Handlebody groups are rigid under measure equivalence.
problem Proving handlebody groups are rigid under measure equivalence.
method Proving superrigidity for measure equivalence of handlebody groups.
result Every countable group measure equivalent to handlebody groups is virtually isomorphic to them.
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.
The paper introduces Patterson-Sullivan systems and proves their rigidity, with applications to random walks and entropy rigidity.
problem Understanding the rigidity of Patterson-Sullivan systems and their applications.
method Generalization of Tukia's measurable boundary rigidity theorem for Patterson-Sullivan systems.
result Entropy rigidity for Anosov groups with Lipschitz limit sets.
Study measures rigidity for random walks and flows via generalized u-Gibbs states.
problem Measure rigidity for stationary measures of random walks and flows.
method Factorization method applied to generalized u-Gibbs states.
result Established extra invariance of generalized u-Gibbs states.
Study rigidity of PSU(1,1) actions on circle via harmonic measures.
problem Rigidity properties of surface group actions on the circle.
method Foliated harmonic measures and curvature estimates.
result Curvature estimate and Gauss--Bonnet formula for S1 connection. Paper introduces new Lp q-torsional measure and solves Minkowski problem.
problem Solving the Minkowski problem for q-torsional rigidity. method Established Lp variational formula and proved existence of solutions. result Existence of solutions to Lp Minkowski problem for specific measures. Maximal representations show strong entropy rigidity.
problem Entropy rigidity for maximal representations.
method Measurable hypertransversality, Gromov product, Bowen-Margulis-Sullivan measure.
result Strong entropy rigidity proved for maximal representations.
Study Busemann spaces with measures under MCP, proving rigidity and structure theorems.
problem Understanding the structure of Busemann spaces with measures.
method Analyzing geodesic completeness and non-collapse assumptions.
result Rigidity and structure theorems for Busemann spaces with MCP.
New method solves generalized Minkowski problem for torsional rigidity.
problem Generalized Minkowski problem for torsional rigidity.
method Flow method
result Existence of solutions for general measures.
Study harmonic measures and rigidity in Seifert 3-manifolds using S1-connections.
problem Rigidity of foliations on Seifert 3-manifolds with maximal Euler number.
method Using S1-connections and harmonic measures, proving the Gauss--Bonnet formula and rigidity results. result A harmonic measure on the suspension bundle of the action with maximal Euler number has rigidity, closely related to the Poisson kernel.
Study graph products of groups, classifying them up to measure equivalence and rigidity.
problem Classifying graph products of groups up to measure equivalence and rigidity.
method Measure-theoretic and structural properties of von Neumann algebras, rigidity theorems.
result Quantified measure equivalence classification and rigidity theorems for graph products.
The paper discusses rigidity results for inequalities on weighted Riemannian manifolds.
problem Rigidity of inequalities on weighted Riemannian manifolds.
method Theorems of rigidity on curvature and measure for the Borell-Brascamp-Lieb inequality, generalizing a theorem by Balogh and Kristály.
result A generalization of the curvature rigidity theorem to the weighted setting.
We prove Obata's rigidity theorem for metric measure spaces that satisfy a Riemannian curvature-dimension condition. Additionally, we show that a lower bound K for the generalized Hessian of a sufficiently regular function u holds if and only if u is K-convex. A corollary is also a rigidity result for higher or…
We establish orbit equivalence rigidity for any ergodic, essentially free and measure-preserving action on a standard Borel space with a finite positive measure of the mapping class group for a compact orientable surface with higher complexity. We prove similar rigidity results for a finite direct product of mapping cl…
The paper proves rigidity and ergodicity of horospherical foliations.
problem Rigidity and ergodicity of horospherical foliations in higher rank.
method Establishes higher rank extensions of rigidity theorems for representations of discrete subgroups of divergence type, using conformal measures and boundary maps.
result Proves conformal measure rigidity and ergodicity of horospherical foliations for hypertransverse subgroups.
Study Poisson boundaries of building lattices and generalize rigidity results.
problem Understanding Poisson boundaries of building lattices and their rigidity properties.
method Proved Poisson boundaries and used them to generalize rigidity results.
result Generalized rigidity results for morphisms and cocycles from lattices in buildings to groups with negative curvature.
This paper solves the dual Minkowski problem for q-torsional rigidity.
problem The dual Minkowski problem for q-torsional rigidity.
method Introduced the p-th dual q-torsional measure and solved the p-th dual Minkowski problem for q-torsional rigidity using a Gauss curvature flow.
result Existence of smooth even and non-even solutions to the p-th dual Minkowski problem for q-torsional rigidity.
Researchers prove rigidity for log-Sobolev inequality on specific metric spaces.
problem Proving rigidity for the logarithmic Sobolev inequality on metric measure spaces.
method Using a new approach to prove the rigidity result.
result Proved that if equality holds in the log-Sobolev inequality, the space must split into a product of a manifold and the Gaussian shrinking soliton.
Artin groups of hyperbolic type are boundary amenable and have rigid properties.
problem Characterizing rigidity and measure equivalence properties of Artin groups.
method Analyzing boundary amenability, measure equivalence, and fixed set graphs.
result Measure equivalent Artin groups of hyperbolic type have isomorphic fixed set graphs.
The article discusses invariant measures outside homogeneous dynamics.
problem Understanding invariant measures in non-homogeneous settings.
method Comparing two recent results on rigidity and invariant measures.
result The importance of invariant and partially invariant measures in rigidity questions.
Study automorphism groups of Artin groups, proving rigidity and classification results.
problem Understanding the structure and automorphisms of Artin groups.
method Computed automorphism groups of intersection graphs, deduced rigidity and classification results.
result Computation of outer automorphism groups and other rigidity properties.
We survey some results on travel time tomography. The question is whether we can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as geometry problems, the boundary rigidity problem and the lens rigidity problem. The boundary r…
Paper solves Minkowski problem for k-torsional rigidity.
problem Solving Minkowski problem for k-torsional rigidity.
method Constructing Hadamard variational formula, proposing k-torsional measure, using curvature flow method.
result Existence of smooth non-even solutions to the Minkowski problem.
Study shows how to detect representation extendability using conformal measures.
problem Detecting extendability of representations using conformal measures.
method Using higher rank conformal measures and self-joinings of groups.
result Affirmative answer to detect extendability of representations.
We prove that a metric measure space equipped with a Dirichlet form admitting an Euclidean heat kernel is necessarily isometric to the Euclidean space. This helps us providing an alternative proof of Colding's celebrated almost rigidity volume theorem via a quantitative version of our main result. We also discuss the c…
In this paper the notion of Measure Equivalence (ME) of countable groups is studied. ME was introduced by Gromov as a measure-theoretic analog of quasi-isometries. All lattices in the same locally compact group are Measure Equivalent; this is one of the motivations for this notion. The main result of this paper is ME r…
We propose a unified computational framework for the problem of deformation and rigidity of submanifolds in a homogeneous space under geometric constraint. A notion of 1-rigidity of a submanifold under admissible deformations is introduced. It measures how a deformation deviates from a one parameter family of motions u…
We show that the mapping class group of a compact orientable surface with higher complexity has the following extreme rigidity in the sense of measure equivalence: if the mapping class group is measure equivalent to a discrete group, then they are commensurable up to finite kernel. Moreover, we describe all lattice emb…
New rigidity theorem for sharp spectral gap in nonnegatively curved spaces.
problem Rigidity of sharp spectral gap in nonnegatively curved spaces.
method Mixing Sobolev theory and singular 1D-localization.
result Rigidity of λ=diam2π2 in compact RCD(0,N) spaces. Let G,H be two Kleinian groups with homeomorphic quotients H3/G and H3/H. We assume that G is of divergence type, and consider the Patterson-Sullivan measures of G and H. The measurable rigidity theorem by Sullivan and Tukia says that a measurable and essentially directly measurable equiv…
Study inequalities involving torsional rigidity and spherical deficit on Riemannian manifolds.
problem Inequalities involving torsional rigidity and spherical deficit on Riemannian manifolds.
method Establish inequalities and derive an integral identity for a Dirichlet problem.
result Characterize metric balls and measure spherical deficit on Riemannian manifolds.
Mostow rigidity proven for special geometric manifolds.
problem Proving rigidity for specific geometric structures.
method Analyzing foliated bundles over closed hyperbolic manifolds with invariant measures.
result Mostow rigidity theorem extended to skew solenoidal manifolds.
We introduce a natural stratification of the space of projective classes of measured laminations on a complete hyperbolic surface of finite area. We prove a rigidity result, namely, the group of self-homeomorphisms of the space of projective measured laminations that preserve such a stratification is in general identif…
Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds.
problem Characterizing and proving rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds.
method Analysis of Riemannian manifolds and metric measure spaces with synthetic lower Ricci curvature bounds, using concentration compactness and Polya-Szego inequalities.
result Closed Riemannian manifolds with optimal Sobolev constant are isometric to the sphere, and almost equality implies close measure Gromov-Hausdorff convergence to a spherical suspension.
Proves existence of sentences to identify homeomorphic manifolds.
problem Identifying homeomorphic manifolds using group properties.
method Defines sentences in group language to match homeomorphic manifolds.
result Existence of sentences to distinguish homeomorphic manifolds.
The paper proves a quantitative rigidity result for spaces with specific curvature bounds.
problem Understanding the rigidity of spaces with almost maximal volume entropy.
method Analyzing Riemannian manifolds and RCD-spaces with specific curvature conditions. result Spaces with almost maximal volume entropy are closely related to hyperbolic space forms.
In this paper we investigate Lott-Sturm-Villani's synthetic lower Ricci curvature bound on Riemannian manifolds with boundary. We prove several measure rigidity results for some important functional and geometric inequalities, which completely characterize CD(K,∞) condition and non-collapsed ${\rm CD}(K, …
We consider the boundary rigidity problem for asymptotically hyperbolic manifolds. We show injectivity of the X-ray transform in several cases and consider the non-linear inverse problem which consists of recovering a metric from boundary measurements for the geodesic flow.
We extend the theory of Patterson-Sullivan measure to any regular covering of a compact manifold using the Busemann compactification and derive an integral formula for the volume entropy. As applications we prove some rigidity theorems for the volume entropy.
We study the boundary rigidity problem with partial data consisting of determining locally the Riemannian metric of a Riemannian manifold with boundary from the distance function measured at pairs of points near a fixed point on the boundary. We show that one can recover uniquely and in a stable way a conformal factor …
Free group automorphisms group rigidity proven.
problem Proving rigidity of Out(F_N).
method Measure equivalence rigidity, new canonical splittings.
result Superrigidity of Out(F_N).
Study positive entropy actions by higher-rank lattices, proving rigidity and conjugacy results.
problem Positive entropy actions by higher-rank lattices in Lie groups.
method Analysis of sub-actions, fiber entropy upper semicontinuity, and conjugacy arguments.
result Actions by higher-rank lattices in SL(n,R) are conjugate to affine actions on (infra-)tori. Consider a closed marked flat surface S of genus g≥2 and area 1 and its universal covering S~. We show that the measure class of the Hausdorff measure of the Gromov boundary of S~ uniquely determines S.
We consider actions of non-compact simple Lie groups preserving an analytic rigid geometric structure of algebraic type on a compact manifold. The structure is not assumed to be unimodular, so an invariant measure may not exist. Ergodic stationary measures always exist, and when such a measure has full support, we show…
Paper solves a new Minkowski problem for a specific type of rigidity.
problem Solving a new Minkowski problem for a specific type of rigidity.
method Developed a nonlinear partial differential equation and used a curvature flow method.
result Existence of smooth non-even solutions to the p-th dual Minkowski problem for p < n-2.
New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.
problem Defining and analyzing surface area measures for ball-convex bodies.
method Introducing Lp relative surface areas, proving invariance and inequalities, and using geometric interpretations. result Established inequalities and a new notion of entropy for ball-convex bodies.
The study proves compactness and existence of entropy minimizers for self-shrinking surfaces.
problem Understanding entropy in higher-codimension mean curvature flow.
method Measure-theoretical techniques and rigidity results for self-shrinkers.
result Existence of entropy minimizers and improved rigidity results.
The paper proves uncertainty principles on Finsler measure spaces.
problem Uncertainty principles on Finsler measure spaces.
method Analyzes Lp-uncertainty principles on Finsler measure spaces with bounded curvatures. result Sharp Lp-uncertainty principles are proven and characterized.