mathrm{…
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…
Loewner inequality proven for curved surfaces.
problem Proving Loewner's inequality for nonpositively curved surfaces.
method Combining Gauss-Bonnet formula with averaging argument using geodesic flow invariance.
result Found a disk with large total curvature around its center, leading to large area.
We define Dirichlet type series associated with homology length spectra of Riemannian, or Finsler, manifolds, or polyhedra, and investigate some of their analytical properties. As a consequence we obtain an inequality analogous to Gromov's classical intersystolic inequality, but taking the whole homology length spectru…
Paper proves constants for Moser-Trudinger inequality on surfaces.
problem Establishing constants for Moser-Trudinger inequality on surfaces.
method Using systole, isoperimetric constant, and curvature as parameters.
result Constants can be chosen to depend on only 3 parameters.
This is an expository essay about systolic geometry. It describes a central theorem in the subject and why the proof is difficult. Then it discusses different metaphors which suggest ways to approach the problem. The metaphors connect the systolic inequality to minimal surfaces, topological dimension, scalar curvature,…
Optimal inequalities for metric surfaces derived from filling minimality.
problem Proving optimal systolic inequalities for metric surfaces.
method Analysis of asymptotic volume growth and minimality of normed planes and hemispheres.
result Optimal constants for tori and real projective planes match Finsler settings.
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) …
The study proves a new positive energy theorem for manifolds with specific curvature properties.
problem Proving a new positive energy theorem for manifolds with specific curvature properties.
method Establishing a systolic inequality relating boundary mean curvature to the systole of the boundary.
result Obtaining a new positive energy theorem, with equality for Horowitz-Myers metrics.
We outline the current state of knowledge regarding geometric inequalities of systolic type, and prove new results, including systolic freedom in dimension 4. Namely, every compact, orientable, smooth 4-manifold X admits metrics of arbitrarily small volume such that every orientable, immersed surface of smaller than un…
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…
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.
We generalize optimal inequalities of C. Loewner and M. Gromov, by proving lower bounds for the total volume in terms of the homotopy systole and the stable systole. Our main tool is the construction of an area-decreasing map to the Jacobi torus, streamlining and generalizing the construction of the first author in col…