arXiv research

Extends Gromov's theorem with amenable covers.

Paper proves vanishing homology groups for certain hyperbolic groups.

The abstract discusses p-harmonic forms and their geometric properties, proving new theorems about Lp-cohomology.

We provide a proof for an inequality between volume and L2-Betti numbers of aspherical manifolds for which Gromov outlined a strategy based on general ideas of Connes. The implementation of that strategy involves measured equivalence relations, Gaboriau's theory of L2-Betti numbers of R-simplicial complexes, and other …

Study $L^{2}$-harmonic forms on almost Kähler manifolds, extending vanishing theorems.

The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in 1982. In order to study its main properties, Gromov himself initiated the dual theory of bounded cohomology, that developed into an active and independent research field. Gromov's theory of bounded cohomology was based on the use of mult…

New proofs and refined theorems on bounded cohomology.

Study on symmetric operators on non-compact manifolds, focusing on their index modulo 2.

Study on scalar curvature bounds and manifold topological complexity.

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

Stability of positive mass theorem proven under Ricci curvature bounds.

We shall prove a new non-vanishing theorem for the stable cohomotopy Seiberg-Witten invariant of connected sums of 4-manifolds with positive first Betti number. The non-vanishing theorem enables us to find many new examples of 4-manifolds with non-trivial stable cohomotopy Seiberg-Witten invariants and it also gives a …

We extend the deep and important results of Lichnerowicz, Connes, and Gromov-Lawson which relate geometry and characteristic numbers to the existence and non-existence of metrics of positive scalar curvature (PSC). In particular, we show: that a spin foliation with Hausdorff homotopy groupoid of an enlargeable manifold…

In this paper, we study some vanishing identities for Gromov-Witten invariants conjectured by K. Liu and H. Xu. We will prove these conjectures in the case that the summation range is large compare to genus. In fact, in such cases, we can obtain a vanishing identity which is stronger than their conjectures. Moreover we…

Convex hypersurfaces in curved spaces bound convex regions.

Paper answers Gromov's compactness question on noncompact manifolds.

We study a metric version of the simplicial volume on Riemannian manifolds, the Lipschitz simplicial volume, with applications to degree theorems in mind. We establish a proportionality principle and a product inequality from which we derive an extension of Gromov's volume comparison theorem to products of negatively c…

We generalize classical theorems due to Lichnerowicz and Hitchin on the existence of Riemannian metrics of positive scalar curvature on spin manifolds to the case of foliated spin manifolds. As a consequence, we show that there is no foliation of positive leafwise scalar curvature on any torus, which generalizes the fa…

We revisit the construction of signature classes in C*-algebra K-theory, and develop a variation that allows us to prove equality of signature classes in some situations involving homotopy equivalences of noncompact manifolds that are only defined outside of a compact set. As an application, we prove a counterpart for …

New theorem using Ricci flow for Gromov almost flat manifolds.

New topological restrictions found for spaces with nonnegative Ricci curvature.

The study proves stability of the positive mass theorem for Kähler manifolds.

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

We prove that the first reduced cohomology with values in a mixing Lp-representation, p larger than 1, vanishes for a class of amenable groups including connected amenable Lie groups. In particular this solves for this class of amenable groups a conjecture of Gromov saying that every finitely generated amenable group h…

Study proves Hirzebruch genus inequality for almost Kähler manifolds with negative curvature.

Paper proves Gromov's conjecture on manifolds with certain group properties.

We show in this short note that if a rational linear combination of Pontrjagin numbers vanishes on all simply-connected $4k$-dimensional closed connected and oriented spin manifolds admitting a Riemannian metric whose Ricci curvature is nonnegative and nonzero at any point, then this linear combination must be a multip…

Let $E$ be a Hermitian vector bundle over a complete Kähler manifold $(X,ω)$, $\dim_{\mathbb{C}}X=n$, with a $d$(bounded) Kähler form $ω$, $d_{A}$ be a Hermitian connection on $E$. The goal of this article is to study the $L^{2}$-Hodge theory on the vector bundle $E$. We extend the results of Gromov's \cite{Gro} to the…

Let $M$ be a closed symplectic manifold of dimension $2n$ with non-ellipticity. We can define an almost Kähler structure on $M$ by using the given symplectic form. Hence, we have a $\G=π_1(M)$-invariant almost Kähler structure on the universal covering, $\ti M$, of $M$. Using Darboux coordinate charts, we globally defo…

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.

We prove a Bochner type vanishing theorem for compact complex manifolds $Y$ in Fujiki class $\mathcal C$, with vanishing first Chern class, that admit a cohomology class $[α] \in H^{1,1}(Y,\mathbb R)$ which is numerically effective (nef) and has positive self-intersection (meaning $\int_Y α^n \,>\, 0$, where $n\,=\,\di…

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

Study Gromov-Hausdorff convergence of metric pairs and tuples.

The paper develops $L^2$-Hodge theory on almost Kähler manifolds and proves the Hopf conjecture.

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…

Analyzes Saito vanishing theorem using $L^{2}$ methods.

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

Study shows local topologies of certain geometric spaces.

Study on simplicial volume and Euler characteristic of aspherical manifolds.

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

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.

In this work we prove convergence results of sequences of Riemannian $4$-manifolds with almost vanishing $L^2$-norm of a curvature tensor and a non-collapsing bound on the volume of small balls. In Theorem 1.1, we consider a sequence of closed Riemannian $4$-manifolds, whose $L^2$-norm of the Riemannian curvature tenso…

The paper reconstructs Lorentzian spacetimes from causal sets.

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…