Derives Weyl law for volume spectrum using parametric inequalities.
problem Deriving the Weyl law for the volume spectrum in compact Riemannian manifolds.
method Proves parametric generalizations of isoperimetric and coarea inequalities to derive the Weyl law.
result Derives the Weyl law for 1-cycles in 3-manifolds.
Sharp 3D Alexandrov inequality applied to volume-preserving flows.
problem Volume-preserving geometric flows in 3D space.
method Sharp quantitative Alexandrov inequality for C2-regular sets. result Established a 3D sharp quantitative version of the Alexandrov inequality.
Generalizes inequality for complete manifolds involving homology classes.
problem Volume and simplicial volume inequality for closed manifolds.
method Extends inequality to ℓ1-norm of homology classes on complete manifolds. result Inequality involving critical exponent and mass of homology classes.
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.
Proves inequalities on curved spaces with positive curvature.
problem Proving inequalities on manifolds with nonnegative Ricci curvature.
method Analyzes manifolds with nonnegative Ricci curvature and Euclidean volume growth.
result Proves Heisenberg-Pauli-Weyl, Hardy-Sobolev, and Caffarelli-Kohn-Nirenberg inequalities.
Paper derives inequalities for eigenvalues of Witten-Laplacian under fixed volume constraint.
problem Eigenvalue inequalities of Witten-Laplacian on bounded domains.
method Rearrangement technique and trial functions under fixed weighted volume constraint.
result Several isoperimetric inequalities for eigenvalues of Witten-Laplacian.
Sharp Sobolev and Michael-Simon inequalities on curved manifolds.
problem Proving inequalities on manifolds with nonnegative curvature.
method Analyzing asymptotic volume ratio and curvature properties.
result Sharp Sobolev and Michael-Simon inequalities established.
Sharp heat kernel and Green's function estimates on Euclidean volume growth manifolds.
problem Estimating heat kernels and Green's functions on specific Riemannian manifolds.
method Analyzing noncompact Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth.
result Sharp Moser-Trudinger inequalities on manifolds with Euclidean volume growth.
This paper proves a curvature entropy inequality for non-symmetric convex bodies.
problem Proving a curvature entropy inequality for non-symmetric convex bodies.
method Demonstrated the log-Minkowski inequality of curvature entropy for general convex bodies in 2D.
result Equivalence of cone-volume measure uniqueness, log-Minkowski volume inequality, and curvature entropy inequality for general convex bodies in 2D.
The paper extends intersection theory for b-divisors, proving monotonicity and volume inequalities.
problem Intersection theory for b-divisors and monotonicity of intersection products.
method Developed general intersection theory of nef b-divisors, defined restricted volume, proved monotonicity.
result Proved quantitative monotonicity of intersection product and new volume inequalities.
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.
Develop an ABP approach to Sobolev and Michael-Simon inequalities beyond Euclidean volume growth.
problem Developing an ABP approach to Sobolev and Michael-Simon inequalities under volume noncollapsing assumptions.
method Using a refinement of Brendle's contact-set argument to derive lower bounds for the volumes of geodesic balls.
result A Michael-Simon type inequality for immersed submanifolds with nonnegative sectional curvature and volume noncollapsing.
Optimal inequalities for systole, inradius, and volume in hyperbolic 3-manifolds
problem Establishing optimal inequalities relating systole, inradius, and volume in hyperbolic 3-manifolds
method Using systole-volume inequalities for extremal manifolds
result Extremal manifolds for systole, inradius, and volume are identified
The Willmore flow preserves surface volume, leading to convergence to a sphere.
problem Long-term behavior of volume-preserving Willmore flow on surfaces.
method Volume-preserving Willmore flow, blow-up analysis, constrained Lojasiewicz-Simon inequality.
result Smooth solutions exist for spherical surfaces with Willmore energy below 8π and converge to a sphere.
At the heart of convex geometry lies the observation that the volume of convex bodies behaves as a polynomial. Many geometric inequalities may be expressed in terms of the coefficients of this polynomial, called mixed volumes. Among the deepest results of this theory is the Alexandrov-Fenchel inequality, which subsumes…
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.
New invariant measures knot geometry, improving volume-volume inequality.
problem Understanding hyperbolic knots through geometry.
method Introducing natural slope based on cusp geometry, using machine learning for detection.
result Improved inequality linking knot signature, volume, and injectivity radius.
Analogues of the classical inequalities from the Brunn-Minkowski theory for rotation intertwining additive maps of convex bodies are developed. Analogues are also proved of inequalities from the dual Brunn-Minkowski theory for intertwining additive maps of star bodies. These inequalities provide generalizations of resu…
Sharp Sobolev inequality derived for Riemannian manifolds with bounded Ricci curvature.
problem Deriving a sharp Sobolev inequality for Riemannian manifolds with bounded Ricci curvature.
method Reduction to functions with small volume support, first order uniform asymptotic expansion of isoperimetric profile, local uniform Sobolev inequality.
result Sharp Sobolev inequality for W1,p(M) into Ln−pnp(M) is derived. The paper establishes a sub-additive inequality for volume and ε-phase-transition spectra of Riemannian manifolds.
problem Analyzing the volume and ε-phase-transition spectra of Riemannian manifolds.
method Using the Almgren-Pitts width and Allen-Cahn approach.
result Proves sub-additive inequalities for volume and ε-phase-transition spectra.
The study establishes inequalities on Finsler manifolds with weighted Ricci curvature.
problem Investigating inequalities on Finsler manifolds with weighted Ricci curvature.
method Volume comparison, Bonnet-Myers theorem, Poincaré-Lichnerowicz inequality.
result Sharp lower bound for the first eigenvalue on Finsler manifolds.
Study intrinsic volume forms on complex hypersurfaces.
problem Computing volume functionals on pseudoconvex hypersurfaces.
method Compute first and second variation formulae, explore infinite dimensional aspects.
result Discuss possible analogues of the affine isoperimetric inequality.
Graphs with nonnegative Bakry-Émery curvature have volume doubling and Poincaré inequalities.
problem Proving properties of graphs with specific curvature conditions.
method Graph-theoretic modified nonlinear heat-flow method, including point-mass consequences and diffusive exit-time control.
result Volume doubling and Poincaré inequalities for graphs with nonnegative Bakry-Émery curvature.
The paper proves a reverse isoperimetric inequality and applies it to analyze surface flows.
problem Analyzing the negative gradient flow of the Willmore energy plus volume.
method Proved a quantitative reverse isoperimetric inequality and applied it to the flow.
result Initial surfaces converge to a round point in finite or infinite time.
This paper develops basic setting for the dual Orlicz-Brunn-Minkowski theory for star bodies. An Orlicz φ-radial addition of two or more star bodies is proposed and related dual Orlicz-Brunn-Minkowski inequality is established. Based on a linear Orlicz φ-radial addition of two star bodies, we derive a f…
Paper uses ABP method to prove logarithmic Sobolev inequalities on curved spaces.
problem Proving logarithmic Sobolev inequalities on manifolds with nonnegative curvature.
method Employing the ABP method developed by Brendle.
result Sharp L2 and Lp logarithmic Sobolev inequalities established. Gromov's universal filling inequalities relate the filling radius and the filling volume of a Riemannian manifold to its volume. The main result of the present article is that in dimensions at least three the optimal constants in the filling inequalities depend only on dimension and orientability, not on the manifold i…
The paper proves nonexistence results for certain parabolic inequalities on Riemannian manifolds.
problem Proving nonexistence results for parabolic inequalities on Riemannian manifolds.
method Using a test function argument and weighted volume growth assumptions.
result Established Liouville-type theorems for (p,q)-Laplacian operator inequalities. 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…
Paper connects Fenchel-Willmore and Sobolev inequalities for submanifolds in curved spaces.
problem Developing inequalities for submanifolds in curved spaces.
method Connecting Fenchel-Willmore and logarithmic Sobolev inequalities for mean-convex submanifolds.
result Established extensions of Fenchel-Willmore inequality and derived new Sobolev-type inequalities.
Sharp isoperimetric inequality for Finsler manifolds with non-negative Ricci curvature.
problem Proving an isoperimetric inequality for Finsler manifolds with specific curvature properties.
method Analyzing measured Finsler manifolds with non-negative Ricci curvature and Euclidean volume growth.
result Sharp isoperimetric inequality and rigidity results for the inequality.
This paper investigates the relationship between the topology of hyperbolizable 3-manifolds M with incompressible boundary and the volume of hyperbolic convex cores homotopy equivalent to M. Specifically, it proves a conjecture of Bonahon stating that the volume of a convex core is at least half the simplicial volume o…
The paper classifies energy-minimizing sets in specific domains.
problem Classifying volume-constraint local energy-minimizing sets.
method Proved a Poincaré-type inequality for stable sets.
result Relative boundary of energy-minimizing sets is smooth.
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 …
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.
We prove inequalities for Laplace eigenvalues on Riemannian manifolds generalising to higher eigenvalues two classical inequalities for the first Laplace eigenvalue - the inequality in terms of the L2-norm of mean curvature, due to Reilly in 1977, and the inequality in terms of conformal volume, due to Li and Yau in…
This paper explains a technique for proving geometric inequalities.
problem Proving various geometric inequalities in different contexts.
method Unified framework based on Alexandrov-Bakelman-Pucci technique.
result Unified approach to proving geometric inequalities.
The article disproves a local systolic inequality and shows a lower bound on filling area.
problem Proving a local systolic inequality and a lower bound on filling area.
method Analyzing Gromov's filling area conjecture and showing a computational mistake.
result The local systolic inequality was disproved and a lower bound on filling area was shown.
Optimal geometric estimates for Kähler manifolds with bounded Nash entropy
problem Optimal geometric estimates for compact Kähler manifolds
method Proving Sobolev-type inequality and local volume noncollapsing with optimal exponents
result Uniformly bounded q-Nash entropy The paper improves CR Sobolev inequalities and classifies minimizers.
problem Higher-order CR Sobolev inequalities on the CR sphere.
method Improvement through vanishing higher order moments of the volume element.
result New direct proof of minimizers' classification and existence of minimizers in C2k(N). Study volume growth in Milnor fibers using real Lagrangians.
problem Volume growth in Milnor fibers of Brieskorn polynomials.
method Investigate real Lagrangians, use Smith inequality for involutions in wrapped Floer homology.
result Uniform lower bound of volume growth for a class of Brieskorn polynomials.
The paper introduces new volume measures and volume comparison inequalities for Lorentzian spaces.
problem Volume comparison in Lorentzian pre-length spaces.
method Introducing modified timelike Hausdorff measures and establishing volume comparison inequalities using timelike Lipschitz maps.
result Coincidence of modified and original volume measures on smooth spacetimes and some pre-length spaces.
We prove a new inequality relating volume to length of closed geodesics on area minimizers for generic metrics on the complex projective plane. We exploit recent regularity results for area minimizers by Moore and White, and the Kronheimer--Mrowka proof of the Thom conjecture.
Study of harmonic functions on infinite penny graphs.
problem Characterizing harmonic functions on infinite penny graphs.
method Proving volume doubling and Poincaré inequalities, analyzing polynomial growth harmonic functions.
result Finite dimensional property of ancient solutions of the heat equation.
Let Σ be a k-dimensional complete proper minimal submanifold in the Poincaré ball model Bn of hyperbolic geometry. If we consider Σ as a subset of the unit ball Bn in Euclidean space, we can measure the Euclidean volumes of the given minimal submanifold Σ and the ideal boundary ∂∞Σ, say $\…
The study proves surfaces with high genus have a specific inequality.
problem Proving surfaces with high genus satisfy a specific inequality.
method Using volume entropy and systolic ratio inequality.
result Every closed surface of genus at least 18 satisfies Loewner's systolic ratio inequality.
We show that the cone-volume measure of a convex body with centroid at the origin satisfies the subspace concentration condition. This implies, among others, a conjectured best possible inequality for the U-functional of a convex body. For both results we provide stronger versions in the sense of stability i…
Sharp Sobolev inequalities proved on manifolds with non-negative Ricci curvature.
problem Proving sharp Sobolev inequalities on noncompact Riemannian manifolds with non-negative Ricci curvature.
method Using Optimal Mass Transportation with quadratic distance cost.
result Sharp Lp-Sobolev and Lp-logarithmic Sobolev inequalities established for p>1 and p=1.