Space of Zoll Finsler metrics on projective plane deformation retracts to round metric.
problem Understanding the structure of Zoll Finsler metrics on projective planes.
method Geodesic flow deformation and curvature flow.
result Space of Zoll Finsler metrics on projective plane is connected and deformation retracts to the round metric.
Study geodesics on positively curved Zoll surfaces.
problem Geodesics on positively curved Zoll surfaces.
method Explicit constructions of Finsler metrics.
result Induction of constant flag curvature metrics.
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.
The systolic ratio of a contact form α on the three-sphere is the quantity \[ ρ_{\mathrm{sys}}(α) = \frac{T_{\min}(α)^2}{\mathrm{vol}(S^3,α\wedge dα)}, \] where Tmin(α) is the minimal period of closed Reeb orbits on (S3,α). A Zoll contact form is a contact form such that all the orbits of the corresponding R…
This article is an exposition of four loosely related remarks on the geometry of Finsler manifolds with constant positive flag curvature. <p> The first remark is that there is a canonical Kahler structure on the space of geodesics of such a manifold. <p> The second remark is that there is a natural way to construct a (…
Characterizes Zoll metrics via min-max values.
problem Characterizing Zoll Riemannian metrics.
method Uses min-max values in a loop space.
result Two min-max values coincide for Zoll metrics.
The isospectral problem for p-widths is solved using Zoll metrics on S^2.
problem Determine if a Riemannian manifold is uniquely determined by its p-widths.
method Construct counterexamples on S^2 using Zoll metrics and properties of geodesic p-widths.
result Many counterexamples exist on S^2, showing uniqueness is not guaranteed.
We strengthen our previous results regarding the moduli spaces of Zoll metrics and Zoll projective structures on S^2. In particular, we describe a concrete, open condition which suffices to guarantee that a totally real embedding of RP^2 in CP_2 arises from a unique Zoll projective structure on the 2-sphere. Our method…
Zoll manifolds with entire Grauert tubes are proven to be standard complex projective spaces.
problem Characterizing Zoll manifolds with entire Grauert tubes.
method Isometric comparison to CPn with the canonical metric. result Zoll manifolds of type CPn with entire Grauert tubes are isometric to CPn. We prove that for Matveev and Shevchishin superintegrable system, with a linear and a cubic integral, the metrics defined on S^2 and on Tannery's orbifold T^2 are either Zoll or Tannery metrics.
New Zoll families of minimal spheres found in spheres and projective spaces.
problem Finding new Zoll families of minimal spheres in various spaces.
method Equivariant constructions using Nash-Moser-Hamilton implicit function theorem.
result First examples of metrics on real projective spaces with Zoll families of minimal projective hyperplanes.
Normal forms prove dynamical results for magnetic fields on surfaces.
problem Existence and rigidity of Zoll flows on surfaces.
method Proved normal forms for strong magnetic fields and used them to derive dynamical results.
result Flow cannot be Zoll unless specific conditions hold.
The question of whether or not the set of Zoll metrics on the 2-sphere is connected is still open. Here we show that a naive application of the Ricci flow is not sufficient to answer this problem.
The paper constructs metrics on spheres with families of minimal hypersurfaces.
problem Finding Riemannian metrics on spheres with specific families of minimal hypersurfaces.
method Used Nash-Moser Inverse Function Theorem in the tame maps setting.
result Generalized Guillemin's theorem for Zoll families of minimal hypersurfaces.
Study on magnetic systems on surfaces with closed orbits.
problem Characterizing and proving rigidity of Zoll magnetic systems.
method Characterization and proof of rigidity for magnetic systems on surfaces of positive genus.
result Only magnetic systems with constant curvature metrics and functions yield Zoll Hamiltonian flows.
We construct a counterexample to a conjectured inequality L<2D, relating the diameter D and the least length L of a nontrivial closed geodesic, for a Riemannian metric on the 2-sphere. The construction relies on Guillemin's theorem concerning the existence of Zoll surfaces integrating an arbitrary infinitesimal odd def…
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.
A theorem proving all geodesics on a sphere are simple and of same length.
problem Characterizing Zoll Riemannian metrics on a 2-sphere.
method Analyzing the simple length spectrum of a 2-sphere.
result All geodesics on a sphere are simple and of the same length.
The article finds non-trivial Zoll magnetic systems for surfaces of any genus.
problem Finding non-trivial Zoll magnetic systems for surfaces of any genus.
method Twistor theoretic approach, constructing holomorphic blow-down maps into ruled surfaces.
result Construction of non-trivial Zoll magnetic systems for surfaces of any genus.
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…
A Zoll metric is a Riemannian metric whose geodesics are all circles of equal length. Via the twistor correspondence of LeBrun and Mason, a Zoll metric on the 2 dimensional sphere corresponds to a family of holomorphic disks in CP_2 with boundary in a totally real submanifold P. In this paper, we show that for a fixed …
Almost Zoll affine surface found on cylinder.
problem Finding surfaces with special geodesic properties.
method Exhibited an affine structure on a cylinder.
result Affine structure on cylinder is almost Zoll.
Study proves certain Zoll manifolds with entire Grauert tubes are isometric.
problem Characterizing Zoll manifolds with entire Grauert tubes.
method Computed algebraic indices of tangent bundles; proved isometry.
result Any Zoll manifold of type HP^2 with entire Grauert tube is isometric to canonical HP^2.
New index characterizes non-smooth Zoll convex bodies.
problem Characterizing non-smooth Zoll convex bodies.
method Defining systolic S1-index and using it to introduce generalized Zoll convex bodies. result Generalized Zoll convex bodies coincide with classical ones under certain conditions.
Three explicit families of spacelike Zoll surface admitting a Killing field are provided. It allows to prove the existence of spacelike Zoll surface not smoothly conformal to a cover of de Sitter space as well as the existence of Lorentzian Möbius strips of non constant curvature all of whose spacelike geodesics are cl…
Optimal Strichartz estimates for Schrödinger on Zoll manifolds.
problem Optimal Strichartz estimates for solutions to the Schrödinger equation on Zoll manifolds.
method Arithmetic properties of the spectrum of the Laplacian and bilinear oscillatory integral estimates.
result Optimal Strichartz estimates for all q≥2 in Lt,xq spaces. Study on Schrödinger operators on Zoll manifolds, focusing on pseudo-spectra.
problem Analyzing the pseudo-spectra of Schrödinger operators on Zoll manifolds.
method Asymptotic analysis of pseudo-spectra and numerical range of non-self-adjoint Schrödinger operators.
result Obtained asymptotic results on the pseudo-spectra of Schrödinger operators.
Study rigidity of Hamiltonians near a minimum in symplectic and magnetic settings.
problem Rigidity of Hamiltonians near a minimum in symplectic and magnetic settings.
method Analyzing Hamiltonian systems near a compact symplectic Morse-Bott minimum, focusing on Zoll flows and magnetic forms.
result A constant curvature quantity characterizes complex space forms among Kähler manifolds.
Let M be a real analytic Riemannian manifold. An adapted complex structure on TM is a complex structure on a neighborhood of the zero section such that the leaves of the Riemann foliation are complex submanifolds. This structure is called entire if it may be extended to the whole of TM. We prove here that the only …
Constructs smooth integrable magnetic systems on a two-torus.
problem Creating smooth magnetic systems on a two-torus with specific properties.
method Uses Nash-Moser implicit function theorem to find zeros of an action functional.
result Characterizes Zoll magnetic systems and proves their existence.
In 3D, Zoll contact forms locally maximize systolic ratio.
problem Maximizing systolic ratio in 3D contact geometry.
method Proving Zoll forms locally maximize the systolic ratio in C3-topology. result Every Zoll form admits a C3-neighborhood where systolic ratio is locally maximized. The paper characterizes Besse and Zoll Reeb flows on specific manifolds.
problem Characterizing Besse and Zoll Reeb flows on different manifolds.
method Using spectral invariants and Ekeland-Hofer capacities.
result Characterizations of Besse and Zoll Reeb flows for specific manifolds.
Study of Lévy flights on Zoll surfaces, revealing geometric information.
problem Understanding the mean first capture time of Lévy flights on Zoll surfaces.
method Analysis of geodesic Lévy processes on Zoll surfaces, focusing on the first correction term.
result The first correction term encodes geometric information, specifically the degree of the conjugate point.
Symplectic capacities of domains near balls are well-defined, but not for all C1-close domains.
problem Understanding symplectic capacities of domains near balls and their stability.
method General theorem about contact forms close to Zoll ones, spectral invariants for contact forms.
result Existence of minimizing geodesics in the space of contact forms.
Characterizes complex Finsler metrics and their properties.
problem Characterize complex Finsler metrics and their geometric properties.
method Defined the canonical connection and investigated holomorphic sectional curvature tensors and Ricci curvatures.
result Characterizes balanced complex Finsler metrics and provides sufficient and necessary conditions.
Study Zoll manifolds with boundary, showing unique geodesic properties.
problem Characterize Zoll manifolds with boundary.
method Analyzing geodesics, boundary conditions, and manifold structure.
result All free boundary geodesics have the same length and Morse index.
Study on special Finsler metrics with conditions for Riemannian and isotropic properties.
problem Characterizing Finsler metrics with specific geometric properties.
method Analyzing conditions for Riemannian and isotropic properties of AR-Finsler metrics.
result Conditions for isotropic S-curvature and mean Landsberg curvature leading to vanishing curvature. Paper studies Landsberg curvature of a specific Finsler metric.
problem Analyzing Landsberg curvature of a particular Finsler metric.
method Derived Landsberg curvature and mean Landsberg curvature of the twisted product Finsler metric.
result Necessary and sufficient conditions for the metric to be Landsberg or weakly Landsberg are established.
Most Finsler metrics have infinite-dimensional holonomy groups.
problem Understanding the holonomy groups of Finsler metrics.
method Analyzing the set of Finsler metrics on a manifold.
result An open dense subset of Finsler metrics have infinite-dimensional holonomy groups.
Monochromatic Finsler metrics are shown to be generalized Berwald metrics.
problem Characterizing Finsler metrics with isomorphic tangent spaces.
method Demonstrating the existence of an affine connection preserving the Finsler function.
result Monochromatic Finsler metrics are generalized Berwald metrics.
Characterizes and simplifies m-th root Finsler metrics.
problem Understanding and simplifying m-th root Finsler metrics. method Characterization and reduction of m-th root Finsler metrics. result Every isotropic mean Berwald curvature m-th root Finsler metric reduces to a weakly Berwald metric. New Finsler metrics constructed from GDW-metrics.
problem Exploring new Finsler metrics within the GDW-metric class. method Constructing new sub-classes of GDW-metrics. result Presented illustrative examples of new Finsler metrics.
Classifies reversible Finsler metrics with positive curvature.
problem Classifying homogeneous reversible Finsler metrics with positive flag curvature.
method Classification based on G-invariant metrics and curvature properties.
result Exceptions exist where homogeneous Finsler metrics with positive flag curvature are not known.
Investigate AR-Finsler metrics for local dual flatness and projective flatness.
problem Locally dually and projectively flat AR-Finsler metrics
method Derive necessary and sufficient conditions and a compatibility relation.
result Establish a rigidity result for AR-Finsler metrics.
Study Ricci curvature of homogeneous Finsler spaces with specific metrics.
problem Curvature properties of homogeneous Finsler spaces with (α,β)-metrics. method Derived explicit formulae for Ricci curvature and found conditions for vanishing S-curvature. result Spaces with vanishing S-curvature and negative Ricci curvature are Riemannian. New definition of naturally reductive Finsler manifolds using geodesic graphs.
problem Defining naturally reductive Finsler manifolds using geodesic graphs.
method Proposed a new geometrical definition using geodesic graphs and constructed examples of Finsler metrics.
result Explicit examples of Finsler naturally reductive metrics constructed.
A theorem on Finsler metrics with special curvature properties.
problem Characterizing Finsler metrics with sectional flag curvature.
method Analyzing the relationship between flag curvature and sectional curvature.
result Locally, Finsler metrics are either Riemannian or have isotropic flag curvature.
New approach connects Finsler geometry's metric and connections.
problem Deriving Finsler geometry's metric and connections from compatibility axioms.
method Compatibility axioms between metric and Finsler connection.
result Metrical formulation of Finsler geometry for field theory.