# Fall 2022 Scalar Curvature and Topology Student Learning Seminar

• 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. Reference: https://link.springer.com/book/10.1007/978-3-319-00819-6
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. Reference: Bounding Scalar Curvature and Diameter along the Kähler Ricci Flow
Oct 24 Aaron Chow Dirac type operators on complete Riemannian manifolds This talk will be a preparation to Gromov-Lawson’s proof on the nonexistence of PSC metrics on enlargeable manifolds. We will begin with a crash course on Dirac type operators and derive the Weitzenbock formula. After that we will study analytic properties of Dirac type operators on complete Riemannian manifolds. Reference: link.springer.com/article/10.1007/BF02953774
Oct 31 Aaron Chow Nonexistence of PSC metrics on enlargeble manifolds We will start by studying the relative index of Dirac type operators on complete Riemannian manifolds. The goal of this talk is to go through Gromov-Lawson’s proof on the nonexistence of PSC metrics on enlargeable manifolds. Reference: link.springer.com/article/10.1007/BF02953774
Nov 7   No seminar due to university holiday
Nov 14 Aaron Chow Nonexistence of PSC metrics on enlargeble manifolds 2 We continue studying the relative index of Dirac type operators on complete Riemannian manifolds and go through Gromov-Lawson’s proof on the nonexistence of PSC metrics on enlargeable manifolds. Reference: link.springer.com/article/10.1007/BF02953774
Nov 21 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 28 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
Dec 5 Jingbo Wan A Generalization of Hawking’s Black Hole Topology Theorem to Higher Dimensions Hawking’s theorem on the topology of black holes asserts that cross sections of the event horizon in 4-dimensional asymptotically ﬂat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres. Geometrically and very roughly, this is analogous to the topological restriction of a stable minimal surface in positively curved 4-manifold. In this reading seminar talk, the speaker is going to talk about Galloway and Schoen’s generalization of Hawking’s theorem to any dimensional Spacetime satisfying the dominant energy condition, asserting that outer apparent horizon is Yamabe positive, except some very special cases. Reference: A Generalization of Hawking’s Black Hole Topology Theorem to Higher Dimensions
Dec 12 Jingbo Wan Maximum Principles and applications In this reading seminar talk, the speaker will present Hamilton’s Maximum Principle for Ricci flow and discuss applications for convergence of Ricci flow under certain positive curvature conditions. Then, as a comparison, the speaker will present a version of Bony’s strict maximum principle for degenerate elliptic equations and discuss its application on rigidity results (where we change the previous positive curvature conditions to non-negative curvature conditions). If time permits, the speaker will present and discuss some results where a version of Hamilton’s Maximum Principle or Bony’s strict maximum principle was applied. Reference: Simon Brendle, “Ricci Flow and the Sphere Theorem”, Chapter 5 & 9.

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