Related papers: Mixed curvature almost flat manifolds

We study the supremum of the total mean curvature on the boundary of compact, mean-convex 3-manifolds with nonnegative scalar curvature, and a prescribed boundary metric. We establish an additivity property for this supremum and exhibit…

We establish a quantitative version of the Gromov compactness theorem for closed genus 0 pseudoholomorphic curves in the setting of a tamed almost complex manifold with bounded geometry.

Following ideas of Gromov we prove scalar and mean curvature comparison results for Riemannian bands with lower scalar curvature bounds in dimension $n\leq7$. The model spaces we use are warped products over scalar-flat manifolds with…

In this paper we study sectional curvature bounds for Riemannian manifolds with density from the perspective of a weighted torsion free connection introduced recently by the last two authors. We develop two new tools for studying weighted…

We obtain generalized Wintgen inequalities for submanifolds in conformally flat manifolds. We give some applications for submanifolds in a Riemannian manifold of quasi-constant curvature. Equality cases are also considered.

Under mild assumptions, we show that a connected weighted graph $G$ with lower Ricci curvature bound $K>0$ in the sense of Bakry-\'Emery and the $d$-th non-zero Laplacian eigenvalue $\lambda_d$ close to $K$, with $d$ being the maximal…

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 a rigidity theorem that shows that, under many circumstances, quasi-isometric embeddings of equal rank, higher rank symmetric spaces are close to isometric embeddings. We also produce some surprising examples of quasi-isometric…

We develop variation formulas for the quantities of extrinsic geometry for adapted variations of metrics on almost-product (e.g. foliated) Riemannian manifolds, and apply them to study the total mixed scalar curvature of a distribution --…

Here I show compatibility of two definition of generalized curvature bounds --- the lower bound for sectional curvature in the sense of Alexandrov and lower bound for Ricci curvature in the sense of Lott--Villani--Sturm.

In this paper, we prove some Willmore-type inequalities for closed hypersurfaces in weighted manifolds with nonnegative Bakry-\'Emery Ricci curvature. In particular, we give a sharp Willmore type inequality in steady gradient Ricci…

We consider almost $\eta$-Ricci solitons in $(LCS)_n$-manifolds satisfying certain curvature conditions. We provide a lower and an upper bound for the norm of the Ricci curvature in the gradient case, derive a Bochner-type formula for an…

We show that the fundamental groups of any two closed irreducible non-geometric graph-manifolds are quasi-isometric. This answers a question of Kapovich and Leeb. We also classify the quasi-isometry types of fundamental groups of…

Jorge-Koutrofiotis and Pigola-Rigoli-Setti proved sharp sectional curvature estimates for extrinsically bounded submanifolds. Alias, Bessa and Montenegro showed that these estimates hold on properly immersed cylindrically bounded…

Inspired by the approach of Ivanisvili and Volberg towards functional inequalities for probability measures with strictly convex potentials, we investigate the relationship between curvature bounds in the sense of Bakry-Emery and local…

For $\rho, v>0$, we say that an $n$-manifold $M$ satisfies local $(\rho,v)$-bound Ricci covering geometry, if Ricci curvature $\text{Ric}_M\ge -(n-1)$, and for all $x\in M$, $\text{vol}(B_\rho(\tilde x))\ge v>0$, where $\tilde x$ is an…

A quantitative version of the scalar lower bound under $C^0$ convergence was conjectured by Gromov. More recently, Mazurowski and Yao proved that a refined form of Gromov's conjecture holds in dimension three. Furthermore, they constructed…

We prove that closed manifolds admitting a generic metric whose sectional curvature is locally quasi-constant are graphs of space forms. In the more general setting of QC spaces where sets of isotropic points are arbitrary, under suitable…

In this paper, we prove a quantitative relative index theorem. It provides a conceptual framework for studying some conjectures and open questions of Gromov on positive scalar curvature. More precisely, we prove a $\lambda$-Lipschitz…

We generalize McCann's theorem of optimal transport to a submanifold setting and prove Michael-Simon-Sobolev inequalities for submanifolds in manifolds with lower bounds on intermediate Ricci curvatures. The results include a variant of the…