The paper characterizes geometric infiniteness for convergence group actions using orbit uniform metrics.
problem Characterizing geometric infiniteness for convergence group actions.
method Introducing orbit uniform metrics and proving properties of discrete orbits.
result Characterization of geometric infiniteness in terms of uncountability of non-conical limit points and existence of escaping sequences.
The study explores maps of 2- and 3-uniform tilings on the torus.
problem Understanding the number of vertex orbits in quotient maps of 2- and 3-uniform tilings.
method Analyzing the quotient maps of 2- and 3-uniform tilings on the torus.
result Bounds on the number of vertex orbits in quotient maps of 2- and 3-uniform tilings.
Paper studies Kähler-Ricci flow convergence on Fano manifolds.
problem Uniform convergence of Kähler-Ricci flow on Fano manifolds.
method Analyzes flow behavior with varied initial metrics and complex structures.
result Proves uniqueness of Kähler-Ricci solitons in diffeomorphism orbits.
Groups can embed uniformly but not act properly on contractible manifolds.
problem Understanding the difference between group actions and embeddings on contractible manifolds.
method Analyzing the relationship between group actions and uniform embeddings on contractible manifolds.
result k-fold products of specific groups do not act on contractible manifolds.
Unified framework for smooth structures on coadjoint orbits.
problem Smooth structures on coadjoint orbits of diffeomorphism groups.
method Decorated and augmented nonlinear Grassmannians, functors, smooth structure.
result Uniform description of coadjoint orbits' smooth structures.
The study finds geodesic orbit metrics on Lie groups from flag manifolds.
problem Investigating geodesic orbit metrics on Lie groups.
method Using generalized flag manifolds to form metrics on simple Lie groups.
result All left-invariant geodesic orbit metrics on simple Lie groups are naturally reductive.
Study on geodesics of Finsler metrics derived from Riemannian metrics.
problem Investigating geodesics in Finsler metrics derived from Riemannian metrics.
method Proved geodesic lemma for a family of Riemannian geodesic orbit metrics, analyzed geodesic graphs.
result Derived Finsler metrics have geodesic orbit property and belong to a new class of metrics.
The study constructs families of nilpotent Lie groups with geodesic orbit metrics.
problem Finding geodesic orbit metrics on nilpotent Lie groups.
method Construction of continuous families of nilpotent Lie groups.
result Continuous families of non-isomorphic nilpotent Lie groups with geodesic orbit metrics.
Geodesic graphs for special Finsler metrics on spheres are studied.
problem Characterizing geodesic orbit Finsler metrics on spheres.
method Explicit constructions and group extensions.
result Not all projective spaces admit invariant Finsler metrics.
Geodesic orbit metrics proven on specific homogeneous spaces.
problem Geodesic orbit metrics on homogeneous spaces.
method Using strongly isotropy irreducible spaces and proving natural reductivity.
result Geodesic orbit metrics are naturally reductive on constructed homogeneous spaces.
Geodesic orbit metrics on real flag manifolds identified.
problem Classifying real flag manifolds with geodesic orbit metrics.
method Investigated invariant metrics on real flag manifolds, focusing on those where geodesics are orbits of one-parameter subgroups.
result Non-trivial geodesic orbit metrics exist on real flag manifolds, unlike in the complex case.
Study geodesic orbit metrics in quaternionic Stiefel manifolds.
problem Characterize geodesic orbit spaces in quaternionic Stiefel manifolds.
method Analyze homogeneous Riemannian spaces (M=G/H,g) with geodesics as orbits of subgroups. result Identify conditions for $(\Sp(n)/\Sp(n_1) imes \cdots imes \Sp(n_s), g)$ to be a geodesic orbit space.
New Einstein metric on G2 is found that's not geodesic orbit.
problem Finding left-invariant Einstein metrics on Lie groups that are not geodesic orbit.
method Developed tools for geodesic orbit Riemannian manifolds; used recent results by I. Chrysikos and Y. Sakane.
result Compact Lie group G2 admits a left-invariant Einstein metric that is not geodesic orbit. The paper classifies geodesic orbit spaces with simple isotropy groups.
problem Classifying geodesic orbit spaces with simple isotropy groups.
method Classifying G-naturally reductive and G-geodesic orbit metrics on M. result Classification of geodesic orbit spaces with simple isotropy groups.
Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
problem Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
method Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
result Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
Study finds homogeneous spaces with geodesic orbits but no integrable distributions.
problem Characterizing homogeneous spaces with specific geometric properties.
method Examined Lie groups with compact stabilizers and classified spaces based on geodesic orbits and integrable distributions.
result Identified homogeneous spaces with geodesic orbits but lacking integrable invariant distributions.
Study pseudo-Riemannian metrics on Jordan superalgebras.
problem No specific problem stated; focus on metrics.
method Coadjoint orbit-like construction for pseudo-Euclidean Jordan superalgebras.
result Investigated canonical pseudo-Riemannian metrics.
Identifying parallel sides of a collection of Euclidean polygons yields a flat surface with cone points of angles multiples of 2 pi, naturally a compact Riemann surface but also an algebraic curve, and a hyperbolic surface. In general two different metrics on a surface have no geodesic arcs in common, but in special ca…
Study geodesic orbit metrics on specific homogeneous spaces.
problem Characterize geodesic orbit spaces in a class of homogeneous bundles.
method Analyze geodesic orbit spaces of compact Lie groups with semisimple subgroups.
result Identify conditions for a metric to be geodesic orbit.
Gradient descent with geometrically adapted metrics drives L2 cost to global minimum at uniform rate.
problem Minimizing L2 cost in deep learning networks. method Adapting gradient descent to output layer metric in deep learning.
result Uniform exponential convergence to global minimum in L2 cost. The article constructs Spin(7) metrics with Aloff--Wallach spaces as orbits.
problem Creating Spin(7) metrics with specific geometric properties.
method Continuous 1-parameter families of non-compact Spin(7) metrics with chiralities, focusing on Aloff--Wallach spaces.
result Construction of Spin(7) metrics with Aloff--Wallach spaces as principal orbits, including geometric transitions.
Researchers classify geodesic orbit spaces for compact Lie groups of rank two.
problem Identifying geodesic orbit spaces for compact Lie groups of specific rank.
method Classification of simply connected geodesic orbit spaces where G is a compact Lie group of rank two.
result Only certain spheres and projective spaces, with metrics induced from Hopf fibrations, are geodesic orbit spaces for compact Lie groups of rank two.
Study on counting orbits and Poincaré series for specific hyperbolic metrics.
problem Counting orbits and analyzing Poincaré series for strongly hyperbolic metrics.
method Combining ergodic theory techniques with topological flows and symbolic dynamics.
result Obtained orbital counting results and described the domain of analyticity for Poincaré series.
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.
We classify all cohomogeneity one manifolds with principal orbit Q^111=SU(2)^3/U(1)^2 or M^110=(SU(3) x SU(2))/(SU(2) x U(1)) whose holonomy is contained in Spin(7). Various metrics with different kinds of singular orbits can be constructed by our methods. It turns out that the holonomy of our metrics is automatically …
The study finds non-uniform lattices with thin Hitchin representations in specific Lie groups.
problem Finding thin Hitchin representations in non-uniform lattices of Lie groups.
method Arithmetic methods to construct thin Hitchin representations.
result Infinitely many orbits of thin Hitchin representations in non-uniform lattices.
We investigate cohomogeneity-one metrics whose principal orbit is an Aloff-Wallach space SU(3)/U(1). In particular, we are interested in metrics whose holonomy is contained in Spin(7). Complete metrics of this kind which are not product metrics have exactly one singular orbit. We prove classification results for metric…
The paper studies geodesic orbit Finsler metrics on Euclidean spaces.
problem Understanding Finsler metrics on Euclidean spaces with specific geometric properties.
method Analyzing Finsler metrics on Euclidean spaces to show they are fiber bundles over symmetric spaces.
result Finsler metrics on Euclidean spaces are fiber bundles over symmetric spaces with specific geometric properties.
Constructs Einstein metrics on manifolds with specific orbits.
problem Finding Einstein metrics on manifolds with given orbits.
method Continuous families of metrics constructed using vector bundles and R4m+4. result Recovery of Spin(7) metrics A8 and B8. 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.
The paper finds new Spin(7) metrics with specific orbits.
problem Existence of Spin(7) metrics with specified orbits. method Construction of three continuous families of non-compact Spin(7) metrics. result Existence of asymptotically conical and locally conical metrics.
Survey on uniformization of metric surfaces, including fractal and topological manifolds.
problem Uniformization of metric surfaces homeomorphic to 2D topological manifolds.
method Various uniformization theorems, including quasisymmetric and quasiconformal approaches.
result Uniformization results for metric spheres and arbitrary metric surfaces.
Study shows infinite Hofer diameter for Lagrangian orbits in cotangent bundles.
problem Determining boundedness of spectral metric on Lagrangian orbit spaces.
method Utilized wrapped Floer cohomology to define spectral invariant and pseudo-metric.
result Proved infinite Hofer diameter for Lagrangian orbits in cotangent bundles.
Study Einstein Lie groups and geodesic orbit manifolds, finding some are not geodesic orbit.
problem Characterizing Einstein Lie groups and geodesic orbit manifolds.
method Characterizing GimesK-invariant geodesic orbit metrics on Lie groups G for regular subgroups K. result Extensive classes of compact simple Einstein Lie groups are not geodesic orbit manifolds.
Uniform K-stability ensures existence of special metrics on toric manifolds.
problem Existence of conformally Kähler, Einstein-Maxwell metrics on toric manifolds.
method Introducing uniform K-stability and showing its equivalence to properness of relative K-energy.
result Uniform K-stability is necessary and sufficient for the existence of f-extremal metrics on toric manifolds. Quasimorphisms and pseudo-Anosov flows
problem connections between quasimorphisms and pseudo-Anosov flows
method showing connections between quasimorphisms and pseudo-Anosov flows
result upper bounds on exponential growth rate of closed orbits in pseudo-Anosov flows
This paper classifies geodesic orbit metrics on compact Lie group G2.
problem Classifying geodesic orbit metrics on compact Lie groups.
method Using representation theory of Lie subgroups, specifically weakly regular subgroups.
result Left-invariant geodesic orbit metrics on compact Lie group G2 are classified. Uniform estimates for Kaehler metrics' diameters and volumes.
problem Estimating diameters and volumes of Kaehler metrics.
method Proving uniform diameter and volume estimates for a family of Kaehler metrics.
result Uniform estimates for diameters and volumes of Kaehler metrics.
Smooth orbit equivalence proves metric equivalence for geodesic flows.
problem Proving metric equivalence for geodesic flows under orbit equivalence.
method Proving metric equivalence for geodesic flows under orbit equivalence.
result Smooth orbit equivalence implies conformal equivalence of metrics.
The paper studies geodesic orbit properties in Finsler spaces.
problem Investigating geodesic orbit properties in homogeneous Finsler spaces.
method Introduced metric operator and defined standard homogeneous Finsler metrics.
result Classified homogeneous manifolds with specific geodesic orbit properties.
Study finds Kähler-Einstein metrics on two Pasquier varieties.
problem Existence of Kähler-Einstein metrics on specific varieties.
method Analyzes Pasquier's two-orbits varieties to find metrics.
result New example of K-unstable Fano manifold with Picard number one.
Study geodesics on SL(n,R) adjoint orbits.
problem Geodesics on adjoint orbits of SL(n,R). method Translate problem into tangent bundle of SO(n)-flag manifolds, use Sasaki metric and Lie Theory. result Explicit description of geodesics families in SL(2,R). Study geodesic orbit Lorentz nilmanifolds, proving structural properties.
problem Understanding the structure of geodesic orbit Lorentz nilmanifolds.
method Analyzing geodesic orbit Lorentz nilmanifolds with reductive decompositions.
result Proves properties of nilpotent subgroups and their nilpotency steps.
The paper finds dense subgroups in certain Lie groups.
problem Finding dense subgroups in Lie groups.
method Constructing dense surface subgroups in specific Lie groups.
result Uniform lattices contain infinitely many dense Hitchin representations.
Classifies geodesic orbit spaces with abelian isotropy subgroups.
problem Characterizing and classifying geodesic orbit spaces with specific isotropy subgroups.
method Simplified study of geodesic orbit metrics on G/S by reducing to submanifolds and generalized flag manifolds, using properties of root systems.
result Geodesic orbit spaces of the form (G/S,g) are naturally reductive.
Proves existence of elliptic Reeb orbit on real projective 3-space using ECH.
problem Existence of elliptic Reeb orbits on real projective 3-space.
method Use of ECH (Embedded Contact Homology) to find distinguished pseudoholomorphic curves.
result Existence of elliptic Reeb orbit proven for some contact forms on RP3. We examine the L2-topology of the gauge orbits over a closed Riemann surface. We prove a subtle local slice theorem based on the div-curl Lemma of harmonic analysis, and deduce local pathwise connectedness and local uniform quasiconvexity of the gauge orbits. Using these, we generalize compactness results for anti-s…
Study of geodesics on Riemannian stacks, measuring distances on orbit spaces.
problem Extending Riemannian geometry to singular spaces.
method Introduce stacky metrics and study stacky curves and geodesics on Riemannian stacks.
result Establish a stacky version of Hopf-Rinow Theorem.