Fall 2022 Scalar Curvature and Topology Student Learning Seminar

3 minute read

  • Time : Monday 6-8 PM
  • Location : Room 507
  • Organizers: Aaron Chow, Shuang Liang, Jingbo Wan, Alex Xu

Scalar curvature is the least understood of all the various notions of curvature: bounded below scalar curvature can tell us something about the topology of the manifold, but a classification of such manifolds remains elusive. And many of techniques used to study them come from many different areas of geometric analysis, such as stable minimal hypersurfaces, spin geometry, harmonic maps, and geometric flows, and in many cases it is unclear whether these techniques are related on some fundamental level or if this is just a strange coincidence. The goal of this seminar is to survey some of these techniques as well as some applications and recent results, using some notes by Gromov as a starting point and branching out based off of participant interest.

Seminar Schedule

Week/Date Speaker Title Abstract
Sept 19 Jingbo Wan Stern’s Bochner formula on compact three-manifolds We give the basic ingredients for Stern’s Bochner formula and apply this formula to harmonic maps \(M^3 \rightarrow S^1\) where \(M\) is a compact three-manifold with or without boundary. This gives a beautiful inequality relating the average Euler characteristics of harmonic map’s level sets and the scalar curvature of \(M\). References: arxiv.org/abs/1908.09754, arxiv.org/abs/1911.06803
Sept 26 Shuang Liang Kähler-Ricci flow and estimates We will cover some basic analytic results concerning the evolution of various curvature tensors along the Kähler-Ricci flow. References: arxiv.org/abs/1508.04823, arxiv.org/abs/1212.3653
Oct 3 Shuang Liang Convergence of Kähler-Ricci flow and examples We aim to prove the convergence of the Kähler-Ricci flow when the first Chern class is negative or zero. If time allows, we will consider the case when the canonical bundle is nef and big and talk about the formulation of singularities. References: Same as last week
Oct 10 Nikita Klemyatin The Kähler-Ricci flow on Fano manifolds Let \((X, \omega)\) be a compact Kähler manifold with \(c_1(X) > 0\). Such manifolds are called Fano manifolds. In this case, there are several obstructions to the existence of the Kähler-Einstein metrics and the convergence and other results are much more involving than in the case of nonpositive $c_1(X)$. We will discuss the basics about Fano manifolds, together with examples of manifolds that do not admit KE metrics. After that, we will study the normalized Kähler-Ricci flow on Fano manifolds, such as the long time existence and preservation of positivity of the bisectional curvature along the NKRF.
Oct 17 Nikita Klemyatin Perelman’s uniform estimates and convergence This talk is the continuation of the previous one. We will prove the uniform diameter and scalar curvature estimates along the normalized Kähler-Ricci flow. If time permits, we will discuss some applications of this result.
Oct 24 Aaron Chow (Gromov Lawson paper) TBA
Oct 31 Aaron Chow (Gromov Lawson paper 2) TBA
Nov 7 Alex Xu Introduction to Seiberg Witten theory This talk will be a crash course on the various aspects of Seiberg-Witten theory most applicable to geometric analysis. We first setup the Seiberg-Witten equations on a closed orientable 4-manifold \(X\) and discuss how existence of irreducible solutions leads to apriori estimates on the total scalar curvature of \(X\). We follow up by defining the Seiberg-Witten invariant and some gluing formulas. If time permits, we discuss the situation on Kähler surfaces where the equations are much more concrete and exhibit some computations of the Seiberg-Witten invariant. References: John Morgan. The Seiberg-Witten equations and applications to the topology of smooth four-manifolds
Nov 14 Alex Xu Total scalar curvature and Seiberg Witten theory In the 90’s Witten noted that solutions to the Seiberg Witten equations gave apriori estimates on the total scalar curvature \(\int_X s^2\). LeBrun later used this observation to prove uniqueness of the Einstein metric on compact quotients of \(\mathbb{CH}^2\) and nonexistence of Einstein metrics on a certain class of manifolds. Applying these estimates to exotic pairs of 4-manifolds leads to the striking observation that \(inf \int_X s^2\) depends on the smooth structure of \(X\). The goal of this talk is to survey these results and the techniques used to derive them. References: arxiv.org/abs/dg-ga/9411005, arxiv.org/abs/dg-ga/9511015, arxiv.org/abs/math/0003068
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Dec 12