Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userExamples of Ricci limit spaces with non-integer Hausdorff dimension
236
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We give the first examples of collapsing Ricci limit spaces on which the Hausdorff dimension of the singular set exceeds that of the regular set; moreover, the Hausdorff dimension of these spaces can be non-integers. This answers a question of Cheeger-Colding about collapsing Ricci limit spaces.
rate research
Read More
We construct a global homeomorphism from any 3D Ricci limit space to a smooth manifold, that is locally bi-Holder. This extends the recent work of Miles Simon and the second author, and we build upon their techniques. A key step in our proof is the construction of local pyramid Ricci flows, existing on uniform regions of spacetime, that are inspired by Hochards partial Ricci flows.
We prove that in two dimensions the synthetic notions of lower bounds on sectional and on Ricci curvature coincide.
A Riemannian n-manifold M has k-dimensional Uryson width bounded by a constant c >0 if there exists a continuous map f from M to an k-dimensional polyhedral space P, such that the pullbacks f^{-1}(p) of all points p in P have diameters bounded by c. We prove that an n-dimensional Riemannian manifold M with at least n-k eigenvalues of the Ricci curvature bounded below by a positive constant (n-1)b has k-dimensional Uryson width bounded by a constant c >0. The constant c depends only on b. In particular, it follows that a Riemannian n-manifold M with scalar curvature S bounded below by a positive constant n (n-1) s has (n-1)-dimensional Uryson width bounded by a constant c >0 depending only on s. This result confirms a conjecture of M. Gromov.
We show that the space of metrics of positive scalar curvature on any 3-manifold is either empty or contractible. Second, we show that the diffeomorphism group of every 3-dimensional spherical space form deformation retracts to its isometry group. This proves the Generalized Smale Conjecture. Our argument is independent of Hatchers theorem in the $S^3$ case and in particular it gives a new proof of the $S^3$ case.
We study closed three-dimensional Alexandrov spaces with a lower Ricci curvature bound in the $mathsf{CD}^*(K,N)$ sense, focusing our attention on those with positive or nonnegative Ricci curvature. First, we show that a closed three-dimensional $mathsf{CD}^*(2,3)$-Alexandrov space must be homeomorphic to a spherical space form or to the suspension of $mathbb{R}P^2$. We then classify closed three-dimensional $mathsf{CD}^*(0,3)$-Alexandrov spaces.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
