arXiv research

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 article disproves a local systolic inequality and shows a lower bound on filling area.

Researchers prove an inequality linking spin manifold fill-ins to hyperspherical radius.

Improved bounds on curve filling areas in Banach spaces, leading to rigidity of Pu's inequality.

We investigate the filling area conjecture, optimal systolic inequalities, and the related problem of the nonvanishing of certain linking numbers in 3-manifolds.

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 $L^\infty$-spaces. This inequality is used in the proof of Gromov's systolic inequality for closed aspherical Riemannian manifolds and is often regarded as the…

Inspired by Gromov's work on 'Metric inequalities with scalar curvature' we establish band width inequalities for Riemannian bands of the form $(V=M\times[0,1],g)$, where $M^{n-1}$ is a closed manifold. We introduce a new class of orientable manifolds we call filling enlargeable and prove: If $M$ is filling enlargeable…

The paper examines slopes and their norms in exceptional Dehn fillings.

Constructs fill-ins with scalar curvature lower bounds for geometric applications.

Proves upper bound on systolic ratio for circle fillings.

We adapt the theory of currents in metric spaces, as developed by the first-mentioned author in collaboration with B. Kirchheim, to currents with coefficients in Z_p. Building on S. Wenger's work in the orientable case, we obtain isoperimetric inequalities mod(p) in Banach spaces and we apply these inequalities to prov…

Study 2-complexes' homology properties and torsion growth.

Solved Cheeger inequalities for simplicial complexes, combining topological and graph theoretic methods.

New rigidity theorems for spin fill-ins with non-negative scalar curvature.

We bound the higher-order Dehn functions and other filling invariants of certain Carnot groups using approximation techniques. These groups include the higher-dimensional Heisenberg groups, jet groups, and central products of two-step nilpotent groups. Some consequences of this work are a construction of groups with ar…

Study gives bounds on filling radius for Riemannian manifolds.

Paper proves flat 3-manifolds with positive mass have unique isoperimetric surfaces.

We use algebraic techniques to study homological filling functions of groups and their subgroups. If $G$ is a group admitting a finite $(n+1)$--dimensional $K(G,1)$ and $H \leq G$ is of type $F_{n+1}$, then the $n^{th}$--homological filling function of $H$ is bounded above by that of $G$. This contrast with known examp…

Linear inequality found for certain curved spaces.

Filling invariants are measurements of a metric space describing the behaviour of isoperimetric inequalities. In this article we examine filling functions and higher divergence functions. We prove for a class of stratified nilpotent Lie groups that in the low dimensions the filling functions grow as fast as the ones of…

Study lens spaces' definite fillings, classifying those with specific inequalities.

Study bounds total mean curvature of fill-ins with scalar curvature constraints.

We prove that every Riemannian metric on the 2-disc such that all its geodesics are minimal, is a minimal filling of its boundary (within the class of fillings homeomorphic to the disc). This improves an earlier result of the author by removing the assumption that the boundary is convex. More generally, we prove this r…

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…

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…

A pair $(α, β)$ of simple closed geodesics on a closed and oriented hyperbolic surface $M_g$ of genus $g$ is called a filling pair if the complementary components of $α\cupβ$ in $M_g$ are simply connected. The length of a filling pair is defined to be the sum of their individual lengths. In \cite{Aou}, Aougab-Huang con…

We derive new inequalities between the boundary capacity of an asymptotically flat 3-manifold with nonnegative scalar curvature and boundary quantities that relate to quasi-local mass; one relates to Brown--York mass and the other is new. We argue by recasting the setup to the study of mean-convex fill-ins with nonnega…

The main subject of this expository paper is a connection between Gromov's filling volumes and a boundary rigidity problem of determining a Riemannian metric in a compact domain by its boundary distance function. A fruitful approach is to represent Riemannian metrics by minimal surfaces in a Banach space and to prove r…

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…

Algorithm finds minimal volume hyperbolic links in 3-manifolds.

Proves homological inequality for cycles in Hadamard spaces of asymptotic rank 2.

New findings on metric spaces with finite Nagata dimension.

New problems on NNSC fill-ins for Bartnik data in high dimensions.

Sharp inequality for compactifying Poincaré-Einstein manifolds.

The free factor complex of rank 4+ fails a combinatorial isoperimetric inequality.

The paper shows how coarse embeddings affect homological Dehn functions.

Develops new oracle inequalities for Gaussian ranking estimators.

The study proves a geometric inequality for surfaces with genus G.

Proves a generalized isoperimetric inequality for spheres in dimensions 4 and above.

We show that the Cheeger constant of compact surfaces is bounded by a function of the area. We apply this to isoperimetric profiles of bounded genus non-compact surfaces, to show that if their isoperimetric profile grows faster than $\sqrt t$, then it grows at least as fast as a linear function. This generalizes a resu…

Paper improves risk bound for MTL with graph-dependent data.

In this paper we provide a framework for the study of isoperimetric problems in finitely generated group, through a combinatorial study of universal covers of compact simplicial complexes. We show that, when estimating filling functions, one can restrict to simplicial spheres of particular shapes, called "round" and "u…

Let $X=G/K$ be a symmetric space of noncompact type and rank $k\ge 2$. We prove that horospheres in $X$ are Lipschitz $(k-2)$--connected if their centers are not contained in a proper join factor of the spherical building of $X$ at infinity. As a consequence, the distortion dimension of an irreducible $\mathbb{Q}$--ran…

We prove a new version of isoperimetric inequality: Given a positive real $m$, a Banach space $B$, a closed subset $Y$ of metric space $X$ and a continuous map $f:Y \rightarrow B$ with $f(Y)$ compact $\inf_FHC_{m+1}(F(X))\leq c(m)HC_m(f(Y))^{\frac{m+1}{m}},$ where $HC_m$ denotes the $m$-dimensional Hausdorff content,…

Paper shows equivalence of two curvature notions on singular surfaces.

The paper provides a new inequality for 4-manifolds and uses it to study knot sliceness and symplectic embeddings.

New spectral Dehn function characterizes word-hyperbolic groups.

We improve bounds for stochastic processes, especially those with heavy tails.