Extends Gromov's optimal systolic inequality to manifolds with specific cohomology properties.
problem Finding optimal systolic inequalities for manifolds with complex cohomology structures.
method Extends Gromov's inequality to manifolds with fundamental cohomology classes as cup products of 2-dimensional classes.
result Provides an optimal systolic inequality for a new class of manifolds.
Paper proves Gromov's cube inequality in all dimensions.
problem Proving Gromov's cube inequality on scalar curvature in all dimensions.
method Used Dirac operator method to prove cube inequality with optimal constant.
result Proved cube inequality in all dimensions with optimal constant.
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.
We give a very short and rather elementary proof of Gromov's filling volume inequality for n-dimensional Lipschitz cycles (with integer and Z_2-coefficients) in L∞-spaces. This inequality is used in the proof of Gromov's systolic inequality for closed aspherical Riemannian manifolds and is often regarded as the…
The paper proves the Singer conjecture for aspherical complex surfaces and refines Gromov's inequality.
problem Proving the Singer conjecture and refining Gromov's inequality for aspherical complex surfaces.
method Proof of the Singer conjecture for aspherical complex surfaces with residually finite fundamental group, and refinement of Gromov's inequality for non-residually finite surfaces.
result Proof of the Singer conjecture for aspherical complex surfaces with residually finite fundamental group.
The paper characterizes hyperbolic manifolds and graphs verifying a specific isoperimetric inequality.
problem Understanding the relationship between hyperbolicity and isoperimetric inequalities in manifolds and graphs.
method Characterization of hyperbolic manifolds and graphs with isoperimetric inequality, using Gromov boundary.
result Having a pole is a necessary condition for verifying the isoperimetric inequality, which can be removed.
We observe after Bayle and Rosales that the Levy-Gromov isoperimetric inequality generalizes to convex manifolds with boundary.
Schwarzschild 3-manifold stability proven for 3D Penrose inequality.
problem Stability of the Schwarzschild 3-manifold in the context of the 3D Riemannian Penrose inequality.
method Pointed measured Gromov-Hausdorff topology, negligible domains and boundary area perturbations.
result Schwarzschild 3-manifold stability proven for 3D Penrose inequality.
The Levy-Gromov inequality states that round spheres have the least isoperimetric profile (normalized by total volume) among Riemannian manifolds with a fixed positive lower bound on the Ricci tensor. In this note we study critical metrics corresponding to the Levy-Gromov inequality and prove that, in two-dimensions, t…
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.
In this article we exhibit the largest constant in a quadratic isoperimetric inequality which ensures that a geodesic metric space is Gromov hyperbolic. As a particular consequence we obtain that Euclidean space is a borderline case for Gromov hyperbolicity in terms of the isoperimetric function. We prove similar resul…
Researchers prove an inequality linking spin manifold fill-ins to hyperspherical radius.
problem Proving an inequality relating the infimum of mean curvature to hyperspherical radius for spin manifolds.
method Combining Hijazi-Montiel-Roldán inequality and Bär's recent theorem; providing an alternative proof based on Bär-Ballmann work.
result Proved inequality linking spin manifold fill-ins to hyperspherical radius.
Potential theory extended to Gromov hyperbolic spaces.
problem Extending potential theory to a new class of spaces.
method Unified framework for Gromov hyperbolic metric measure spaces.
result Boundary Harnack inequalities and complete classification of positive harmonic functions.
In this paper we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary. Furthermore, we…
Extends K-cowaist inequality to manifolds with boundary.
problem Applying K-cowaist inequality to manifolds with boundary.
method Extended K-cowaist inequality for generalized Dirac operators on manifolds with boundary.
result New applications of K-cowaist inequality to manifolds with boundary.
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…
Paper proves new inequalities for Einstein-Maxwell data sets.
problem Establishing area-charge inequalities for Einstein-Maxwell initial data sets.
method Applying Gromov's μ-bubble technique in a new geometric context.
result Novel rigidity theorems for noncompact Einstein-Maxwell data sets.
Study of Gaussian distributions using entropic Gromov-Wasserstein and inner product Gromov-Wasserstein.
problem Optimal transportation between Gaussian distributions with different dimensions.
method Entropic Gromov-Wasserstein and inner product Gromov-Wasserstein, with closed-form expressions and von Neumann's trace inequality.
result Closed-form expressions for the entropic IGW and its unbalanced variant between Gaussian distributions.
Proves compactness for timed-metric spaces using new distance and maps.
problem Weak convergence of space-times using timed-Hausdorff distance.
method Uses Gromov's original compactness theorem and introduces addresses.
result Establishes compactness theorem for intrinsic timed-Hausdorff convergence.
Two new proofs of Gromov's non-squeezing theorem using curve reparametrization and gradient bounds.
problem Gromov's non-squeezing theorem in symplectic geometry.
method Reparametrization of pseudo-holomorphic curves and application of mean value inequality or Gromov-Schwarz lemma.
result Uniform bounds on the gradient of pseudo-holomorphic curves leading to compactness of moduli space.
Derives generalizations of the long neck principle and spectral width inequality.
problem Understanding the spectral width of geodesic collar neighborhoods.
method Spinorial Callias operator approach and relative Gromov-Lawson pair.
result Generalizations of the long neck principle and spectral width inequality.
The study refines known counterexamples in 4D to satisfy certain inequalities.
problem Addressing counterexamples in Gromov's and Rosenberg's conjectures.
method Analyzing simply connected and non-simply connected four manifolds up to homeomorphism.
result Gromov's and Rosenberg's conjectures hold for simply connected four manifolds up to homeomorphism.
Solved Cheeger inequalities for simplicial complexes, combining topological and graph theoretic methods.
problem Extend Cheeger inequalities to simplicial complexes and their higher order Laplacians.
method Combining constructions from simplicial topology, signed graphs, Gromov filling radii, and interpolating between 1-Laplacians and 2-Laplacians.
result Developed a general theory for p-Laplacians on simplicial complexes and proved Cheeger-type inequalities.
Proves a quantitative index theorem for positive scalar curvature metrics.
problem Studying conjectures and open questions on positive scalar curvature.
method Quantitative relative index theorem and λ-Lipschitz rigidity theorem. result Positive answers to Gromov's open questions on scalar curvature.
Upper bound found for systolic geometry on manifolds with positive scalar curvature.
problem Relationship between systolic geometry and positive scalar curvature.
method Spinorial methods combined with geometric measure theory and curvature estimates.
result Upper bound for the two-dimensional stable systole on certain manifolds.
Gromov's Conjecture states that for a closed n-manifold M with positive scalar curvature the macroscopic dimension of its universal covering M~ satisfies the inequality dimmcM~≤n−2\cite{G2}. We prove this inequality for totally non-spin n-manifolds whose fundamental group is a virtual duali…
Shows flexible sheaves as fibrant objects for Gromov's h-principle.
problem Applying the h-principle to partial differential relations.
method Interprets flexible sheaves as fibrant objects in a model structure.
result Flexible sheaves can be understood as fibrant objects.
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.
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.
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…
We revisit classical eigenvalue inequalities due to Buser, Cheng, and Gromov on closed Riemannian manifolds, and prove the versions of these results for the Dirichlet and Neumann boundary value problems. Eigenvalue multiplicity bounds and related open problems are also discussed.
Constructs foliations for 3-manifolds with positive scalar curvature.
problem Finding surfaces in 3-manifolds with positive scalar curvature.
method Singular foliations of compact three-manifolds with controlled properties.
result Extends Urysohn and Gromov-Lawson waist inequalities.
We discuss the behavior of (λ1.p(M))1/p with respect to the Gromov-Hausdorff topology and the variable p, where λ1,p(M) is the first positive eigenvalue of the p-Laplacian on a compact Riemannian manifold M. Applications include new estimates for the first eigenvalues of the p-Laplacian on Rieman…
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.
We define Dirichlet type series associated with homology length spectra of Riemannian, or Finsler, manifolds, or polyhedra, and investigate some of their analytical properties. As a consequence we obtain an inequality analogous to Gromov's classical intersystolic inequality, but taking the whole homology length spectru…
We give a general lower bound for the normal Gromov norm of genuine laminations in terms of the topology of the complementary regions. In the special case of 3-manifolds, this yields a generalization of Agol's inequality from incompressible surfaces to tight laminations. In particular, the inequality excludes the exist…
Note on relation between K-cowaist and A-cowaist invariants.
problem Relation between K-cowaist and A-cowaist invariants on manifolds.
method Detailed proof of inequality involving these invariants.
result Established inequality K-cw2(M) ≤ c * A-cw2(M).
We define a capacity which measures the size of Weinstein tubular neighbourhoods of Lagrangian submanifolds. In symplectic vector spaces this leads to bounds on the codisc radius for any closed Lagrangian submanifold in terms of Viterbo's isoperimetric inequality. Moreover, we prove a generalization of Gromov's packing…
New example shows non-compact manifolds can lack Lp-Calderón-Zygmund inequalities.
problem Exploring Lp-Calderón-Zygmund inequalities on non-compact manifolds. method Developed a concrete example using local deformations of metrics.
result Found a non-compact manifold without Lp-Calderón-Zygmund inequalities. We prove that on a Riemannian manifold, a smooth differential form has a primitive with a given (functional) upper bound provided the necessary weighted isoperimetric inequalities implied by Stokes are satisfied. We apply this to prove a comparison predicted by Gromov between the cofilling function and the filling area…
Paper shows limits of Heisenberg manifolds are flat tori.
problem Understanding limits of sub-Riemannian Heisenberg manifolds.
method Analyzes collapsed Gromov--Hausdorff limits of compact Heisenberg manifolds.
result Collapsed limits are isometric to flat tori.
Study of irreversible metric-measure spaces, proving convergence and stability results.
problem Understanding Gromov-Hausdorff convergence and stability in noncompact irreversible metric-measure spaces.
method Introducing a nondecreasing function to bound reversibility of larger balls, proving convergence/stability results in Gromov-Hausdorff topology.
result Satisfactory convergence/stability results in Gromov-Hausdorff topology for various irreversible spaces, including Finsler manifolds.
In his work on singularities, expanders and topology of maps, Gromov showed, using isoperimetric inequalities in graded algebras, that every real valued map on the n-torus admits a fibre whose homological size is bounded below by some universal constant depending on n. He obtained similar estimates for maps with va…
The report explores conditions for positive or non-negative scalar curvature in 3-manifolds and weak Ricci curvature bounds.
problem Conditions for positive or non-negative scalar curvature in 3-manifolds and weak Ricci curvature bounds on non-smooth spaces.
method Description of results in dimension 3, exploration of weak forms of Ricci curvature, use of volume entropy and Bishop-Gromov inequality.
result Recent results on weak Ricci curvature bounds and conditions for positive or non-negative scalar curvature in 3-manifolds.
Let X be a finite 2-complex with unfree fundamental group. We prove lower bounds for the area of a metric on X, in terms of the square of the least length of a noncontractible loop in X. We thus establish a uniform systolic inequality for all unfree 2-complexes. Our inequality improves the constant in M. Gromov's inequ…
Proves the Weyl law for 1-cycles in manifolds.
problem Proving the Weyl law for the volume spectrum of 1-cycles in n-dimensional manifolds.
method Using parametric versions of the coarea inequality and isoperimetric inequality, along with a localized approximation method.
result Proves the Weyl law for 1-cycles in manifolds.
Sharp inequalities for Kähler manifolds' systolic invariants are established.
problem Understanding systolic invariants of Kähler manifolds.
method Analyzing metrics with positive scalar curvature on Kähler manifolds and their products.
result Bounds for systolic invariants attain equality for specific manifolds.
We give the definition of Lp-convergence of tensor fields with respect to the Gromov-Hausdorff topology and several fundamental properties of the convergence. We apply this to establish a Bochner-type inequality which keeps the term of Hessian on the Gromov-Hausdorff limit space of a sequence of Riemannian manifolds…