Convergence and Stratification Workshop
May 14, 2015
365 Fifth Avenue, NY NY 10016, Room 4214:03
This impromptu unfunded workshop is aimed at doctoral students and postdocs
working on projects involving Gromov-Hausdorff convergence.
All talks will be blackboard and chalk to allow for interaction.
Registration is free. Please email Christina Sormani to register.
Schedule:
10:00-11:00 P. Solorzano (Universidade Federal de Santa Catarina)
"GH Convergance and Waning"
11:30-12:30 R. Perales (Stony Brook University)
"Glued Limit Spaces of Sequences of Manifolds"
1:00-2:00 M. Munn (New York University)
"Intrinsic flat convergence with bounded Ricci curvature"
3:00-4:00 C. Breiner (Fordham University)
"Quantitative Stratification and Higher Regularity for Biharmonic Maps"
4:30-5:30 Z. Zhang (Capital Normal University)
"L^p version of Cheeger-Colding Theory"
Abstracts:
10 am Pedro Solorzano "GH Convergence and Waning"
Abstract: Given any sequence of pointed complete riemannian manifolds converging in the pointed GH sense to a limit metric space, there is a subsequence of the associated Sasaki metrics on the tangent bundles that converges and the natural projection converges to a submetry. We will analyse the case when the sequence is given by re-scalings of the fibres of a principal bundle with a Kaluza-Klein metric. In particular we will discuss the structure of the limit submetry with respect to the vertical and horizontal directions. As part of this we will see how this relates to the general (non limiting) case.
11:30 am Raquel Perales "Glued Limit Spaces of Sequences of Manifolds"
Abstract: In this talk we consider open Riemannian manifolds M endowed with the length metric, d_M. The boundary of M is defined as the metric completion of M minus M. By avoiding the boundary we define the \delta-inner region of M, M^\delta \subset M. Gromov-Hausdorff compactness theorems are proven for sequences of \delta_i inner regions, then their Gromov-Hausdorff limit spaces are glued together into a single metric space which we call a glued limit space. This space might exist even if the original sequence of manifolds does not converge in Gromov-Hausdorff sense. We prove that under non-collapsing conditions, volume bounded above, diameter bounds and nonnegative Ricci curvature the glued limit of a sequence of manifolds has positive lower density everywhere.
(joint work with C. Sormani)
1:00 pm Mike Munn "Intrinsic flat convergence with bounded Ricci curvature"
Abstract: In their work introducing the intrinsic flat distance, Sormani-Wenger address (among many other things) the relationship between Gromov-Hausdorff limits and intrinsic flat limits of complete Riemannian manifolds. In particular, they show that for a sequence of Riemannian manifolds with nonnegative Ricci curvature, a uniform upper bound on diameter, and non-collapsed volume, the intrinsic flat limit exists and agrees with the Gromov-Hausdorff limit. This can be viewed as a non-cancellation theorem showing that for such sequences points don't cancel each other out in the limit. In this work, we extend this result to show that there is no cancellation when replacing the assumption of nonnegative Ricci curvature with a two-sided bound on the Ricci curvature.
3:00 pm Christine Breiner "Quantitative Stratification and Higher Regularity for Biharmonic Maps"
Abstract: We consider $u \in W^{2,2}(\Omega^m,N)$, $m\geq 4$, which are critical for the biharmonic energy $E(u):= \int_\Omega |\Delta u|^2$. Using the techniques of quantitative stratification developed by Cheeger-Naber, we prove quantitative regularity results for minimizers. As an application, we prove that every minimizing map is in $W^{4,p}$ for $1\leq p<5/4$. This work is joint with Tobias Lamm.
4:30 pm Zhenlei Zhang "L^p version of Cheeger-Colding theory "
Abstract: Cheeger-Colding regularity theory for manifolds of bounded Ricci curvature is a fundamental theory in Gromov-Hausdorff convergence. It has many geometric applications. In this talk, I will present an L^p verstion of the regularity theory, namely the regularity for manifiolds with L^p bounded Ricci curvature. A crucial assumption is the volume noncollapsing in all scales. The theory is built on the work of Petersen-Wei and Dai-Wei. It is based on a joint work with Professor Tian.
Organizers:
This impromptu unfunded workshop is aimed at doctoral students and postdocs
working on projects involving intrinsic flat convergence and/or Ricci curvature .
Registration is free. Please email Christina Sormani to register.