The study shows that ergodic measures are not generic on non-positively curved manifolds.
problem Determining the genericity of ergodic measures on non-positively curved Riemannian manifolds.
method Investigates the existence of an open isometric embedding of a product manifold with a factor isometric to S1. result The closure of the set of ergodic measures does not encompass all invariant measures, indicating the failure of genericity.
We study unimodular measures on the space Md of all pointed Riemannian d-manifolds. Examples can be constructed from finite volume manifolds, from measured foliations with Riemannian leaves, and from invariant random subgroups of Lie groups. Unimodularity is preserved under weak* limits, and under certain…
Improved Kalman filter for Stiefel manifold measurements.
problem Improving accuracy in measurements on Stiefel manifolds.
method Generalization of extended Kalman filter for Stiefel manifold-valued measurements.
result Significant improvement over raw measurements.
Extends rigidity results to non-homogeneous manifolds.
problem Measure and topological rigidity in dynamical systems.
method From homogeneous to general manifolds.
result Measure and topological rigidity results extended.
Paper proves convergence of Kalman filter on Stiefel manifolds with measurement errors.
problem Filtering constant particle with measurement errors on Stiefel manifolds.
method Extended Kalman filter applied to Stiefel manifold-valued observations.
result Convergence of the extended Kalman filter proved for constant system process.
Measuring foliations at infinity of quasi-Fuchsian manifolds, proving filling pairs and showing realisation.
problem Understanding foliations at the boundary of quasi-Fuchsian manifolds.
method Proving filling pairs and showing realisation of measured foliations.
result Proved that measured foliations at infinity of quasi-Fuchsian manifolds are filling when close to being Fuchsian.
The paper connects geodesic flows and limit sets on visibility manifolds.
problem Understanding dynamics and ergodic properties on non-compact visibility manifolds.
method Analyzing geodesic flows and Patterson-Sullivan measures on visibility manifolds without conjugate points.
result The positivity of the Patterson-Sullivan measure of the Myrberg limit set is equivalent to the conservativity of the geodesic flow.
Locally homogeneous RCD spaces are shown to be smooth manifolds.
problem Understanding the structure of RCD spaces.
method Adapting existing results to new spaces.
result Locally homogeneous RCD spaces are isometric to smooth manifolds.
Measure contraction property is a synthetic Ricci curvature lower bound for metric measure spaces. We consider Sasakian manifolds with non-negative Tanaka-Webster Ricci curvature equipped with the metric measure space structure defined by the sub-Riemannian metric and the Popp measure. We show that these spaces satisfy…
New foliations at infinity for quasi-Fuchsian manifolds near the Fuchsian locus are uniquely determined.
problem Determining foliations at infinity for quasi-Fuchsian manifolds near the Fuchsian locus.
method Inspired by Bonahon's method, uses measured bending laminations on the boundary of convex cores.
result Measured foliations at infinity of quasi-Fuchsian manifolds can be uniquely realized for small t. Compact embeddings for invariant functions in metric-measure spaces.
problem Embedding functions with symmetry in metric-measure spaces.
method Analyzing H-invariant functions in compact metric-measure spaces, extending to Riemannian manifolds. result Obtained compact Sobolev embeddings for critical exponents.
The paper estimates variance of random sections on complex manifolds.
problem Estimating variance of random holomorphic sections on compact Kahler manifolds.
method Analyzes a sequence of smooth Hermitian holomorphic line bundles on a compact Kahler manifold X, considering specific probability measures.
result Provides variance estimates for various measures including Gaussian and Fubini-Study measures.
Geometrically proves majorizing measure theorem on Hadamard manifolds.
problem Volume size relation between random process index space and its convex hull.
method Assumed Hadamard manifold, derived upper bound for volume ratio, applied to prove majorizing measure theorem.
result Upper bound for volume ratio between index space and convex hull.
Study equilibrium measures on manifolds without conjugate points with visibility covering.
problem Uniqueness and properties of equilibrium measures on manifolds without conjugate points.
method Analysis of geodesic flows, study of equilibrium measures, ergodic properties, and pressure gap.
result Equilibrium measures satisfy a weak pressure gap under certain conditions.
Approximates measures on curved spaces using Dirac measures.
problem Topology of invariant measures on curved manifolds.
method Introducing weakly regular vectors and approximating measures by Dirac measures.
result Ergodicity is a generic property in the space of invariant measures supported on weakly regular vectors.
This article explores some properties of universal covers of compact Kahler manifolds, under the assumption of Caratheodory measure hyperbolicity. In particular, by comparing invariant volume forms, an inequality is established between the volume of canonical bundle of a compact Kahler manifolds and the Caratheodory me…
Study Finsler metric measure manifolds' concentration properties.
problem Understanding concentration properties in Finsler metric measure manifolds.
method Established relationships with observable diameter, isoperimetric inequalities, and first eigenvalue.
result Derived a Cheng type upper bound estimate for the first closed eigenvalue.
In this paper, we show that the Euler characteristic of an even dimensional closed projectively flat manifold is equal to the total measure which is induced from a probability Borel measure on RP^n invariant under the holonomy action, and then discuss its consequences and applications. As an application, we show that t…
Continuous solutions found for complex geometry equations.
problem Finding solutions to complex geometry equations on Hermitian manifolds.
method Proving existence of continuous quasi-plurisubharmonic solutions for specific measures.
result Existence of continuous quasi-plurisubharmonic solutions for measures dominated by capacity.
Extends Dirac operator results to foliations with invariant measures.
problem Positive scalar curvature on foliations with invariant measures.
method Relative measured index theorem for foliations with invariant transverse measures.
result Space of positive scalar curvature metrics has infinitely many path components.
Study KMS measures in Poisson geometry, focusing on b-Poisson manifolds.
problem Characterize KMS measures in Poisson geometry.
method Generalize symplectic results to b-Poisson manifolds. result Complete characterization of KMS measures on b-Poisson manifolds. Solves Calabi-Yau equation on symplectic manifolds using measurable Kahler metrics.
problem Solving the Calabi-Yau equation on symplectic manifolds.
method Global deformation of almost complex structures compatible with symplectic form, constructing measurable Lipschitz Kahler metric.
result Existence theorem for solutions to the one-form type Calabi-Yau equation on closed symplectic manifolds.
In this paper we consider non-compact non-flat simply connected harmonic manifolds. In particular, we show that the Martin boundary and Busemann boundary coincide for such manifolds. For any finite volume quotient we show that (up to scaling) there is a unique Patterson-Sullivan measure and this measure coincides with …
The tangent space is constructed in sub-Finsler geometry, leading to the failure of the CD condition in 3D-contact manifolds.
problem The failure of the CD condition in sub-Finsler geometry.
method Construction of the tangent space in the measured Gromov-Hausdorff sense, application of nilpotent approximation.
result The CD condition fails in 3D-contact sub-Finsler manifolds.
Study SRB measures for Anosov actions on manifolds.
problem Characterize SRB measures for Anosov actions.
method Use Ruelle-Taylor resonances and properties of Sinai-Ruelle-Bowen measures.
result SRB measures have properties like smooth disintegrations, positive basins, and are unique under certain conditions.
Study shows observability from a measurable set for Gevrey functions.
problem Determining observability from a subset for Gevrey functions.
method Used measurable sets and inequalities for Gevrey regular functions.
result Established observability estimates from measurable sets for Gevrey functions.
The study shows finite measure-preserving isometry groups for certain metric measure spaces.
problem Understanding the structure of isometry groups in metric measure spaces.
method Analyzing synthetic negative Ricci curvature and Bakry-Émery Ricci curvature.
result The measure-preserving isometry group is finite for compact metric measure spaces with specific curvature conditions.
Study geodesic flows on hyperbolic manifolds without conjugate points, proving unique measure of maximal entropy.
problem Proving uniqueness of measure of maximal entropy for geodesic flows on specific manifolds.
method Analyzing geodesic flows on closed Riemannian manifolds without conjugate points, using properties of Gromov hyperbolic and residually finite groups.
result Proves geodesic flow has a unique measure of maximal entropy under appropriate assumptions.
Wassmap reduces image complexity while preserving key features.
problem Global nonlinear dimensionality reduction in imaging.
method Wassmap uses Wasserstein space and pairwise distances to create isometric embeddings.
result Wassmap can recover parameters of image manifolds like translations and dilations.
Unified positive mass theorem and Dirac operator study on weighted manifolds.
problem Establishing a unified positive mass theorem for weighted manifolds and smooth metric measure spaces.
method Analyzing Dirac operators on warped product manifolds and applying results to the positive mass theorem.
result Equivalence of weighted positive mass theorem to usual positive mass theorem.
Recover manifold homology from a sample in Euclidean space.
problem Recovering the homology of an immersed manifold from a sample.
method Estimate tangent bundle, apply measure-based filtrations for persistent homology, use normal reach.
result Construction consistent and stable, does not require manifold dimension.
Inequalities linking entropy, Fisher info, Stein discrepancy, and Wasserstein distance on Riemannian manifolds.
problem Linking entropy, Fisher info, Stein discrepancy, and Wasserstein distance on Riemannian manifolds.
method Deriving inequalities linking these measures on Riemannian manifolds.
result Strengthening and extending existing inequalities to Riemannian manifolds.
Unified theory of measure-preserving diffusions on manifolds.
problem Deriving a complete recipe for measure-preserving diffusions on manifolds.
method Developed a geometric theory that unifies and generalizes previous constructions, relying on intrinsic geometry of the target measure.
result The completeness result is a direct consequence of manifold topology and target measure geometry.
The study defines divergence for multivector fields on infinite-dimensional manifolds.
problem Defining divergence for multivector fields on infinite-dimensional manifolds.
method Definition of divergence consistent with finite-dimensional geometry, properties transferred from finite to infinite dimensions.
result Natural properties of divergence are preserved in infinite dimensions.
Let (M,ω) be a Kähler manifold and let K be a compact group that acts on M in a Hamiltonian fashion. We study the action of KC on probability measures on M. First of all we identify an abstract setting for the momentum mapping and give numerical criteria for stability, semi-stability and polystabili…
Measuring wave sources uniquely identifies manifold properties.
problem Determining Riemannian manifold structure from wave observations.
method Semilinear wave equation measurements at a single point.
result Topological, differential, and geometric structure can be inferred.
We study the generic invariant probability measures for the geodesic flow on connected complete nonpositively curved manifolds. Under a mild technical assumption, we prove that ergodicity is a generic property in the set of probability measures defined on the unit tangent bundle of the manifold and supported by traject…
Paper introduces a new distance measure for Gaussian Mixture Models.
problem Developing a new distance measure for Gaussian Mixture Models.
method Embedding K-component Gaussian Mixture Models into the manifold of symmetric positive definite matrices and calculating a lower bound for the Fisher-Rao metric.
result Demonstrated effectiveness through experiments on standard datasets.
Proves the bending map is proper for hyperbolic 3-manifolds.
problem Properness of the bending map in hyperbolic 3-manifolds.
method Analyzes geometric properties and isotopy classes of homeomorphisms.
result Proving the bending map is proper for hyperbolic 3-manifolds.
In this paper we introduce a general notion of weak extension property for embeddings induced by a group actions. As an example, for the group H(M, m) of measure-preserving homeomorphisms of a noncompact manifold M, we deduce weak type extension theorems, and as an application we exhibit the local contractibility of th…
The paper provides precise estimates for isoperimetric inequalities on weighted manifolds.
problem Quantitative isoperimetric inequalities on weighted Riemannian manifolds.
method Analyzes L1, Lp, and W2 estimates for the push-forward of measures. result Close approximation of the guiding function's push-forward to Gaussian measure.
The paper studies random systems of holomorphic sections on compact Kähler manifolds and proves equidistribution results.
problem Estimating the distribution of zeros of random holomorphic sections on compact Kähler manifolds.
method Asymptotic variance estimate for smooth linear statistics, equidistribution result derivation.
result Smooth positive closed form ω^k can be approximated by currents of integration along analytic subsets of X.
Study foliations at infinity and constant mean curvature surfaces in quasi-Fuchsian manifolds.
problem Understanding foliations at infinity and constant mean curvature surfaces in quasi-Fuchsian manifolds.
method Using measured foliations and quasi-Fuchsian manifolds, proving the existence and uniqueness of foliations by constant mean curvature surfaces.
result For quasi-Fuchsian manifolds close to the Fuchsian locus, measured foliations at infinity can be uniquely realized and foliated by constant mean curvature surfaces.
We study a "div-grad type" sub-Laplacian with respect to a smooth measure and its associated heat semigroup on a compact equiregular sub-Riemannian manifold. We prove a short time asymptotic expansion of the heat trace up to any order. Our main result holds true for any smooth measure on the manifold, but it has a spec…
Proves bounded subsolution theorem for complex Monge-Ampère equation on compact Hermitian manifolds.
problem Complex Monge-Ampère equation with positive Radon measure on compact Hermitian manifolds.
method Proves bounded subsolution theorem.
result Establishes bounded subsolution theorem for complex Monge-Ampère equation.
In this short note we compare the weighted Laplacians on real and complex (Kähler) metric measure spaces. In the compact case Kähler metric measure spaces are considered on Fano manifolds for the study of Kähler-Einstein metrics while real metric measure spaces are considered with Bakry-Émery Ricci tensor. There are tw…
The paper improves quantization error estimates on Riemannian manifolds.
problem Improving quantization error estimates on Riemannian manifolds.
method Using covering growth estimates of spheres instead of curvature bounds.
result Provides a more general integral condition for quantization error.
Characterizing the appearance of real-world surfaces is a fundamental problem in multidimensional reflectometry, computer vision and computer graphics. For many applications, appearance is sufficiently well characterized by the bidirectional reflectance distribution function (BRDF). We treat BRDF measurements as sample…