Paper determines Assouad-Nagata dimension for all minor-closed metrics.
problem Understanding the Assouad-Nagata dimension of minor-closed metrics.
method Using edge-weighted graphs and edge-deletion/contraction to model minor-closed metrics, determining their Assouad-Nagata dimension.
result Determined the Assouad-Nagata dimension for every minor-closed metric.
The article classifies G2-structures with conformally flat metrics.
problem Identifying G2-structures with specific geometric properties.
method Classifying closed G2-structures with conformally flat metrics.
result Any closed G2-structure with conformally flat metric is locally equivalent to one of three explicit examples.
Classifies Einstein metrics on 4-manifolds with specific symmetry groups.
problem Classifying Einstein metrics on 4-manifolds with certain symmetry properties.
method Analyzes cohomogeneity-one Einstein metrics and uses symmetry properties.
result Locally symmetric or homothetic to the Page metric on CP2♯CP2. Paper shows regions close to negatively curved metrics are minimal fillings and rigid.
problem Boundary rigidity and minimality of metrics near negatively curved ones.
method Generalizes previous work on filling volume minimality and boundary rigidity for almost hyperbolic metrics.
result Regions with metrics close to a negatively curved symmetric metric are strict minimal fillings and boundary rigid.
New metrics found with specific curvature properties on 4D manifolds.
problem Constructing metrics with specific curvature properties on closed manifolds.
method Using Aubin's deformation method to find metrics with pinched Bach tensor and scalar curvature.
result Existence of metrics with Bach tensor pinched by scalar curvature on 4D manifolds.
We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces and prove the nonexistence of quadratically-superintegrable metrics of nonconstant curvature on closed surfaces
Paper finds conditions for two geodesics on complex manifolds.
problem Existence of two distinct closed geodesics on manifolds with infinite fundamental group.
method Topological and metric conditions for existence of geodesics in Riemannian and Finsler metrics.
result Generic Finsler metrics have two distinct closed geodesics.
We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces, prove the nonexistence of quadratically-superintegrable metrics of nonconstant curvature on closed surfaces, and prove t…
Researchers prove existence of metrics maximizing Laplace eigenvalue on all closed surfaces.
problem Proving the existence of metrics maximizing the first Laplace eigenvalue on closed surfaces.
method By contradiction and refinement of techniques, proving strict monotonicity under surface modifications.
result Existence of metrics maximizing the area-normalized first eigenvalue on all closed surfaces.
Paper proves flat metrics from holomorphic quadratic differentials can be identified by length spectrum.
problem Identifying flat metrics from holomorphic quadratic differentials.
method Proved using length spectrum on closed oriented surfaces.
result Flat metrics from holomorphic quadratic differentials can be distinguished by their length spectrum.
In every conformal class of Finsler (or Riemannian) metrics on a closed manifold there exists a residual subset of Finsler metrics, such that, with respect to the residual Finsler metrics, in any non-trivial homotopy class of free loops there is precisely one shortest geodesic loop.
Study on flat metrics on 3D and 4D manifolds, focusing on topology and algebra.
problem Topology and algebraic structure of flat metrics on manifolds.
method Algebraic and topological descriptions of moduli spaces.
result Algebraic description and topology of moduli spaces for 4D manifolds with a single holonomy generator.
Generalizes Thurston's asymmetric metric to flat metrics.
problem Defining an asymmetric metric on flat metrics.
method Defined an asymmetric metric on the space of unit-area flat metrics.
result Discussed two different topologies from the asymmetry.
In this paper we prove that the space of flat metrics (nonpositively curved Euclidean cone metrics) on a closed, oriented surface is marked length spectrally rigid. In other words, two flat metrics assigning the same lengths to all closed curves differ by an isometry isotopic to the identity. The novel proof suggests a…
Study describes flat metric moduli spaces on 4D manifolds.
problem Understanding flat metrics on 4D closed manifolds.
method Algebraic and topological description of moduli spaces.
result Algebraic and topological description of moduli spaces of flat metrics.
Paper proves rigidity theorems for geodesically reversible Finsler metrics.
problem Understanding geodesically reversible Finsler metrics in closed manifolds.
method Applied theory of volumes and areas on Finsler spaces to establish rigidity theorems.
result Partial explanation of the scarcity of geodesically reversible Finsler metrics in closed manifolds.
We construct non-trivial continuous isospectral deformations of Riemannian metrics on the ball and on the sphere in Rn for every n≥9. The metrics on the sphere can be chosen arbitrarily close to the round metric; in particular, they can be chosen to be positively curved. The metrics on the ball are both Diric…
Topological entropy decreases strictly along Ricci flow near hyperbolic metrics.
problem Understanding entropy changes in flows near hyperbolic metrics.
method Analysis of geodesic flow on Riemannian manifolds with variable negative curvature.
result Topological entropy strictly decreases along normalized Ricci flow near hyperbolic metrics.
Homoclinic orbits found in geodesic flows on surfaces.
problem Existence of homoclinic orbits in geodesic flows.
method Kupka-Smale metric on closed surfaces.
result Homoclinic orbits for all hyperbolic geodesics.
We show that every closed symplectic four-dimensional manifold admits compatible almost Kaehler metrics of negative scalar curvature.
The study proves the existence of many geodesics on complex manifolds.
problem Existence of closed geodesics on manifolds with non-trivial first Betti number.
method Combining Mañé's theorem with a new theorem about minimal geodesics and transverse homoclinic points.
result Proves the existence of infinitely many closed geodesics of arbitrary large length on manifolds with non-trivial first Betti number.
We define enlargeable length-structures on closed topological manifolds and then show that the connected sum of a closed n-manifold with an enlargeable Riemannian length-structure with an arbitrary closed smooth manifold carries no Riemannian metrics with positive scalar curvature. We show that closed smooth manifold…
Estimates gradients of solutions on closed surfaces.
problem Gradient estimates for solutions on closed surfaces.
method Considered a new metric g′=e2ug with bounded integral curvature, derived gradient estimates for g′, and used these to obtain gradient estimates for u. result Gradient estimates for solutions on closed surfaces are established.
Unique conformal metrics found on certain manifolds.
problem Finding unique conformal metrics with constant Q-curvature.
method Proving uniqueness on manifolds with positive scalar curvature.
result Only metrics of the form λg with λ>0 are constant Q-curvature.
Study on geodesics in Kropina metrics with applications.
problem Existence of connecting and closed geodesics in Kropina metrics.
method Analytical proofs and applications to null geodesics and navigation problems.
result Proves existence of geodesics in Kropina metrics.
Concrete example of 2-sphere with constant curvature 1 and closed geodesics.
problem Constructing a 2-sphere with constant curvature 1 and closed geodesics.
method Constructing a concrete example of a Finsler metric on the 2-sphere.
result A 2-sphere with constant Gauss curvature 1 and all geodesics closed.
In this paper, we study general (α,β)-metrics which α is a Riemannian metric and β is an one-form. We have proven that every weak Landsberg general (α,β)-metric is a Berwald metric, where β is a closed and conformal one-form. This show that there exist no generalized unicorn metric in this class of general $(…
We prove that for each closed smooth spin 4-manifold M there exists a closed smooth 4-manifold N such that the connected sum M # N admits a conformally flat Riemannian metric.
Let Σg be a closed hyperbolic surface of genus g and let Ham(Σg) be the group of Hamiltonian diffeomorphisms of Σg. The most natural word metric on this group is the autonomous metric. It has many interesting properties, most important of which is the bi-invariance of this metric. In this work we show that $…
Paper examines stability of minimizing metrics on manifolds with boundary.
problem Stability of minimizing Yamabe metrics on compact manifolds with boundary.
method Investigates stability in the sense introduced by Escobar, showing closeness of nearly minimizing metrics.
result Quantitative closeness of nearly minimizing metrics to minimizing Yamabe metrics within their conformal class.
We prove that the L^2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold induces a metric space structure. As the L^2 metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove several results on the exponentia…
We prove that a riemannian metric on the 2-sphere or the projective plane can be C2-approximated by a smooth metric whose geodesic flow has an elliptic closed geodesic.
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.
The study improves bounds on the number of closed geodesics and logarithmic improvements in the Weyl law.
problem Estimating the number of closed geodesics and improving logarithmic bounds in the Weyl law.
method Study of non-degeneracy properties of nearly closed orbits for predominant sets of metrics.
result Logarithmic improvements in the Weyl law and exponential bounds on the number of closed geodesics.
Quantitative stability for nearly minimizing Yamabe metrics.
problem Understanding the stability of nearly minimizing metrics in Riemannian geometry.
method Proving quantitative closeness of nearly minimizing metrics to minimizing metrics in a specific sense.
result The distance between nearly minimizing metrics and minimizing metrics is controlled quadratically by the Yamabe energy deficit.
Optimizes sharp curvature inequality on spheres, proving near-minimizers are close to standard metric.
problem Optimizing total σ2-curvature on spheres with positive scalar curvature. method Analyzes metrics conformal to the standard sphere, uses Sobolev norms to measure closeness.
result Near-minimizers of total σ2-curvature are almost the standard metric (up to Möbius transformations). The existence of two geometrically distinct closed geodesics on an n-dimensional sphere Sn with a non-reversible and bumpy Finsler metric was shown independently by Duan--Long [7] and the author [27]. We simplify the proof of this statement by the following observation: If for some N∈N all closed ge…
Study on metrics maximizing eigenvalues of Paneitz operator on 4-manifolds.
problem Investigating metrics maximizing eigenvalues of Paneitz operator.
method Critical points of eigenvalues of Paneitz operator on Riemannian metrics with fixed volume.
result Critical metrics associated with extrinsic conformal-harmonic maps into round spheres.
For almost all Riemannian metrics (in the C∞ Baire sense) on a closed manifold Mn+1, 3≤(n+1)≤7, we prove that the union of all closed, smooth, embedded minimal hypersurfaces is dense. This implies there are infinitely many minimal hypersurfaces thus proving a conjecture of Yau (1982) for generic …
We investigate the geometry of word metrics on fundamental groups of manifolds associated with the generating sets consisting of elements represented by closed geodesics. We ask whether the diameter of such a metric is finite or infinite. The first answer we interpret as an abundance of closed geodesics, while the seco…
New theorem shows metrics of certain groups are close if their lengths are identical.
problem Identifying metrics of relatively hyperbolic groups from their lengths.
method Proved rigidity for relatively hyperbolic groups using coarse marked length spectrum.
result Metrics of relatively hyperbolic groups are uniformly close if lengths are identical.
We show that any closed biquotient with finite fundamental group admits metrics of positive Ricci curvature. Also, let M be a closed manifold on which a compact Lie group G acts with cohomogeneity one, and let L be a closed subgroup of G which acts freely on M. We show that the quotient N := M/L carries metrics of nonn…
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. We prove that compact Kähler manifolds whose sectional curvatures are close to 1/4-pinched have ratios of Chern numbers close to the corresponding ratios of a complex hyperbolic space form. We deduce that the Mostow-Siu surfaces (and their three-dimensional analogues constructed by the first author) do not admit Kähler…
Unique extremal Kähler metric found near a divisor.
problem Uniqueness of extremal Kähler metric near a smooth divisor.
method Analyzes Poincaré type extremal Kähler metric with cusp singularity.
result Uniqueness of extremal Kähler metric up to holomorphic transformations.
We prove that any metric with curvature ≤−1 (in the sense of A. D. Alexandrov) on a closed surface of genus >1 is isometric to the induced intrinsic metric on a space-like convex surface in a Lorentzian manifold of dimension (2+1) with sectional curvature −1. The proof is done by approximation, using a resu…
Constructs a function to count closed geodesics on Riemannian manifolds.
problem Counting closed geodesics on Riemannian manifolds.
method Defines a locally constant geodesic count function and investigates the weight of compact open subsets of closed geodesics.
result Constructs a function to count closed geodesics on Riemannian manifolds.
In this paper, we establish first the resonance identity for non-contractible homologically visible prime closed geodesics on Finsler n-dimensional real projective space (RPn,F) when there exist only finitely many distinct non-contractible closed geodesics on (RPn,F), where the integer $n\geq2…