Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCurvature bounds via Ricci smoothing
69
0
0.0
(
0
)
Authors
Vitali Kapovitch
Ask ChatGPT about the research
No Arabic abstract
We give a proof of the fact that the upper and the lower sectional curvature bounds of a complete manifold vary at a bounded rate under the Ricci flow.
rate research
Read More
We survey some recent developments in the study of collapsing Riemannian manifolds with Ricci curvature bounded below, especially the locally bounded Ricci covering geometry and the Ricci flow smoothing techniques. We then prove that if a Calabi-Yau manifold is sufficiently volume collapsed with bounded diameter and sectional curvature, then it admits a Ricci-flat Kahler metrictogether with a compatible pure nilpotent Killing structure: this is related to an open question of Cheeger, Fukaya and Gromov.
Measure contraction property is a synthetic Ricci curvature lower bound for metric measure spaces. We consider Sasakian manifolds with non-negative Tanaka-Webster Ricci curvature equipped with the metric measure space structure defined by the sub-Riemannian metric and the Popp measure. We show that these spaces satisfy the measure contraction property $MCP(0,N)$ for some positive integer $N$. We also show that the same result holds when the Sasakian manifold is equipped with a family of Riemannian metrics extending the sub-Riemannian one.
In this paper we analyze the behavior of the distance function under Ricci flows whose scalar curvature is uniformly bounded. We will show that on small time-intervals the distance function is $frac12$-Holder continuous in a uniform sense. This implies that the distance function can be extended continuously up to the singular time.
For Riemannian manifolds with a smooth measure $(M, g, e^{-f}dv_{g})$, we prove a generalized Myers compactness theorem when Bakry--Emery Ricci tensor is bounded from below and $f$ is bounded.
In this paper we study regularity and topological properties of volume constrained minimizers of quasi-perimeters in $sf RCD$ spaces where the reference measure is the Hausdorff measure. A quasi-perimeter is a functional given by the sum of the usual perimeter and of a suitable continuous term. In particular, isoperimetric sets are a particular case of our study. We prove that on an ${sf RCD}(K,N)$ space $({rm X},{sf d},mathcal{H}^N)$, with $Kinmathbb R$, $Ngeq 2$, and a uniform bound from below on the volume of unit balls, volume constrained minimizers of quasi-perimeters are open bounded sets with $(N-1)$-Ahlfors regular topological boundary coinciding with the essential boundary. The proof is based on a new Deformation Lemma for sets of finite perimeter in ${sf RCD}(K,N)$ spaces $({rm X},{sf d},mathfrak m)$ and on the study of interior and exterior points of volume constrained minimizers of quasi-perimeters. The theory applies to volume constrained minimizers in smooth Riemannian manifolds, possibly with boundary, providing a general regularity result for such minimizers in the smooth setting.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
