arXiv research

Proves compactness for timed-metric spaces using new distance and maps.

New theorem on flat tori stability using harmonic maps and Ricci flow.

It is well known that quasi-isometric embeddings of Gromov hyperbolic spaces induce topological embeddings of their Gromov boundaries. A more general question is to detect classes of functions between Gromov hyperbolic spaces that induce continuous maps between their Gromov boundaries. In this paper we introduce the cl…

Constructs Serre spectral sequence for bounded cohomology.

We propose localization techniques for computing Gromov-Witten invariants of maps from Riemann surfaces with boundaries into a Calabi-Yau, with the boundaries mapped to a Lagrangian submanifold. The computations can be expressed in terms of Gromov-Witten invariants of one-pointed maps. In genus zero, an equivariant ver…

The paper connects geometric and topological concepts to bound distances between metric spaces.

Thanks to a theorem of Brock on comparison of Weil-Petersson translation distances and hyperbolic volumes of mapping tori for pseudo-Anosovs, we prove that the entropy of a surface automorphism in general has linear bounds in terms of Gromov norm of its mapping torus from below and in bounded geometry case from above. …

We study Thom Transversality Theorem using a point of view, suggested by Gromov, which allows to avoid the use of Sard Theorem and gives finer informations on the structure of the set of non-transverse maps.

New proofs and refined theorems on bounded cohomology.

Defines super stable maps and proves quotient superorbifolds for genus zero.

A theorem on odd dimensional noncompact manifolds shows curvature bounds.

We give a proof of the Gromov compactness theorem using the language of stable curves (i.e. cusp-curve of Gromov, or stable maps of Kontsevich and Manin) in general setting: An almost complex structure on a target manifold is only continuous and can vary; the curves are only assumed to have fixed ``topological type'', …

By Gromov's mapping theorem for bounded cohomology, the projection of a group to the quotient by an amenable normal subgroup is isometric on group homology with respect to the $\ell^1$-semi-norm. Gromov's description of the diffusion of cycles also implicitly produces efficient cycles in this situation. We present an e…

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…

New theorem using Ricci flow for Gromov almost flat manifolds.

Develops analysis of Hölder continuous mappings on Heisenberg groups.

Proves regularity of harmonic maps into Euclidean buildings and applies to superrigidity of algebraic groups.

Survey on random walks on mapping class groups and their properties.

Random walks on hyperbolic spaces show linear growth in translation lengths.

Let $k$ and $k'$ be two knots in 3-sphere. Say $k$ 1--dominates $k'$, if there is a proper degree 1 map $f\co E(k)\to E(k')$, between knot exterior of $k_i$. Theorem: Suppose that any companion of $k$ is prime. If $k$ 1--dominates $k'$ with the same Gromov volume, then $k'$ can be obtained from $k$ by finitely many de-…

We prove a version of Gromov's compactness theorem for pseudo-holomorphic curves which holds locally in the target symplectic manifold. This result applies to sequences of curves with an unbounded number of free boundary components, and in families of degenerating target manifolds which have unbounded geometry (e.g. no…

Proof of Gromov's theorem on convex polytopes with acute angles.

The study proves super-rigidity of Gromov's random monster group for various types of groups.

We give a version of Gromov's compactess theorem for pseudoholomorphic curves in the case of quasiregular mappings between closed manifolds. More precisely we show that, given $K\ge 1$ and $D\ge 1$, any sequence $(f_n \colon M \to N)$ of $K$-quasiregular mappings of degree $D$ between closed Riemannian $d$-manifolds ha…

Extends Gromov's theorem with amenable covers.

In the 1970s and again in the 1990s, Gromov gave a number of theorems and conjectures motivated by the notion that the real homotopy theory of compact manifolds and simplicial complexes influences the geometry of maps between them. The main technical result of this paper supports this intuition: we show that maps of di…

Two new proofs of Gromov's non-squeezing theorem using curve reparametrization and gradient bounds.

New proof shows no Hölder embeddings into Heisenberg group.

Proves a quantitative index theorem for positive scalar curvature metrics.

Extended Otal-Peigné's Theorem to Gromov-hyperbolic spaces.

Study Gromov-Hausdorff convergence of metric pairs and tuples.

The paper extends a vanishing theorem for hypersurfaces in aspherical manifolds.

Enhanced Bishop-Gromov theorem for homogeneous and inhomogeneous spaces.

In 1969 M. Gromov in his PhD thesis greatly generalized Smale-Hirsch-Phillips immersion-submersion theory by proving what is now called the h-principle for invariant open differential relations over open manifolds. Gromov extracted the original geometric idea of Smale and put it to work in the maximal possible generali…

Compactness theorem for quasiregular curves proves normality and resolves nodal points.

Prove Gromov's Euclidean endpoint $C^0$ rigidity conjecture for positive mass theorem.

New proof of Llarull's rigidity theorem in odd dimensions via spectral flow.

This is the second of three papers about the Compression Theorem. We give proofs of Gromov's theorem on directed embeddings [M Gromov, Partial differential relations, Springer--Verlag (1986); 2.4.5 C'] and of the Normal Deformation Theorem [The compression theorem I; 4.7], arxiv:math.GT/9712235.

The paper reconstructs Lorentzian spacetimes from causal sets.

In a recent paper~\cite{DDL10} we studied basic properties of partial immersions and partially free maps, a generalization of free maps introduced first by Gromov in~\cite{Gro70}. In this short note we show how to build partially free maps out of partial immersions and use this fact to prove that the partially free map…

This note explores comparison geometry concepts and theorems.

The Gromov-Eliashberg theorem says that the group of symplectomorphisms of a symplectic manifold is C^0-closed in the group of diffeomorphisms. This can be translated into a statement about the Lagrangian submanifolds which are graphs of symplectomorphisms. It is also known that such Lagrangian submanifolds are locally…

Researchers describe the Gromov boundary of a graph related to surfaces.

We will give a new proof for the Gromov's theorem on almost flat manifolds, which is an inductive proof on dimension.

Compactness theorem for timed-metric spaces established.

Study uses equivariant topology to measure distances between G metric spaces.

By Cheeger-Colding's almost splitting theorem, if a domain in a Ricci flat manifold is pointed-Gromov-Hausdorff close to a lower dimensional Euclidean domain, then there is a harmonic almost splitting map. We show that any eigenfunction of the Laplace operator is almost constant along the fibers of the almost splitting…

Improved non-squeezing theorem for calibrated geometries proved.