Study noncommutative Sobolev inequalities using quantum state metrics.
problem Establishing Sobolev inequalities in noncommutative settings.
method Generalizing monotone metrics in quantum states.
result Developed new matrix-valued Beckner inequalities.
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 a Harnack inequality for heat equations on Finsler metric measure manifolds.
problem Proving a Harnack inequality for positive solutions to heat equations on Finsler metric measure manifolds.
method Volume comparison theorem, weighted Poincaré inequality, local uniform Sobolev inequality, mean value inequalities.
result Derives a Harnack inequality for positive solutions to heat equations.
The paper derives inequalities for submanifolds in quaternionic Kaehler manifolds.
problem Analyzing submanifolds in quaternionic Kaehler manifolds.
method Established Chen's and generalized Casorati curvature inequalities.
result Derived inequalities for submanifolds in quaternionic Kaehler manifolds.
Sharp inequality in spaces with non-negative Ricci curvature.
problem Proving a sharp isoperimetric inequality in metric measure spaces.
method Using volume entropy in non-compact metric measure spaces with non-negative synthetic Ricci curvature.
result Proved a sharp dimension-free isoperimetric inequality.
The paper studies Harnack inequalities on Finsler metric measure spaces.
problem Analyzing Harnack inequalities on Finsler metric measure spaces.
method Using weighted Ricci curvature and distortion conditions, the authors derive an elliptic p-Harnack inequality.
result The paper establishes an elliptic p-Harnack inequality and derives Hölder continuity and gradient estimates for positive harmonic functions.
Sharp inequalities for J functional on Kahler metrics.
problem Asymptotic behavior of J functional on Kahler metrics. method Sharp inequalities derived for d1 metric. result Sharp inequalities between J functional asymptotics. Paper establishes inequality for submanifolds in real space forms with semi-symmetric non-metric connection.
problem Deriving a sharp lower bound for Ricci curvature of submanifolds.
method Using semi-symmetric non-metric connection, derive a lower bound for Ricci curvature in terms of mean curvature vector and second fundamental form.
result Established Hineva inequality for submanifolds with semi-symmetric non-metric connection.
Uniform Poincaré inequalities established for various metric spaces.
problem Establishing uniform Poincaré inequalities on different metric spaces.
method Proper geodesic metric spaces equipped with a Borel measure. Local Poincaré inequality and volume conditions are used to derive uniform Poincaré inequalities.
result Uniform Poincaré inequalities are established for various metric spaces including hyperbolic spaces and covers of compact spaces.
Study fine Pólya-Szegő inequalities in metric spaces with applications.
problem Fine Pólya-Szegő rearrangement inequalities in metric spaces.
method Theory of Sobolev and BV functions, synthetic Ricci bounds, isoperimetric inequality.
result New geometric and functional inequalities under Ricci lower bounds.
The paper examines inequalities for Chern numbers on specific 4D Kähler manifolds.
problem Investigating Chern number inequalities on 4D Kähler manifolds with deformed Hermitian-Yang-Mills metrics.
method Analyzing 4D Kähler manifolds with deformed Hermitian-Yang-Mills metrics under the condition θ^∈(π,2π). result Established Chern number inequalities for the specified manifolds.
Paper derives inequalities for submanifolds in a specific geometric space.
problem Chen's inequalities for submanifolds in (κ,μ)-contact space form. method Using generalized semi-symmetric non-metric connections.
result Derives new inequalities for submanifolds.
Proves equality in Minkowski inequality for static, flat manifolds.
problem Proving equality in Minkowski inequality for static, flat manifolds.
method Analyzes quasi-spherical metrics and static manifolds.
result Equality in Minkowski inequality achieved only by Schwarzschild space slices.
Uniform Sobolev inequality for Kähler metrics with entropy bound.
problem Establishing Sobolev inequalities for Kähler metrics with entropy bound.
method Uniform Sobolev inequality for Kähler metrics with entropy bound and no lower Ricci curvature bound.
result Derive various geometric estimates for Kähler-Einstein currents.
New findings on metric spaces with finite Nagata dimension.
problem Understanding isoperimetric properties in subsets of metric spaces.
method Analyzing quasiconvex subsets with finite Nagata dimension and applying isoperimetric inequalities.
result Quasiconvex subsets of metric spaces with finite Nagata dimension are isoperimetrically undistorted up to a certain dimension.
Paper finds inequalities for eigenvalues of buckling problems on special metric spaces.
problem Eigenvalue inequalities for buckling problems of drifting Laplacian.
method Investigated on bounded domains in complete smooth metric measure spaces (SMMSs) with special functions.
result General inequalities for eigenvalues derived under curvature constraints.
Two lectures on metric geometry of manifolds.
problem Understanding metric geometry properties of manifolds.
method Discussion of specific inequalities and concepts.
result Exploration of metric geometry properties of manifolds.
Sharp inequalities found for orbifold metrics.
problem Bounding systolic ratios on rotationally symmetric orbifolds.
method Analyzing spindle orbifolds and Besse metrics.
result Upper bounds on systolic ratios are attained at Besse metrics.
The paper proves a Moser-Trudinger inequality on metric spaces with curvature-dimension conditions.
problem Proving a Moser-Trudinger inequality on metric measure spaces.
method Rearrangement of functions on CD(k,n)-spaces satisfying a Polya-Szegö type inequality.
result Characterization of manifolds with lower bounded Ricci curvature admitting a Moser-Trudinger inequality.
The paper proves inequalities on Finsler manifolds under Ricci curvature bounds.
problem Proving (p,q)-Sobolev and Nash inequalities on Finsler metric measure manifolds. method Global p-Poincaré inequality, (p,q)-Sobolev inequality, Nash inequality derivation. result Established global optimal (p,q)-Sobolev inequality with a sharp constant. Given a probability measure μ supported on a convex subset Ω of Euclidean space (Rd,g0), we are interested in obtaining Poincaré and log-Sobolev type inequalities on (Ω,g0,μ). To this end, we change the metric g0 to a more general Riemannian one g, adapted in a certain sense to μ, and perform…
Study Poincaré inequality in metric spaces via separating sets.
problem Geometric characterization of Poincaré inequality in metric spaces.
method Properties of separating sets and various notions of energy.
result Equivalence of conditions for 1-Poincaré inequality.
Using an inverse system of metric graphs as in: J. Cheeger and B. Kleiner, "Inverse limit spaces satisfying a Poincaré inequality", we provide a simple example of a metric space X that admits Poincaré inequalities for a continuum of mutually singular measures.
The paper proves Gromov hyperbolicity of certain metrics using isoperimetric inequalities.
problem Investigating Gromov hyperbolicity of specific metrics.
method Using isoperimetric inequalities to characterize Gromov hyperbolicity.
result Characterization of domains where these metrics are Gromov hyperbolic.
Study Penrose inequality for metrics with singular sets.
problem Penrose inequality for metrics with singular sets.
method Analysis of Penrose inequality for metrics with singular sets of dimension less than n-1, without additional conditions.
result Complement existing results by studying Penrose inequality for metrics with singular sets of lower dimension.
We present in this note a lower bound for the Calabi functional in a given Kähler class. This yields an integral inequality for constant scalar curvature metrics, which can be viewed as a refined version of Yau's Chern number inequality.
The paper establishes Sobolev inequalities between Riemannian metrics and their distance functions.
problem Establishing a theory of Sobolev inequalities for Riemannian metrics and distance functions.
method Analyzing the sub-critical case $p < rac{m}{2}$, proving a Sobolev inequality linking $L^{ rac{p}{2}}$ bounds on metrics to Lq bounds on distance functions. result A Sobolev inequality exists between Riemannian metrics and their distance functions, leading to a convergence theorem.
Deviation inequalities and limit laws for random walks on metric spaces.
problem Understanding random walks on metric spaces with contracting isometries.
method Adapting Gouëzel's pivotal time construction to establish deviation inequalities.
result Exponential bounds and limit laws for random walks on mapping class groups and CAT(0) spaces.
The paper shows how Sobolev maps affect currents in metric spaces.
problem Understanding how Sobolev maps affect currents in metric spaces.
method Proving that a Sobolev map pushes almost every compactly supported integral current to an Ambrosio-Kirchheim integral current.
result The paper proves an isoperimetric inequality for Sobolev mappings relative to bounded, closed, and additive cochains.
Proves Hölder-type inequality for Lagrangians' distance.
problem Understanding the symplectic geometry of Lagrangians.
method Developed methods from previous works to establish the inequality.
result Established a Hölder-type inequality for the Hausdorff distance between Lagrangians.
Optimal inequalities found between Riemannian and Hilbert metrics in convex projective domains.
problem Finding optimal bounds between Riemannian and Hilbert metrics in convex projective domains.
method Optimal control techniques applied to Riemannian metrics induced by centro-affine hypersurface immersions.
result Optimal inequalities between Riemannian and Hilbert metrics for a class of convex projective domains.
New neural nets respect triangle inequality, improving graph and reinforcement learning performance.
problem Neural nets lack inductive bias for certain subadditive distances.
method Introduced novel architectures that universally approximate norm-induced metrics.
result Neural nets with triangle inequality inductive bias outperform existing approaches.
The paper extends entropy formulas to super Ricci flows on metric measure spaces.
problem Entropy formulas for super Ricci flows on metric measure spaces.
method Extending Perelman's W-entropy and Shannon entropy power to super Ricci flows. result Equivalence between volume non-local collapsing property and lower boundedness of W-entropy on RCD(0,N) spaces. Transforms metric space geometry into persistent homology.
problem Geometric properties of metric spaces encoded by curvature inequalities.
method Persistent homology induced by Čech filtration.
result Translation of geometric properties into topological representation.
Sharp isoperimetric inequality on Finsler manifolds with non-negative Ricci curvature.
problem Proving an isoperimetric inequality on Finsler metric measure manifolds.
method Defining volume entropy and second Cheeger constant, proving sharp inequality.
result Sharp isoperimetric inequality involving volume entropy and weighted Ricci curvature.
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.
We provide an isoperimetric inequality for critical metrics of the volume functional with nonnegative scalar curvature on compact manifolds with boundary. In addition, we establish a Weitzenböck type formula for critical metrics of the volume functional on four-dimensional manifolds. As an application, we obtain a clas…
Reverse Hölder inequalities on Fano metrics with applications to geodesics and singularities.
problem Establishing reverse Hölder inequalities on Kähler metrics of Fano varieties.
method Using log-concavity and properties of Ricci potentials, the inequality is proven for Fano manifolds with log terminal singularities.
result The inequality holds for Fano varieties with log terminal singularities and the constant depends only on p and the dimension of X.
Lipschitz maps on metric surfaces are rigid if they preserve area.
problem Understanding the rigidity of Lipschitz maps on metric surfaces.
method Established a coarea inequality for continuous Sobolev functions on metric surfaces.
result Proved that 1-Lipschitz maps from a closed metric surface to a closed Riemannian surface preserving area are isometries.
Miyaoka-Yau inequality proven for smooth minimal models.
problem Proving the Miyaoka-Yau inequality for smooth minimal models.
method Existence of cscK metrics in a neighborhood of the canonical class.
result Miyaoka-Yau inequality holds for compact Kähler manifolds with nef canonical bundle.
In the asymptotically locally hyperbolic setting it is possible to have metrics with scalar curvature at least -6 and negative mass when the genus of the conformal boundary at infinity is positive. Using inverse mean curvature flow, we prove a Penrose inequality for these negative mass metrics. The motivation comes fro…
In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli-Kohn-Nirenberg inequality with same exponent n(n>1), then it has exactly n-dimensional volume growth. As application, we obtain geometric and topological properties of Alexandrov space, Riemannian manifold …
We prove that n-dimensional (n⩾3) complete and non-compact metric measure spaces with non-negative weighted Ricci curvature in which some Caffarelli-Kohn-Nirenberg type inequality holds are close to the model metric measure n-space (i.e., the Euclidean metric n-space).
Study on stability of ALE Ricci-flat metrics using a modified Perelman's λ-functional.
problem Stability and instability of ALE Ricci-flat metrics.
method Use of a modified Perelman's λ-functional and Lojasiewicz inequality.
result Demonstrates dynamical instability of ALE Ricci-flat metrics.
A local cut point is by definition a point that disconnectes its sufficiently small neighborhood. We show that there exists an upper bound for the degree of a local cut point in a metric measure space satisfying the generalized Bishop--Gromov inequality. As a corollary, we obtain an upper bound for the number of ends o…
Nonexistence results for semilinear parabolic and hyperbolic inequalities on metric graphs
problem Nonexistence of solutions to semilinear parabolic and hyperbolic inequalities on metric graphs
method Construction of a new pseudo-metric and space-time test functions
result All solutions must be identically zero
The Funk metric connects billiards, projective geometry, and convex geometry.
problem Exploring the Funk metric's invariants and inequalities.
method Using the Funk metric, extending results from projective geometry and convex geometry.
result General affine inequalities and volume maximizers in Funk geometry.
A carpet is a metric space homeomorphic to the Sierpinski carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincaré inequalities. Our results yield new examples of compact doubling metric measure spaces supporting Poinca…