Schedule for: 22w5066 - Metric measure Spaces with Symmetry and Lower Ricci Curvature Bounds

In this two-part talk, we will present basic results on compact Lie group actions and discuss structure and classification results for metric spaces with lower curvature bounds, with an emphasis on Riemannian manifolds with positive or non-negative sectional curvature. In the first talk, we will discuss basic notions of isometric Lie group actions and recall different measures for the ``size'' of a compact Lie group action. We will then present fundamental structure and classification results for isometric Lie group actions on Riemannian manifolds with positive curvature. In the second talk, we will discuss results for manifolds with (almost) non-negative sectional curvature Throughout, we will point out generalizations to more general spaces, such as Alexandrov and RCD spaces, along with open problems.

The plan of this presentation is to review the beginning of the CD/RCD theory. We will cover the main properties of CD spaces, the most recent ones and motivate the introduction of RCD spaces. The presentation is intended for a general audience.

In this two-part talk, we will present basic results on compact Lie group actions and discuss structure and classification results for metric spaces with lower curvature bounds, with an emphasis on Riemannian manifolds with positive or non-negative sectional curvature. In the first talk, we will discuss basic notions of isometric Lie group actions and recall different measures for the ``size'' of a compact Lie group action. We will then present fundamental structure and classification results for isometric Lie group actions on Riemannian manifolds with positive curvature. In the second talk, we will discuss results for manifolds with (almost) non-negative sectional curvature Throughout, we will point out generalizations to more general spaces, such as Alexandrov and RCD spaces, along with open problems.

I will review some of the developments of the structure theory of RCD spaces in the last ten years, mostly motivated by the theory of Ricci limit spaces. In the first part I will consider the general case and outline the rectifiable structure with constant dimension away from sets of measure zero. In the second part of the talk I will focus on the ``non collapsed’’ case and illustrate how additional assumptions on the reference measure lead to further regularity, both at the topological and at the metric level.

It has proved useful both in Riemannian and Alexandrov geometry to study spaces with symmetries. Naturally one would like to do the same in the more general setting of RCD-spaces. If 𝐺 is the group of isometries of a given RCD space (𝑋,𝑑,𝔪) one of the first things one would like to prove is that it is a Lie group. We will discuss that thanks to structural results available for RCD-spaces this is indeed the case. Next, we will also show that if we further assume that the group 𝐺 is compact then we are able to modify the reference measure 𝔪 in such a way that we end up with a 𝐺−invariant reference measure that satisfies the same curvature dimension bounds as the original 𝔪. Finally, we will focus on a space that plays an important role in the geometry of a metric measure space: the Wassertein space (ℙ2(𝑋),𝕎2). Since this is a metric space on its own it is interesting to study the relationship between the symmetries of these two spaces. More concretely one can ask whether every isometry of ℙ2(𝑋) comes from one of 𝑋.

In Riemannian geometry, it is common to study the existence of nonnegatively Ricci curved metrics on a manifold. If such a metric exists, it is then interesting to describe such metrics' space, i.e. the moduli space of nonnegatively Ricci curved metrics. In 2017, Tuschmann and Wiemeler published the first result on these moduli spaces' homotopy groups. In the talk, we will study the singular analogue of the problem described above, i.e. we will replace nonnegatively Ricci curved Riemannian manifolds with RCD(0,N)-spaces. First, we will construct the Albanese and soul maps on moduli spaces of RCD(0,N)-structures (which reflect how structures on the universal cover split). Our main result is the continuity of these maps. Then, as a first application, we will prove the analogue of Tuschmann and Wiemeler's result in the RCD setting (providing a family of moduli spaces with non-trivial higher homotopy groups). Finally, as a second application, we will describe the situation in dimension 2. More precisely, we will show that moduli spaces of compact RCD(0,2)-structures are contractible.

Kapovitch, Lytchak, and Petrunin introduced the notion of metric measure boundary in an attempt to solve a long-standing conjecture of Perelman-Petrunin about the existence of infinite geodesics on Alexandrov spaces. They proved that the conjecture is satisfied provided the metric measure boundary of any Alexandrov space without boundary is vanishing. However, the latter turned out to be a difficult question and remained open. In this talk, I will present recent work in collaboration with A. Mondino and D. Semola, where we show that the metric measure boundary is vanishing on any RCD space without boundary.

It was shown by Gromov and Gallot that for a fixed dimension 𝑛 there exists a positive number 𝜀(𝑛) so that any 𝑛-dimensional closed Riemannian manifold (𝑀,𝑔) satisfying Riccidiam2≥−𝜀(𝑛) must have first Betti number smaller than or equal to 𝑛. Later on, Cheeger and Colding showed that if the first Betti number of 𝑀 equals 𝑛 then (𝑀,𝑔) has to be bi-Hölder homeomorphic to a flat torus. In this talk we will generalize the previous results to the case of RCD(𝐾,𝑁) spaces, which is the synthetic notion of riemannian manifolds satisfying Ricci≥𝐾 and dim≤𝑁. This class of spaces include Ricci limit spaces and Alexandrov spaces. Joint work with I. Mondello and A. Mondino.

By Gromov's precompactness theorem, any sequence of n-dim manifolds with uniform Ricci curvature lower bound has a convergent subsequence. The limit spaces are referred to as Ricci limit spaces. Cheeger-Colding-Naber developed great regularity and geometric properties for Ricci limit spaces. However, unlike Alexandrov spaces, these spaces could locally have infinite topological types. About twenty years ago, joint with C. Sormani, we gave the first topological result by showing the universal cover of Ricci limit spaces exists. Here the universal cover is in the sense of a universal covering map (need not be simply connected). I will present a series of recent works of J. Pan-G.Wei, J.Pan-J.Wang and J.Wang showing that the Ricci limit spaces are semilocally simply connected, therefore the universal covers are simply connected.

We give some examples of open manifolds with positive Ricci curvature. These examples give negative answers to two open questions. One is about the properness of Busemann function at a point, and the other one regards the singular set of Ricci limit spaces. This is joint work with Guofang Wei.

I will give an overview of some results about circle actions on Alexandrov spaces and their use in proving classification results for Alexandrov spaces. I will discuss in some more detail recent work with Searle classifying positively curved 4-dimensional spaces with circle actions. It is expected that these spaces are equivariantly homeomorphic to spherical suspensions of spherical 3-manifolds or finite quotients of weighted projective spaces. However, this project is incomplete -- I will introduce a particularly interesting problem about the geometry of isometric circle actions on positively curved 3-spaces which is key to making further progress.

In this talk, we show the stability of the curvature-dimension condition for negative values of the generalized dimension parameter under a suitable notion of convergence. We start by presenting an appropriate setting to introduce the CD(K, N)-condition for 𝑁<0, allowing metric measure structures in which the reference measure is quasi-Radon. Then in this class of spaces we introduce the distance 𝑑𝗂𝖪𝖱𝖶, which extends the already existing notions of distance between metric measure spaces. Finally, we prove that if a sequence of metric measure spaces satisfying the CD(K, N)-condition with 𝑁<0 is converging with respect to the distance 𝑑𝗂𝖪𝖱𝖶 to some metric measure space, then this limit structure is still a CD(K, N) space.

In this talk I will discuss regularity results on Regular Lagrangian Flows (RLFs) in the RCD(K,N) setting. RLFs are generalizations of flows via vector field to the nonsmooth setting and an understanding of its spatial regularity has applications in generalizing results from the Riemannian setting where smooth flows are used. Time permitting, I will discuss some applications of this in studying the structure and topology of RCD spaces. Parts of this talk are joint work with Elia Brué and Daniele Semola, and upcoming work with Dimitri Navarro.