Related papers: On Gromov's dihedral extremality and rigidity conj…

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…

We prove a conjecture of Gromov's to the effect that manifolds with isotropic curvature bounded below by 1 (after possibly rescaling) are macroscopically 1-dimensional on the scales greater than 1. As a consequence we prove that compact…

We introduce the concept of boundary rigidity for Gromov hyperbolic spaces. We show that a proper geodesic Gromov hyperbolic space with a pole is boundary rigid if and only if its Gromov boundary is uniformly perfect. As an application, we…

In this article, we develop a new index theory for noncompact manifolds endowed with an admissible exhaustion by compact sets. This index theory allows us to provide examples of noncompact manifolds with exotic positive scalar curvature…

In this paper we introduce two new notions of sectional curvature for Riemannian manifolds with density. Under both notions of curvature we classify the constant curvature manifolds. We also prove generalizations of the theorems of…

We show that closed surfaces with minimal total absolute curvature in Cartan-Hadamard 3-manifolds bound flat convex bodies. This generalizes Chern-Lashof's theorem for surfaces in Euclidean space and solves a problem posed by Gromov in…

We show that the strong cohomological rigidity conjecture for Bott manifolds is true. Namely, any graded cohomology ring isomorphism between two Bott manifolds is induced by a diffeomorphism.

Sectional curvature bounds are of central importance in the study of Riemannian manifolds, both in smooth differential geometry and in the generalized synthetic setting of Alexandrov spaces. Riemannian metrics along with metric spaces of…

We establish enhanced bounds on Cheeger-Gromov rho-invariants for general 3-manifolds and yet stronger bounds for special classes of 3-manifold. As key ingredients, we construct chain null-homotopies whose complexity is linearly bounded by…

We develop a new approach to, and small extension of, results of Cheeger, Colding and Tian concerning the $L^{k/2}$ norm of the curvature of a Riemannian manifold Gromov-Hausdorff close to a codimension $k$ singularity.

We prove some rigidity results for compact manifolds with boundary. In particular for a compact Riemannian manifold with nonnegative Ricci curvature and simply connected mean convex boundary, it is shown that if the sectional curvature…

In this short note we show how the higher index theory can be used to prove results concerning the non-existence of complete riemannian metric with uniformly positive scalar curvature at infinity. By improving some classical results due to…

We use Gromov's K--area to define a generalized homology theory on compact smooth manifolds. In fact, this theory collects obstructions to the enlargeability of the manifold and its nontrivial submanifolds. Moreover, using the K--area…

Discrete analogs of extrema of curvature and generalizations of the four-vertex theorem to the case of polygons and polyhedra are suggested and developed. For smooth curves and polygonal lines in the plane, a formula relating the number of…

In the article we introduce new conformal and smooth invariants on compact, oriented four-manifolds with boundary. In the first part, we show that "positivity" conditions on these invariants will impose topological restrictions on…

We observe after Bayle and Rosales that the Levy-Gromov isoperimetric inequality generalizes to convex manifolds with boundary.

In this paper, we use the viewpoint of Gromov-Haustorff convergence to give some new comprehension of well known theorem,it is Huber's classification theorem\cite{Huber}\cite{MS}for complete Riemannian surfaces immersed in $\mathbb{R}^n$…

We investigate extremal metrics at which various types of rigidity theorems involving scalar curvatures hold. The rigidity we discuss here is related to the rigidity theorems presented by Mario Listing in his previous preprint. More…

The Gromov-Lawson-Rosenberg-conjecture for a group G states that a closed spin manifold M^n (n>4) with fundamental group G admits a metric with positive scalar curvature if and only if its C^*-index A(M) in KO_n(C^*_r(G)) vanishes. We prove…

Inspired by a recent work of Grove-Petersen in [GP18], where the authors studied Alexandrov spaces with largest possible boundary. We study Alexandrov spaces with lower curvature bound 1 and with small boundary. When the radius of X is…