Related papers: Intermediate Ricci curvatures and Gromov's Betti n…

The comparison theory for the Riccati equation satisfied by the shape operator of parallel hypersurfaces is generalized to semi-Riemannian manifolds of arbitrary index, using one-sided bounds on the Riemann tensor which in the Riemannian…

For a noncollapsed Gromov-Hausdorff convergent sequence of Riemannian manifolds with a uniform bound of Ricci curvature, we establish two spectral convergence. One of them is on the Hodge Laplacian acting on differential one-forms. The…

Let $M$ be a domain enclosed between two principal orbits on a cohomogeneity one manifold $M_1$. Suppose $T$ and $R$ are symmetric invariant (0,2)-tensor fields on $M$ and $\partial M$, respectively. The paper studies the prescribed Ricci…

The Colding-Gromov gap theorem asserts that an almost non-negatively Ricci curved manifold with unit diameter and maximal first Betti number is homeomorphic to the flat torus. In this paper, we prove a parametrized version of this theorem,…

Consider an essentially nonbranching metric measure space with the measure contraction property of Ohta and Sturm, or with a Ricci curvature lower bound in the sense of Lott, Sturm and Villani. We prove a sharp upper bound on the inscribed…

We study the boundary behaviors of a complete conformal metric which solves the $\sigma_k$-Ricci problem on the interior of a manifold with boundary. We establish asymptotic expansions and also $C^1$ and $C^2$ estimates for this metric…

We consider Ricci flow invariant cones C in the space of curvature operators lying between nonnegative Ricci curvature and nonnegative curvature operator. Assuming some mild control on the scalar curvature of the Ricci flow, we show that if…

We explore existence of invariant metrics with positive intermediate Ricci curvature on closed, low-dimensional cohomogeneity one manifolds. For a certain cohomogeneity one $\mathsf{Spin}(4)$-action on $S^3 \times \mathbb{C}\mathrm{P}^2$,…

In Riemannian geometry the prescribed Ricci curvature problem is as follows: given a smooth manifold $M$ and a symmetric 2-tensor $r$, construct a metric on $M$ whose Ricci tensor equals $r$. In particular, DeTurck and Koiso proved the…

We study Betti numbers of sequences of Riemannian manifolds which Benjamini-Schramm converge to their universal covers. Using the Price inequalities we developed elsewhere, we derive two distinct convergence results. First, under a negative…

In this paper we obtain a simple upper bound for the infimum of the Ricci curvatures of a complete Riemannian manifold with nonzero injectivity radius i(M) depending only on of the i(M). In case of rigidity the Riemannian manifold must be…

In this paper we address the relationship between Gromov-Hausdorff limits and intrinsic flat limits of complete Riemannian manifolds. In \cite{SormaniWenger2010, SormaniWenger2011}, Sormani-Wenger show that for a sequence of Riemannian…

For Riemannian manifolds with a measure $(M,g, e^{-f} dvol_g)$ we prove mean curvature and volume comparison results when the $\infty$-Bakry-Emery Ricci tensor is bounded from below and $f$ is bounded or $\partial_r f$ is bounded from…

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…

We prove that if $(X,\mathsf{d},\mathfrak{m})$ is a metric measure space with $\mathfrak{m}(X)=1$ having (in a synthetic sense) Ricci curvature bounded from below by $K>0$ and dimension bounded above by $N\in [1,\infty)$, then the classic…

In this paper, we prove fibration theorems for manifolds with almost nonnegative Ricci curvature and certain extra regularity assumptions. We show that a closed $n$-manifold $M$ satisfying $\mathrm{diam}(M)^2\mathrm{sec}_M \geq -\kappa$ and…

We extend a positive Ricci curvature gluing theorem of Perelman to a range of positive intermediate curvature conditions, ranging from positive scalar curvature up to (and including) positive sectional curvature. As an application of this,…

We give new estimates on the lower bounds for the first closed or Neumann eigenvalue for a compact manifold with positive Ricci curvature in terms of the diameter and the lower bound of Ricci curvature. The results improve the previous…

We show that, for any $n\geq 2$, there exists a homogeneous space of dimension $d=8n-4$ with metrics of $\mathrm{Ric}_{\frac{d}{2}-5}>0$ if $n\neq 3$ and $\mathrm{Ric}_6>0$ if $n=3$ which evolve under the Ricci flow to metrics whose Ricci…

We study Riemannian manifolds $(M^n,g)$ with mean-convex boundary whose Ricci curvature is nonnegative in a spectral sense. Our first main result is a sharp spectral extension of a rigidity theorem by Kasue: we prove that under the…