Related papers: Medium-scale curvature at larger radii in finitely…

We investigate the topology of the compact hypersurfaces in round spheres whose Ricci curvature satisfies an appropriate bound that only depends on the mean curvature of the submanifold. In this paper, the use of the Bochner technique…

The investigation of Ricci solitons is the focus of this work. We have proved triviality results for compact gradient Ricci soliton under certain restriction. Later, a rigidity result is derived for a compact gradient shrinking Ricci…

In this work, we investigate compact K\"ahler manifolds with non-negative or quasi-positive mixed curvature coming from a linear combination of the Ricci and holomorphic sectional curvature, which covers various notions of curvature…

For Einstein four-manifolds with positive scalar curvature, we derive relations among various positivity conditions on the curvature tensor, some of which are of great importance in the study of the Ricci flow. These relations suggest…

We are concerned with the study of different notions of curvature on graphs. We show that if a graph has stronger inner-outer curvature growth than a model graph, then it has faster volume growth too. We also study the relationhips of…

In this paper we generalize the monotonicity formulas of [C] for manifolds with nonnegative Ricci curvature. Monotone quantities play a key role in analysis and geometry; see, e.g., [A], [CM1] and [GL] for applications of monotonicity to…

In this survey article, we present some panorama of groups acting on metric spaces of non-positive curvature. We introduce the main examples and their rigidity properties , we show the links between algebraic or analytic properties of the…

In this paper, we demonstrate certain curvature estimates on complete non-compact steady and expanding gradient Ricci solitons in higher dimensions. In the expanding case, we prove that if the Ricci curvature decays at least quadratically,…

In this second part of our overview of the different metric curvatures and their various applications, we concentrate on the Ricci curvature and flow for polyhedral surfaces and higher dimensional manifolds, and we largely review our…

In this paper, we introduce the first Aeppli-Chern class for complex manifolds and show that the $(1,1)$- component of the curvature $2$-form of the Levi-Civita connection on the anti-canonical line bundle represents this class. We…

In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature. We show that a complete noncompact manifold with asymptoticaly nonnegative Ricci curvature and sectional…

Let the Ricci curvature of a compact Riemannian manifold be greater, at every point, than the Lie derivative of the metric with respect to some fixed smooth vector field. It is shown that the fundamental group then has only finitely many…

We construct non-trivial elements in the homotopy groups of the observer moduli space of positive sectional curvature metrics on $\mathbb{C}P^n$ and non-trivial elements in the homotopy groups of the observer moduli space of positive scalar…

We study graphs with nonnegative Bakry-\'Emery curvature or Ollivier curvature outside a finite subset. For such a graph, via introducing the discrete Gromov-Hausdorff convergence we prove that the space of bounded harmonic functions is…

We prove that for many degrees in a stable range the homotopy groups of the moduli space of metrics of positive scalar curvature on S^n and on other manifolds are non-trivial. This is achieved by further developing and then applying a…

We derive sub-Riemannian Ricci curvature tensor for sub-Riemannian manifolds. We provide examples including the Heisenberg group, displacement group, and Martinet sub-Riemannian structure with arbitrary weighted volumes, in which we…

We study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds. We construct, in particular, the first classes of manifolds for which these moduli spaces have…

The observer moduli space of Riemannian metrics is the quotient of the space $\mathcal{R}(M)$ of all Riemannian metrics on a manifold $M$ by the group of diffeomorphisms $\mathrm{Diff}_{x_0}(M)$ which fix both a basepoint $x_0$ and the…

We classify all cubic graphs with either non-negative Ollivier-Ricci curvature or non-negative Bakry-\'Emery curvature everywhere. We show in both curvature notions that the non-negatively curved graphs are the prism graphs and the M\"obius…

In this paper, we derive curvature estimates for 4-dimensional complete gradient expanding Ricci solitons with nonnegative Ricci curvature (outside a compact set $K$). More precisely, we prove that the norm of the curvature tensor $Rm$ and…