Convergence or Scalar Curvature Seminar
This seminar is organized by Brian Allen, Edward Bryden, and Demetre Kazaras. The seminar will run twice a month for Spring 2022 alternating with the Not Only Scalar Curvature (GNOSC) seminar organized by Misha Gromov, Bernhard Hanke, Christina Sormani, and Guoliang Yu (See the GNOSC website for titles and abstracts of talks for the GNOSC seminar). Those interested in speaking in the COSC seminar can send an email to brianallenmath@gmail.com with a proposed topic and a link to an arxiv paper which will be forwarded to the organizers for consideration. In order to attend the COSC you should register for zoom access here. You can find recordings of the talks here.
January 21- GNOSC Misha Gromov 9:00am EST and Rudolph Zeidler 10:15am EST
January 28- COSC Demetre Kazaras 9:00am EST
Title: Introduction to mass in general relativity for geometers
Abstract: Many contemporary ideas in the study of lower scalar curvature bounds have their roots in Schoen-Yau's investigation of mass in general relativity from the late 70's. In this talk, we will give some geometric background on the notion of total (ADM) mass for asymptotically flat manifolds and discuss the problem of mass positivity. There will be a brief overview of the many proofs of mass positivity and their pure-geometry applications. This talk is aimed at graduate students and people new to the area -- no familiarity with physics or relativity will be assumed.
Demetre Kazaras 10:00am EST
Title: Total mass in terms of harmonic functions
Abstract: This talk will build on the previous talk, giving details on a recent formula for the ADM mass of asymptotically flat manifolds (in 3 spatial dimensions) using harmonic functions and related objects. As a consequence, we will see a rather transparent proof of the positive mass theorem. We will briefly discuss applications of these ideas to the questions of flat space's stability among scalar non-negative asymptotically flat manifolds: How flat is a manifold with very little total mass?
February 4- GNOSC Christina Sormani 9:00am EST and Chao Li 10:15 am EST
February 11-COSC Brian Allen 9:00am EST
Title: Introduction to Metric Geometry Notions of Convergence
Abstract: Metric geometry notions of convergence present themselves when studying geometric stability questions for Riemannian manifolds involving Ricci and scalar curvature. In this talk we will give definitions of Gromov-Hausdorff (GH) convergence and Sormani-Wenger Intrisic Flat (SWIF) convergence and explore many examples of sequences of Riemannian manifolds. Compactness theorems for GH and SWIF convergence and methods for estimating these distances will be discussed. This talk is aimed at graduate students and people new to the area.
Brian Allen 10:00am EST
Title: From $L^p$ to Metric Geometry Notions of Convergence For Riemannian Manifolds
Abstract: This talk will build on the previous talk by giving several theorems which allow one to combine geometric hypotheses with $L^p$ convergence to imply metric geometry notions of convergence. We give sharp characterizations of which metric geometry notions of convergence are implied by different choices of $p$. Many examples will be discussed which demonstrate the sharp characterizations and illuminate general results. We will finish by mentioning applications to geometric stability results involving scalar curvature.
February 18-GNOSC Shmuel Weinberger 9:00am EST and Guoliang Yu 10:15am EST
February 25-COSC Georg Frenck 9:00am EST
Title: Spaces of positive scalar curvature metrics
Abstract: This talk will be a survey about spaces of Riemannian metrics of positive scalar curvature. I will motivate their study and highlight classical as well as recent results about them.
Francesca Oronzio 10:00am EST
Title: A Green's function proof of the positive mass theorem
Abstract: In this talk, we describe a new monotonicity formula holding along the level sets of the Green’s function of a complete one–ended asymptotically flat manifold of dimension 3 with nonnegative scalar curvature. Using such a formula, we obtain a simple proof of the celebrated positive mass theorem. In the same context, and for 1<p<3, a Geroch-type calculation is performed along the level sets of p-harmonic functions, leading to a new proof of the Riemannian Penrose Inequality in some case studies. These results are obtained in collaboration with V. Agostiniani and L. Mazzieri.
March 4-GNOSC Bernhard Hanke 9:00am EST and Johannes Ebert 10:15am EST
March 11-COSC Edward Bryden 9:00am EST
Title: Stability of the spacetime positive mass theorem for rotationally symmetric initial data
Abstract: When studying the stability of the positive mass theorem, much of the attention has focused on the stability of the Riemannian positive mass theorem. It is thus interesting to take a first stab at what a stability result might look like for the full positive mass theorem. There are two ways of doing this. On the one hand, one might try to tackle the formidable task of adapting techniques from geometric measure theory to Lorentzian metrics, and then attempting to prove stability in a manner analogous to the stability results obtained for the Riemannian case. On the other hand, one can study the stability of the positive mass theorem as it applies to initial data sets. If one focuses on initial data, then the usual techniques of geometric measure theory apply to the base Riemannian metric in the initial data. However, one must reconsider what stability means, since there are many initial data sets which produce Minkowski space. In this talk I will present a formulation for the stability of the positive mass theorem for initial data sets, and will show that it holds for rotationally symmetric initial data.
Sajjad Lakzian 10:00am EST
Title: The rigidity of sharp spectral gap in non-negatively curved spaces
Abstract: In the smooth setting, non-negative Ricci curvature on a compact manifold (without boundary or with a convex one) implies sharp lower bounds (Pi squared over the diameter squared) on the lowest positive Neumann eigenvalue (the principal eigenvalue); a result which is due to Zhong and Yang who strengthened the Li and Yau's bounds (half the sharp bound). There is a rigidity when the non-negatively curved manifold attains this bound namely it can only be 1 dimensional. This dates back to Payne-Weinberger for convex domains in Euclidean space and is proven for smooth manifolds by Hang and Wang; also in the Finsler setting, the same sharp bounds and rigidity is achieved by Xia. In the singular setting of metric and measure spaces, sharp bounds, spectral comparison results as well as Obata type rigidity results in RCD spaces has been established by various authors. In this talk, we outline the proof of rigidity for sharp spectral gap with nonnegative weak Ricci curvature and show the space has to be 1 dimensional. Due to the lack of the convexity that would otherwise be provided by positive curvature, our proof incorporates some new and stronger techniques to establish the rigidity and along the way some interesting regularity results in the singular setting. It is worth mentioning that we also provide a second proof for the rigidity in the weighted manifold setting with nonnegative Bakry-\'Emery Ricci tensor by modifying and strengthening the proof of Hang and Wang. So altogether, this provides the rigidity for non-negatively curved Bakry-'Emery manifolds, Alexandrov spaces and other RCD spaces. This is a joint work with C. Ketterer and Y. Kitabeppu.
March 18-GNOSC Natasa Sesum 9:00am EDT and Paula Burkhardt-Guim 10:15am EDT
March 25-COSC Jianchun Chu 9:00am EDT
Title: Kahler tori with almost non-negative scalar curvature
Abstract: For a sequence of Kahler metrics with almost non-negative scalar curvature on complex torus, under some assumptions, I will show that after passing to a subsequence, it will converge to flat torus weakly. This is a joint work with Man-Chun Lee.
Man-Chun Lee 10:00am EDT
Title: Convergence with scalar curvature lower bound and examples
Abstract: In this talk, we will consider Riemannian metrics with scalar curvature bounded from below and discuss the possible behavior of the sequence under various non-collapsing assumption. This is based on joint works with A. Naber, R. Neumayer, P. Topping and J. Chu.
April 1-GNOSC Zhizhang Xie 9:00am EDT and Jinmin Wang 10:15am EDT
April 8-GNOSC Weiping Zhang 9:00am EDT and Guangxiang Su 10:15am EDT
April 15-COSC Lina Chen 9:00am EDT
Title: Segment inequality and almost rigidity structures for integral Ricci curvature
Abstract：In this talk, we will show some almost rigidity structure results for integral Ricci curvature in the collapsing case. And to derive these results we will give the Cheeger-Colding segment inequality for manifolds with an integral Ricci curvature bound and then use a similar method as in [Cheeger-Colding, 1996].
Bo Zhu 10:00am EDT
Title: Geometry of positive scalar curvature on complete manifolds
Abstract: In this talk, we first introduce some conjectures related to scalar curvature in the context of size geometry. Then, we will particularly discuss how the strictly positive scalar curvature affects the volume growth of the geodesic ball in complete manifolds with non-negative Ricci curvature.
April 22-GNOSC Yuguang Shi 9:00am EDT and Jintian Zhu 10:15am EDT
April 29-GNOSC Christina Bar 9:00am EDT and Simone Cecchini 10:15 EDT
May 13-GNOSC Bernd Ammann 9:00am EDT and Claude LeBrun 10:15am EDT
May 20-COSC Shouhei Honda 9:00am EDT
Title: Topological stability theorem from nonsmooth to smooth spaces with Ricci curvature bounded below.
Abstract: In this talk, inspired by a recent work of Bing Wang and Xinrui Zhao, we prove that for a fixed n-dimensional closed Riemannian manifold $(M^n, g)$, if an $RCD(K, n)$ space $(X, d, m)$ is Gromov-Hausdorff close to $M^n$, then there exists a homeomorphism $F$ from $X$ to $M^n$ such that $F$ is Lipschitz continuous and $F^{-1}$ is H¨older continuous, where the Lipschitz constant of $F$, the H¨older exponent and the H{\"o}lder constant of $F^{-1}$ can be chosen arbitrary close to $1$. Moreover if $X$ is smooth, then such a map $F$ can be chosen as a diffeomorphism. This is a joint work with Yuanlin Peng (Tohoku University). The corresponding preprint can be found in arXiv:2202.06500.
May 27-GNOSC Pengzi Miao 9:00am EDT and Jeff Jauregui 10:15am EDT
June 3- COSC Daniel Rade 9:00am EDT
Title: Scalar and mean curvature comparison for Riemannian bands
Abstract: We use $\mu$-bubbles to compare Riemannian bands in scalar curvature, mean curvature and width to warped products over scalar flat manifolds with $\log$-concave warping functions. Furthermore we explain how these comparison results can be used to prove two conjectures due to Gromov resp. Rosenberg and Stolz in dimensions $\leq 7$. Part of this talk is based on joint work with Cecchini and Zeidler.
Renato Bettiol 10:15am EDT
Title: Extremality and rigidity for scalar curvature in dimension 4
Abstract: Following Gromov, a Riemannian manifold is called area-extremal if any modification that increases scalar curvature must decrease the area of some tangent 2-plane. We prove that large classes of compact 4-manifolds, with or without boundary, with nonnegative sectional curvature are area-extremal. We also show that all regions of positive sectional curvature on 4-manifolds are locally area-extremal. These results are obtained using twisted spinor methods and the Finsler--Thorpe trick. This is joint work with McFeely Jackson Goodman (UC Berkeley).
June 17-GNOSC Christine Breiner 9:00am EDT and Daniel Stern 10:15am EDT