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.
Investigates conditions for non-rigidity in extremal metrics involving scalar curvature.
problem Rigidity of extremal metrics involving scalar curvature.
method Analyzes sufficient conditions for non-rigidity and provides examples.
result Provides sufficient conditions for metrics not to be rigid.
Study recovers Lorentzian metrics from boundary data, proving local rigidity.
problem Recovering a Lorentzian metric from scattering data on a boundary.
method Analyzes jet and real analyticity of metrics near lightlike points.
result Metric can be recovered up to gauge transformations near lightlike strictly convex points.
Study on recovering Lorentzian metrics from scattering data.
problem Recovering Lorentzian metrics from scattering data on a boundary.
method Analyzing the role of boundary distance functions and linearizing the light ray transform.
result Scattering rigidity can be reduced to boundary rigidity of magnetic systems.
Proves rigidity for maps between manifolds using degree theory and current developments.
problem Lipschitz-volume rigidity for maps between metric manifolds and Riemannian manifolds.
method Degree theory and recent developments of Lipschitz-volume rigidity for integral currents.
result Proves a Lipschitz-volume rigidity result for 1-Lipschitz maps.
Einstein manifolds are rigid under certain metric deformations.
problem Characterizing Einstein manifolds that resist volume-preserving metric deformations.
method Various characterizations and constructions of mass-decreasing perturbations.
result Constructs mass-decreasing perturbations of specific metrics.
Boundary rigidity proven for non-reversible Finsler metrics.
problem Recovering non-reversible Finsler metrics from boundary distance data.
method Sum of reversible Finsler norm and closed 1-form, boundary rigidity results.
result 1-form can be uniquely recovered from boundary distance data.
Totally geodesic subvarieties in moduli space are locally rigid.
problem Understanding rigidity of subvarieties in moduli space.
method General rigidity result for orbifold maps to moduli space.
result Covering constructions and totally geodesic subvarieties are locally rigid.
Study rigidity of Einstein metrics as critical points of curvature functionals.
problem Characterize Einstein metrics as critical points of quadratic curvature functionals.
method Analyze pointwise inequalities involving Weyl curvature and traceless Ricci curvature.
result Provide rigidity results for Einstein metrics and locally conformally flat critical metrics.
Characterizes rigid and flexible hyperbolic cone metrics and billiards.
problem Understanding the rigidity and flexibility of hyperbolic cone metrics and their billiard dynamics.
method Characterization through Liouville currents and deformation spaces.
result Generic rigidity and parameterization of deformation spaces for flexible metrics.
We analyze sub-Riemannian and lightlike metrics from the point of view of their rigidity as geometric structures. Following Cartan's and Gromov's formal definitions, they are never rigid, yet, in generic cases, they naturally give rise to rigid geometric structures!?
Local rigidity results for Bergman and Kähler Carathéodory metrics on domains.
problem Characterizing domains with specific metric properties.
method Analyzing Carathéodory and Bergman metrics on strictly pseudoconvex domains.
result Domains with specific metric properties are biholomorphically equivalent to balls.
The paper examines rigidity of Einstein metrics using curvature functionals.
problem Characterizing rigidity of Einstein metrics.
method Critical points of quadratic curvature functionals and integral inequalities involving Weyl curvature, trace-less Ricci curvature, and Sobolev constant.
result Rigidity results for Einstein metrics on complete manifolds.
Rigidity theorem for special metrics on 4-manifolds.
problem Rigidity of Bach-flat metrics on manifolds with boundary.
method Critical point analysis of Weyl energy with boundary conditions.
result Rigidity of critical metrics on upper hemisphere.
Study on spherical Finsler metrics with isotropic curvature rigidity.
problem Characterizing and understanding spherically symmetric Finsler metrics with isotropic E-curvature. method Provided the correct formula for mean Berwald curvature, established differential equations for projective and dual flatness, and derived a rigidity result.
result Rigidity result on spherically symmetric Finsler metrics with isotropic E-curvature. 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.
In \cite{BK02}, M. Bonk and B. Kleiner proved a rigidity theorem for expanding quasi-Möbius group actions on Ahlfors n-regular metric spaces with topological dimension n. This led naturally to a rigidity result for quasi-convex geometric actions on CAT(−1)-spaces that can be seen as a metric analog to the "entrop…
The paper proves rigidity of Einstein metrics as critical points of quadratic curvature functionals.
problem Characterizing Einstein metrics as critical points of quadratic functionals.
method Analyzing Einstein metrics on closed manifolds using quadratic curvature functionals and point-wise inequalities.
result Rigidity results for Einstein metrics involving Weyl curvature, trace-less Ricci curvature, and Yamabe invariant.
Local rigidity shown for certain spherical conical metrics.
problem Deformation of spherical conical metrics with cone angles > 2π.
method Synthetic geometry approach.
result Local rigidity in the choice of cone positions for a specific family of metrics.
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.
Paper proves rigidity for Einstein metrics in high dimensions.
problem Einstein metrics on high-dimensional manifolds.
method Liouville type rigidity result for asymptotically hyperbolic metrics.
result Established a rigidity theorem for d≥5. Bi-invariant metrics on Lie groups and homogeneous spaces are extremal and rigid.
problem Gromov's question on extremality of bi-invariant metrics on compact Lie groups.
method Proving rigidity of bi-invariant metrics on compact Lie groups and homogeneous spaces.
result Bi-invariant metrics on compact Lie groups and homogeneous spaces are extremal and rigid.
The paper examines rigidity of metric constructions in Wasserstein spaces.
problem Isometric rigidity of metric constructions in Wasserstein spaces.
method Analyzes spaces like Hilbert, rays, half-cylinders, and spherical suspensions.
result Different spaces exhibit varying levels of isometric rigidity in Wasserstein spaces.
Revisits Koiso's rigid metrics on complex projective spaces.
problem Computing obstructions to integrability of deformations.
method Elementary complex differential geometry.
result Computes Koiso's obstruction on CPnimesCP1. 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…
Smooth metrics satisfying Penrose inequality are necessarily smooth.
problem Rigidity of Penrose inequality with singular metrics.
method Showed suitable singular metrics attaining the optimal value in the Riemannian Penrose inequality are smooth in specified coordinates.
result Smooth metrics satisfying Penrose inequality are necessarily smooth.
The paper proves Weyl projective rigidity for sub-Riemannian metrics and shows genericity of such metrics.
problem Investigating the Weyl projective rigidity of sub-Riemannian metrics.
method Analytic and smooth category proofs for specific distributions with minimal order complex abnormal extremals.
result Genericity of Weyl projectively rigid sub-Riemannian metrics and distributions.
The paper examines properties of generalized τ-quasi Ricci-harmonic metrics and proves rigidity results.
problem Characterizing and proving rigidity of generalized τ-quasi Ricci-harmonic metrics.
method Exploring conditions for harmonic-Einstein metrics, obtaining rigidity results, and proving gap theorems.
result Rigidity results for compact generalized τ-quasi Ricci-harmonic metrics.
Proves infinitesimal rigidity of Hermitian gravitational instantons.
problem Understanding the moduli space of Hermitian gravitational instantons.
method Proof of infinitesimal rigidity and integrability using boundary conditions and conformal Kähler properties.
result Completes the understanding of Hermitian gravitational instantons, both compact and non-compact.
The paper explores gaps in curvature-related metrics and rigidity.
problem Understanding gaps in curvature-related metrics and rigidity.
method Analyzes three types of gaps: spectral, metric-rigidity, and topological-rigidity.
result Proposes open problems in the field.
Hermitian metrics with zero second Chern Ricci curvature are rigid and exist on specific manifolds.
problem Characterizing Hermitian metrics with vanishing second Chern Ricci curvature.
method Analyzing the rigidity of the second Chern Ricci curvature on compact complex manifolds.
result Characterization of second Chern Ricci-flat Hermitian metrics and non-existence results.
The paper develops techniques to study dynamical systems with Carnot metrics.
problem Understanding smooth dynamical systems in the presence of Carnot metrics.
method Employing techniques from Margulis-Mostow, Métivier, Mitchell, and Pansu on tangent cones, the paper establishes resonances between Lyapunov exponents.
result Local rigidity properties of higher hyperbolic rank metrics and uniform lattice actions on quaternionic and octonionic symmetric spaces.
Paper proves rigidity of inversive distance circle packings.
problem Proving rigidity of inversive distance circle packings.
method Variational principles and combinatorial curvature study.
result Global rigidity of inversive distance circle packings proved.
Proves rigidity of sphere metrics with subsets removed.
problem Scalar curvature rigidity of spheres with subsets removed.
method Techniques involving wrapping property and L∞ metrics. result Proves scalar rigidity for L∞ metrics on Sn\Σ. Solves Besse conjecture on 3D manifolds, proving metric rigidity.
problem Besse conjecture on 3D compact manifolds
method Analytical proof of critical point equation
result Proves rigidity of Miao-Tam metric
Flexible metrics found on a genus 2 surface.
problem Identifying non-rigid hyperbolic cone metrics on a genus 2 surface.
method Using a theorem by Erlandsson, Leininger, and Sadanand.
result Nine mapping class group orbits of non-rigid metrics found.
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.
New rigidity results for critical metrics of a quadratic curvature functional.
problem Proving uniqueness of critical metrics for a specific curvature functional.
method Analyzing complete, possibly non-compact, critical metrics of the quadratic curvature functional.
result Critical metrics with finite energy are scalar flat (global minima) for dimensions n≥10.
New rigidity results for critical metrics with curvature pinching.
problem Understanding critical metrics with curvature pinching conditions.
method Proving rigidity for metrics defined on closed smooth manifolds that are critical for a quadratic functional.
result Bach-flat metrics with constant scalar curvature satisfying Sec > 1/48 R are Einstein and isometric to specific spaces.
3D spherical caps are rigid under certain perturbations.
problem Rigidity of 3D spherical caps under specific perturbations.
method Gromov's μ-bubble technique
result 3D spherical caps are rigid under perturbations that maintain metric, scalar curvature, and mean curvature.
New findings on flatness of certain metrics with fast decay.
problem Rigidity of positive mass theorem under fast metric decay.
method Considered metrics with nonnegative scalar curvature and rapid decay at infinity.
result Any such metric is necessarily flat in dimensions 4 and higher if decay rate exceeds Schwarzschild metric.
Scattering rigidity of a Riemannian manifold allows one to tell the metric of a manifold with boundary by looking at the directions of geodesics at the boundary. Lens rigidity allows one to tell the metric of a manifold with boundary from the same information plus the length of geodesics. There are a variety of results…
Study on projective and affine equivalence of sub-Riemannian metrics on manifolds.
problem Classifying projective and affine equivalence of sub-Riemannian metrics.
method Analyzing geodesics, integrals, and product structures on cotangent bundles.
result Generic sub-Riemannian metrics are projectively conformally rigid.
The paper studies Einstein metrics on specific manifolds and their rigidity properties.
problem Investigating rigidity of Einstein metrics on homogeneous Gray manifolds.
method Computing coindex and analyzing infinitesimal deformations of Einstein metrics.
result Infinitesimal Einstein deformations on F1,2=SU(3)/T2 are not integrable. Critical metrics in Levy-Gromov inequality are rigid in 2D.
problem Understanding critical metrics in Levy-Gromov inequality.
method Analyzing critical points of the isoperimetric profile.
result Criticality condition characterizes only round spheres and projective planes in 2D.
New rigidity results for warped product domains.
problem Scalar curvature rigidity of domains in warped products.
method Developed a new connection on a twisted spinor bundle and associated Dirac operator.
result Obtained Llarull type scalar curvature rigidity for a general class of domains in a warped product.
Rigidity theorem for curved manifolds with boundary.
problem Proving that certain curved manifolds are uniquely determined by their lens data.
method Analyzing lens data and using properties of Anosov type metrics.
result Local lens rigidity holds for negatively-curved manifolds and metrics of Anosov type.
Study proves rigid spectral properties of planets with metric discontinuities.
problem Establishing spectral rigidity for spherically symmetric planets with discontinuities.
method Novel trace formula applied to two wave types in spherically symmetric manifolds with boundary and interior interfaces.
result Spectral rigidity of spherically symmetric planets with discontinuities is proven.