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.
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 4π/3. 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.
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.
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.
The Fubini-Study metric minimizes a volume-normalized holomorphic systole in CPn.
problem Finding metrics with minimal holomorphic systoles in complex projective spaces.
method Introduced holomorphic k-systole and used Gauduchon metrics to establish minimization. result The Fubini-Study metric locally minimizes the volume-normalized holomorphic (n−1)-systole. The paper proves combinatorial versions of systolic inequalities for manifolds.
problem Establishing inequalities for combinatorial structures of manifolds.
method Using triangulations and Riemannian metrics, the paper establishes combinatorial systolic inequalities.
result A class of manifolds satisfies a systolic inequality if and only if it satisfies a combinatorial systolic inequality.
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.
Characterizes the largest two-systole in real projective spaces.
problem Finding the largest two-systole in real projective spaces.
method Integral-geometric formula for minimal two-spheres, characterizing metrics with largest two-systole.
result Each homogeneous metric on the three-dimensional real projective space is the unique metric with the largest possible two-systole among metrics with the same volume.
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.
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.
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…
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.
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…
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.
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.
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.
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.
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.
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…
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 (systole)n/volume. Its supremum on the set of all the riemannian metrics, is known to be finite for a large class of manifolds, including …
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) …
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.
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. We prove the 3-manifold $\RP^3 \# \RP^3$ is of Z2-coefficient homology (1,2)-systolic freedom. Given a Riemannian metric on $\RP^{3}\# \RP^{3}$, we define Z2-coefficient homology 1-systole as the infimum of lengths of all nonseparating geodesic loops representing nontrivial classes in $H_{1}(\RP^3\#\…
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…
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.
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.
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.
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…
Flat surfaces with finitely many singularities solve systolic extremal problem.
problem The regularity of systolically extremal surfaces in nonpositive curvature.
method Developed a hands-on approach using Alexandrov surfaces and combinatorial estimates.
result Every systolically extremal nonpositively curved surface is piecewise flat with finitely many conical singularities.
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.
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…
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.
The paper establishes a uniform Lipschitz bound on the square root of the systole function in Teichmüller space.
problem Uniform Lipschitz bounds on geometric functions in Teichmüller space.
method Injectivity radius analysis and Lipschitz bounds on systole function.
result Uniform Lipschitz constant for the square root of the systole function on Teichmüller space.
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.
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.
Positive scalar curvature implies small 2-systoles in Kähler manifolds
problem Finding topologically non-trivial 2-spheres with small area in Kähler manifolds
method Using index theoretic methods
result Quantitative upper bounds on the 2-systole
The study examines the systole of 3-manifolds with positive scalar curvature.
problem Analyzing the systole of 3-manifolds with positive scalar curvature.
method Local-to-global approach using capillary prisms and Coxeter gluing.
result Estimates the systole of 3-manifolds with positive scalar curvature.
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…
Multiplicative relations in the cohomology ring of a manifold impose constraints upon its stable systoles. Given a compact Riemannian manifold (X,g), its real homology H_*(X,R) is naturally endowed with the stable norm. Briefly, if h\in H_k(X,R) then the stable norm of h is the infimum of the Riemannian k-volumes of re…
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 estimate for 2-systole on Kähler surfaces with positive scalar curvature.
problem Estimating the 2-systole on compact Kähler surfaces with positive scalar curvature.
method Combining classification of positive scalar curvature Kähler surfaces with Stern's level set method adapted to Kähler setting.
result Proved the sharp estimate minXS(ω)⋅sys2(ω)≤12π. A long-standing open problem in systolic geometry asks whether a Riemannian metric on the real projective space whose volume equals that of the canonical metric, but is not isometric to it, must necessarily carry a periodic geodesic of length smaller than π. A contact-geometric reformulation of systolic geometry and th…
Study confirms a 2-sphere metric with three geodesics of minimal length.
problem Understanding the systolic, width, and Gromov-Guth metrics on a 2-sphere.
method Classical min-max and hyperbolic geometry tools.
result Figure-eight geodesics achieve the systolic, width, and Gromov-Guth metrics on a 2-sphere.
Sharp inequalities found for Reeb flows on 3-sphere.
problem Finding systolic inequalities for Reeb flows on the 3-sphere.
method Analyzing the systolic ratio of contact forms on the 3-sphere.
result Sharp systolic inequalities for Reeb flows on the 3-sphere, with equality at Zoll contact forms.
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…
Decomposes geodesic currents on surfaces into measured laminations or positive systole components.
problem Decomposing geodesic currents on surfaces of finite type.
method Topological decomposition and analysis of intersection functions.
result Currents with positive systole are bilipschitz equivalent to length functions under hyperbolic metrics.