The paper bounds eigenvalue multiplicities for hyperbolic surfaces using short geodesics.
problem Bounding the multiplicity of Laplacian eigenvalues for hyperbolic surfaces.
method Using the number of short closed geodesics and surface genus.
result Upper bounds on eigenvalue multiplicities, showing sublinear behavior under certain conditions.
Study on sphere widths and geodesic multiplicity.
problem Understanding the width of spheres and geodesic multiplicity.
method Computed sphere widths for k=1 to 8 and used min-max critical varifolds.
result Unstable geodesics can arise with multiplicity.
Study on geodesics proving index and intersection bounds, with examples of multiplicity.
problem Understanding the index and intersections of min-max geodesics on surfaces.
method Proof of tangent cone structure, construction of metrics with multiplicity.
result Upper bounds on index and intersections, examples of multiplicity.
Only finitely many rational multiples of π are angles between geodesics on hyperbolic surfaces.
problem Characterizing angles between geodesics on hyperbolic surfaces.
method Analyzing the set of angles between closed geodesics on hyperbolic surfaces of finite type.
result There are only finitely many rational multiples of π in the set of angles between geodesics.
Partial regularity theorem for geodesic networks in R^n.
problem Proving regularity for geodesic networks in R^n.
method Partial ε-regularity theorem for sequences of integer multiplicity stationary geodesic networks converging to a multiplicity k line. result Proves partial regularity for geodesic networks.
The paper proves the existence of multiple closed geodesics on certain Finsler manifolds.
problem Proving the existence of multiple closed geodesics on positively curved Finsler manifolds.
method Analyzing Finsler manifolds with specific curvature conditions to prove the existence of closed geodesics.
result On Finsler manifolds with certain curvature conditions, there exist multiple closed geodesics.
Study on geodesics in spacetime, proving properties of multiple maximizing paths.
problem Characterizing multiple maximizing geodesics in globally hyperbolic spacetimes.
method Analyzing topological properties of the locus of multiple maximizing geodesics.
result The set of multiple maximizing geodesics is locally contractible.
The paper finds at least N orthogonal Finsler geodesic chords in a disk-like manifold.
problem Existence and multiplicity of orthogonal Finsler geodesic chords in a disk-like manifold.
method Study of Finsler geodesic chords under reversibility assumption.
result At least N orthogonal Finsler geodesic chords found in a disk-like manifold.
A bound on surface area in Riemannian manifolds without totally geodesic surfaces.
problem Bounding the area of surfaces in Riemannian manifolds without totally geodesic surfaces.
method Using extrinsic curvature energy to bound surface area.
result The area of any complete surface immersed into M is bounded by a multiple of its extrinsic curvature energy. We prove the existence of multiple closed geodesics on non-compact cylindrica manifolds.
Study of multiplicative connections in Lie groupoids.
problem Defining and understanding multiplicative connections in Lie groupoids.
method Definition and study of multiplicative connections satisfying compatibility with the groupoid structure.
result Identification of the obstruction to the existence of a multiplicative connection.
Study geodesics entering a fixed cusp neighborhood multiple times.
problem Understanding geodesics entering a specific cusp neighborhood multiple times.
method Investigate reciprocal geodesics entering a fixed cusp neighborhood a fixed number of times.
result Characterized the class of reciprocal geodesics entering a fixed cusp neighborhood a fixed number of times.
We prove a strong multiplicity one theorem for the length spectrum of compact even dimensional hyperbolic spaces i.e. if all but finitely many closed geodesics for two compact even dimensional hyperbolic spaces have the same length, then all closed geodesics have the same length.
If all prime closed geodesics on (Sn,F) with an irreversible Finsler metric F are irrationally elliptic, there exist either exactly 2[2n+1] or infinitely many distinct closed geodesics. As an application, we show the existence of three distinct closed geodesics on bumpy Finsler (S3,F) if a…
We prove that for every $\Q$-homological Finsler 3-sphere (M,F) with a bumpy and irreversible metric F, either there exist two non-hyperbolic prime closed geodesics, or there exist at least three prime closed geodesics.
The paper proves unique geodesics on hyperbolic surfaces and finds lower bounds.
problem Characterizing geodesics on hyperbolic surfaces.
method One-parameter Allen-Cahn min-max constructions.
result Every geodesic occurs with multiplicity one and provides uniform sharp lower bounds.
Study geodesic string counts on Riemann-Finsler manifolds, linking to KAM theory.
problem Counting geodesic strings on Riemann-Finsler manifolds.
method Using Fuller index and KAM theory, derive product formula for counts.
result Arithmetic constraints on geodesic string counts and existence of negative curvature metrics.
Given two points of a Generalized Robertson-Walker spacetime, the existence, multiplicity and causal character of geodesic connecting them is characterized. Conjugate points of such geodesics are related to conjugate points of geodesics on the fiber, and Morse-type relations are obtained. Applications to bidimensional …
The paper finds lower bounds for volumes of complex geometric structures.
problem Estimating the volume of complex geometric structures.
method Reduction to a counting problem in the unit tangent bundle, solved using exponential multiple mixing for the geodesic flow.
result First known lower bound for the volume of these manifolds in terms of curve length.
The study reveals conditions for infinite closed geodesics on specific surfaces.
problem Conditions for infinite closed geodesics on complete surfaces.
method Analyzes geometric and homological properties of closed geodesics on cylinders and planes.
result Proves that complete cylinders with isolated geodesics have zero, one, or infinitely many homologically visible geodesics.
The paper finds geodesics on specific Finsler spheres with unique properties.
problem Identifying geodesics on Finsler spheres with given curvature constraints.
method Analyzes Finsler 4-spheres with specific curvature conditions to determine geodesic properties. result Proves existence of at least four prime closed geodesics under certain conditions.
The study finds the number of closed geodesics on a specific type of manifold.
problem Determining the number of closed geodesics on a manifold with elliptic prime geodesics.
method Analyzes a compact manifold with a specific cohomology structure and a bumpy Finsler metric.
result There are either exactly 2dn(n+1) or (d+1) distinct closed geodesics, or infinitely many. In this paper, we prove the existence of at least two distinct closed geodesics on every compact simply connected irreversible or reversible Finsler (including Riemannian) manifold of dimension not less than 2.
The paper proves a conjecture about the minimum number of closed geodesics on a Finsler 3-sphere.
problem Proving the Anosov conjecture for bumpy Finsler 3-spheres.
method Analyzing the Morse index of prime closed geodesics.
result Established the conjecture for Finsler 3-spheres with nonzero Morse index.
In this paper we prove that for every bumpy Finsler metric F on every rationally homological n-dimensional sphere Sn with n≥2, there exist always at least two distinct prime closed geodesics.
Study on shortest geodesics crossing multiple times on hyperbolic surfaces with cusps.
problem Finding shortest geodesics crossing multiple times on hyperbolic surfaces.
method Investigates closed geodesics on hyperbolic surfaces with at least one cusp, focusing on minimal length and self-intersection numbers.
result For large enough k, self-intersection numbers are exactly k for geodesics crossing at least k times. Paper introduces MPPGA for integrating multiple PGA models on Riemannian manifolds.
problem Challenges in dimensionality reduction on Riemannian manifolds with multiple modalities.
method Develops a mixture probabilistic principal geodesic analysis (MPPGA) model.
result Demonstrates improved clustering and shape analysis using MPPGA.
In this paper we give a proof of the existence of an orthogonal geodesic chord on a Riemannian manifold homeomorphic to a closed disk and with concave boundary. This kind of study is motivated by the link of the multiplicity problem with the famous Seifert conjecture (formulated in 1948) about multiple brake orbits for…
The paper finds minimal-area metrics on polygons with multiple crossing geodesics.
problem Finding minimal-area metrics on polygons with specific geodesic conditions.
method Applied convex programs to polygons with length conditions on curves.
result The extremal metric coincides with the conformal extremal metric on regular polygons.
Stable nets on convex hypersurfaces maintain their shape under small perturbations.
problem Maintaining the shape of nets on convex surfaces under slight changes.
method Constructing stable geodesic nets on convex hypersurfaces.
result Stable geodesic nets on convex hypersurfaces do not change shape under small perturbations.
We prove that on any closed Riemannian manifold (M1×M2,g), with $\rank\Hom_1(M_1)\neq0$ and dim(M2)≥2, every isometry homotopic to the identity admits infinitely many isometry-invariant geodesics.
The paper proves conditions for the existence of multiple non-contractible closed geodesics on Finsler projective spaces.
problem Existence of non-contractible closed geodesics on Finsler projective spaces.
method Fadell-Rabinowitz index theory and equivariant Poincaré series.
result Proves conditions for the existence of at least n−1 non-contractible closed geodesics. Study proves existence of multiple geodesics in a specific metric space.
problem Existence of multiple geodesics in a manifold with a Randers-Kropina metric.
method Lusternik-Schnirelman theory applied to a homotopy type of solutions of an affine control system.
result Proves existence of infinitely many geodesics between two points in a non-contractible manifold.
Using geodesic length functions, we define a natural family of real codimension 1 subvarieties of Teichmüller space, namely the subsets where the lengths of two distinct simple closed geodesics are of equal length. We investigate the point set topology of the union of all such hypersurfaces using elementary methods. Fi…
The paper finds infinitely many magnetic geodesics on non-compact manifolds.
problem Existence and multiplicity of periodic orbits of magnetic flows.
method Morse theory applied to non-compact manifolds with energy levels above the Mañé critical value.
result Infinitely many noncontractible closed magnetic geodesics found.
Paper proves existence of at least two non-contractible geodesics on Finsler space forms.
problem Existence of non-contractible closed geodesics on Finsler compact space forms.
method Established resonance identity and proved existence of geodesics.
result Existence of at least two non-contractible closed geodesics on Finsler compact space forms.
Study proves existence of closed geodesics on spheres and projective spaces.
problem Proving the existence of closed geodesics on Finsler metrics.
method Topological methods and Lusternik-Schnirelmann-type approach, using spherical complexities.
result Existence of multiple closed geodesics and upper bounds on their lengths.
In this paper, we construct families of nonisometric hyperbolic orbifolds that contain the same isometry classes of nonflat totally geodesic subspaces. The main tool is a variant of the well-known Sunada method for constructing length-isospectral Riemannian manifolds that handles totally geodesic submanifolds of multip…
Rich geometry on disk with special geodesics and integer spectrum.
problem Developing a unique geometric structure on the disk.
method Introducing a Riemannian structure with specific properties.
result Eigenfunctions are orthogonal polynomials with integer spectrum.
A general class of Lorentzian metrics, M0xR2, ds2=<.,.>+2dudv+H(x,u)du2, with (M0,<.,.> any Riemannian manifold, is introduced in order to generalize classical exact plane fronted waves. Here, we start a systematic study of their main geodesic properties: geodesic completeness, geodesic connected…
We give existence and nonuniqueness results for simple planar curves with prescribed geodesic curvature.
In this paper, we prove that for every Finsler n-dimensional sphere (Sn,F) with reversibility $\lm$ and flag curvature K satisfying $\left(\frac{\lm}{1+\lm}\right)^2<K\le 1$, either there exist infinitely many closed geodesics, or there exist at least two elliptic closed geodesics and each linearized Poincaré …
Surveying recent results on the geometry of Jacobian loci.
problem Understanding the extrinsic geometry of Jacobian loci.
method Analyzing the Torelli map as a multiplication map and studying totally geodesic subvarieties.
result Relation between totally geodesic subvarieties and Hodge loci.
The study of solutions with fixed energy of certain classes of Lagrangian (or Hamiltonian) systems is reduced, via the classical Maupertuis--Jacobi variational principle, to the study of geodesics in Riemannian manifolds. We are interested in investigating the problem of existence of brake orbits and homoclinic orbits,…
Proves min-max theory for constant geodesic curvature curves on closed surfaces.
problem Prescribing mean curvature on surfaces with constant geodesic curvature.
method Min-max theory applied to classify blowups and ensure almost embedded solutions.
result Produces a solution with constant geodesic curvature c on closed surfaces. This paper calculates interaction strength for translation surfaces with multiple singularities.
problem Computing the interaction strength of translation surfaces with multiple singularities is challenging.
method The authors study interaction strength of specific families of translation surfaces, including regular polygons and Bouw-Möller surfaces.
result The paper provides exact computations of KVol on translation surfaces with multiple singularities.
In the recent paper \cite{LoD1}, we classified closed geodesics on Finsler manifolds into rational and irrational two families, and gave a complete understanding on the index growth properties of iterates of rational closed geodesics. This study yields that a rational closed geodesic can not be the only closed geodesic…
Eigenvalues of 1-form Laplacian on hyperbolic manifolds relate to geodesic cycles.
problem Understanding the geometry of small eigenvalues on hyperbolic manifolds.
method Relating eigenvalues to cycle complexity and geodesics.
result Small eigenvalues correspond to closed geodesics with low genus surfaces.