arXiv research

We study collapsed manifolds with Ricci bounded covering geometry i.e., Ricci curvature is bounded below and the Riemannian universal cover is non-collapsed or consists of uniform Reifenberg points. Via Ricci flows' techniques, we partially extend the nilpotent structural results of Cheeger-Fukaya-Gromov, on collapsed …

The study extends convergence theorems for Ricci-limit spaces with bounded curvature.

Generalizes tools for studying collapsed manifolds to new geometry.

The paper extends a theorem to manifolds with local Ricci bounded covering geometry.

Study collapsing geometry with Ricci curvature, proving Kähler metrics and Killing structures.

The study defines a canonical nilpotent structure for certain collapsed manifolds.

Study the topology of Ricci limit spaces using Gromov-Hausdorff limits.

We study the long time behaviour of Ricci flow with bubbling-off on a possibly noncompact $3$-manifold of finite volume whose universal cover has bounded geometry. As an application, we give a Ricci flow proof of Thurston's hyperbolisation theorem for $3$-manifolds with toral boundary that generalizes Perelman's proof …

We study various covering spectra for complete noncompact length spaces with universal covers (including Riemannian manifolds and the pointed Gromov Hausdorff limits of Riemannian manifolds with lower bounds on their Ricci curvature). We relate the covering spectrum to the (marked) shift spectrum of such a space. We de…

New principles prove precompactness of domains with lower Ricci curvature bound.

Torus covers have controlled volume and diameter under curvature and diameter bounds.

The energy of any $C^1$ representative of a homotopy class of maps from a compact and connected Riemannian manifold with nonnegative Ricci curvature into a complete Riemannian manifold with no conjugate points is bounded below by a constant determined by the asymptotic geometry of the target, with equality if and only …

Sharp upper diameter limit found for Ricci solitons.

We study the size of the isometry group Isom(M, g) of Riemannian manifolds (M, g) as g varies. For M not admitting a circle action, we show that the order of Isom(M, g) can be universally bounded in terms of the bounds on Ricci curvature, diameter, and injectivity radius of M. This generalizes results known for negativ…

Study reveals fundamental group properties of manifolds with specific curvature and growth.

Ricci flow stability on manifolds with bounded geometry ensures convergence to hyperbolic metrics.

Study shows no new eigenvalues in specific finite coverings.

The paper proves properties of Lipschitz spacetimes with bounded Ricci curvature.

Unified study of Riemannian and sub-Riemannian geometries with synthetic Ricci curvature bounds.

We analyze Lorentzian spacetimes subject to curvature-dimension bounds using the Bakry-Émery-Ricci tensor. We extend the Hawking-Penrose type singularity theorem and the Lorentzian timelike splitting theorem to synthetic dimensions $N\le 1$, including all negative synthetic dimensions. The rigidity of the timelike spli…

Let (M,g) be a complete noncompact riemannian manifold with bounded geometry and parallel Ricci curvature. We show that some operators, "affine" relatively to the Ricci curvature, are locally invertible, in some classical Sobolev spaces, near the metric g.

The study shows how nonnegative Ricci curvature and metric cones imply the existence of abelian subgroups in the fundamental group of open manifolds.

We prove that if $Y$ is the Gromov-Hausdorff limit of a sequence of complete manifolds, $M^n_i$, with a uniform lower bound on Ricci curvature then $Y$ has a universal cover.

We prove that complete Riemannian manifolds with polynomial growth and Ricci curvature bounded from below, admit uniform Poincaré inequalities. A global, uniform Poincaré inequality for horospheres in the universal cover of a closed, $n$-dimensional Riemannian manifold with pinched negative sectional curvature follows …

Manifolds with nonnegative Ricci curvature have nilpotent subgroups with specific geometric properties.

The paper proves rigidity results for Einstein manifolds with specific geometric constraints.

Study compares manifolds with boundary under weighted Ricci curvature bounds.

Let $(M,g)$ be a compact manifold with Ricci curvature almost bounded from below and $π:\bar{M}\to M$ be a normal, Riemannian cover. We show that, for any nonnegative function $f$ on $M$, the means of $føπ$ on the geodesic balls of $\bar{M}$ are comparable to the mean of $f$ on $M$. Combined with logarithmic volume est…

Global regularity proved for 4D Ricci flow with scalar curvature integral bound.

This paper proves a new, more precise version of Cheng's maximal diameter theorem for manifolds with positive Ricci curvature.

This survey introduces synthetic timelike Ricci curvature bounds in Lorentzian spaces.

For a Riemannian covering $π\colon M_1\to M_0$, the bottoms of the spectra of $M_0$ and $M_1$ coincide if the covering is amenable. The converse implication does not always hold. Assuming completeness and a lower bound on the Ricci curvature, we obtain a converse under a natural condition on the spectrum of $M_0$.

In the study manifolds of Ricci curvature bounded below, a stumbling obstruction is the lack of links between large-scale geometry and small-scale geometry at a fixed reference point. There have been few links (volume, dimension) when the unit ball at the point is not collapsed, that is, $\mathrm{vol}(B_1(p))\ge v>0$. …

This is the second paper of two in a series under the same title ([CRX]); both study the quantitative volume space form rigidity conjecture: a closed $n$-manifold of Ricci curvature at least $(n-1)H$, $H=\pm 1$ or $0$ is diffeomorphic to a $H$-space form if for every ball of definite size on $M$, the lifting ball on th…

The paper proves a quantitative rigidity result for spaces with specific curvature bounds.

Study compares spectral properties of a specific tensor in geometry.

The study examines rigidity properties of noncompact manifolds with nonnegative Ricci curvature.

Let $(M^n, g)$ be a compact $n$-dim ($n\geq 2$) manifold with nonnegative Ricci curvature, and if $n\geq 3$ we assume that $(M^n, g)\times \mathbb{R}$ has nonnegative isotropic curvature. The lower bound of the Ricci flow's existence time on $(M^n, g)$ is proved. This provides an alternative proof for the uniform lower…

The paper proves a gap theorem for almost non-negatively curved manifolds.

In this lecture notes, we aim at giving an introduction to the Kähler-Ricci flow (KRF) on Fano manifolds. It covers some of the developments of the KRF in its first twenty years (1984-2003), especially an essentially self-contained exposition of Perelman's uniform estimates on the scalar curvature, the diameter, and th…

Estimates Kähler metric diameters with entropy bound alone.

In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized Ricci flow of model geometries. In every Bianchi class, we estimate the derivative of the eigenvalue. Then we construct monotonic quantities under the Ricci flow and obtain upper and lower bounds for…

Study on 4D compact Ricci solitons and their geometric properties.

Comparison theorems in centro-affine differential geometry

The study proposes conjectures on limit spaces of Riemannian manifolds with Ricci curvature.

We study Riemannian manifolds with boundary under a lower weighted Ricci curvature bound. We consider a curvature condition in which the weighted Ricci curvature is bounded from below by the density function. Under the curvature condition, and a suitable condition for the weighted mean curvature for the boundary, we ob…

Overview of recent results on isoperimetric inequalities on manifolds with Ricci lower bounds.

Extends Ricci flow theory to Kato-type curvature bounds, proving manifold properties.