The seminar usually takes place in 515, Cosmology Building at 4-5PM every Thursday, unless noted otherwise.
Organizers: Nicolau Sarquis Aiex 艾尼克 (NTNU), Siao-Hao Guo 郭孝豪 (NTU), Ulrich Menne 孟悟理 (NTNU), Chung-Jun Tsai 蔡忠潤 (NTU)
via Zoom
Jérémie Szeftel (Sorbonne Université): The nonlinear stability of Kerr for small angular momentum
I will introduce the celebrated black hole stability conjecture according to which the Kerr family of metrics are stable as solutions to the Einstein vacuum equations of general relativity. I will then discuss the history of this problem, including a recent work on the resolution of the black hole stability conjecture for small angular momentum.
via Zoom
Jérémie Szeftel (Sorbonne Université): The nonlinear stability of Kerr for small angular momentum
In the first hour, I will introduce the Einstein equations and the corresponding evolution problem. I will then review some of the technics used in the stability of Minkowski. I will end the first hour by presenting Kerr black holes and the Kerr stability conjecture. In the second hour, I will recall the history of the Kerr stability conjecture. I will then focus on a recent work on the resolution of the black hole stability conjecture for small angular momentum.
Kuan-Hui Lee 李冠輝 (University of California, Irvine): Linear stability and moduli space of generalized Ricci solitons
The generalized Einstein-Hilbert action is an extension of the classic scalar curvature energy and Perelman’s F-functional which incorporates a closed three-form. The critical points are known as generalized Ricci solitons, which arise naturally in mathematical physics, complex geometry, and generalized geometry. In this talk, we would like to study the local behavior of the generalized Ricci solitons and compute the second variation formula of the generalized Einstein-Hilbert action. As an application we show that all Bismut-flat manifolds are linearly stable critical points, and admit nontrivial deformations arising from Lie theory. Furthermore, this leads to extensions of classic results of Koiso, Podesta, Spiro, Kröncke to the moduli space of generalized Ricci solitons.
Wei-Ting Kao 高尉庭 (National Taiwan University): The existence of the solution of the singular Yamabe equation on CR manifolds and its asymptotic behavior near boundary
The solution of the singular Yamabe problem gives us a tool to study the conformal invariant of a hypersurface. For example, Graham used renormalizing the volume of the singular Yamabe metric to get a conformal invariant energy of a hypersurface which is a generalization of the Willmore energy. For CR manifolds, we also can discuss singular Yamabe problem on CR manifolds. Inspired by Graham, Cheng, Yang, and Zhang used the similar method to get the CR invariant energy E2 of a hypersurface on 3 dim’l strongly pseudoconvex (spc) CR manifold, assuming such solution exists and some behaviors near the hypersurface. In this talk, I will briefly prove the existence of the solution of singular Yamabe equation on (2n+1)-dim’l spc CR manifolds with nonsingular smooth boundary and its behavior near boundary. To prove it, we will use some estimate on subelliptic operator with H\"ormander condition and some technique in the proof for Riemannian case by Andersson, Chru\’sciel, and Friedrich. This result also helps us to study the CR invariant of a hypersurface for higher dimension case.
Tongrui Wang 王童瑞 (Westlake University): Free boundary minimal hypersurfaces in locally wedge-shaped manifolds
Given a compact Riemannian manifold $M^{n+1}$ with smooth boundary $\partial M$, a free boundary minimal hypersurface (FBMH) in M is a critical point for the area functional with respect to the variations that constrain its boundary to lie in $\partial M$ but be otherwise free to vary. When the ambient manifold $M$ has a stratified singular structure (e.g. a polyhedron), a natural question is whether there is a FBMH in $M$ with a compatible stratified singular structure (e.g. a minimal polygon whose k-skeleton lies in $M$'s (k+1)-skeleton). In this talk, I will introduce related concepts of FBMHs in a class of spaces we call locally wedge-shaped manifolds, whose boundaries are formed by faces and edges. By extending Almgren-Pitts min-max theory, we show the existence of a $C^{2,\alpha}$ FBMH in any locally wedge-shaped manifolds of dimension $3 \leq n+1 \leq 6$ with either acute wedge angle or right wedge angle coupled with a certain additional assumption. This talk is based on the joint work with Liam Mazurowski.
Kai-Hsiang Wang 望開翔 (Northwestern University): Collapsing Ricci Limit Spaces with No Manifold Structure
It is known in the literature that a non-collapsing Ricci limit space admits a manifold structure on an open dense subset (J. Cheeger and T. H. Colding, '97). It has been an open question about how regular can a collapsing Ricci limit space be (A. Naber, '20). In our joint work with Erik Hupp and Aaron Naber, we provide an example of collapsing Ricci limits that admit no manifold structure (nowhere Euclidean).
Willie Wong (Michigan State University): Can Big Bangs Survive Smaller Bangs?
Much of our current understanding of the early universe is based on cosmological solutions to the Einstein field equations. These solutions are obtained after imposing a spatial homogeneity assumption, which reduces the dynamics to a finite system of coupled ordinary differential equations. An open question, however, is whether such a homogeneous reduction is valid: while one may expect the cosmological principle to hold on the largest scales, our everyday experience does suggest some amount of inhomogeneity in the universe. The validity of the homogeneous reduction hinges on whether such inhomogeneities can compound nonlinearly and lead to a significant departure from the spatially homogeneous solutions on the cosmological timescale. In this talk, I will present some recent joint work with Shih-Fang Yeh, where we invalidate the homogeneous ansatz for a class of big bang solutions to the Einstein-Euler equations in which the matter content of the universe is modeled by a compressible fluid. Our result exploits a combination of two features: the tendency for compressible fluids to develop shock waves and the conformal structure of the big bang solutions near the primordial singularity.
Franco Vargas Pallete (Yale University): On the topology and index of minimal and Bryant framed surfaces
We study framed surfaces, which are a class of Euclidean minimal and hyperbolic CMC-1 surfaces that generalize immersed Euclidean minimal surfaces and Bryant surfaces. For this class we prove a lower bound on the (unrestricted) Morse index by a linear function of the genus, number of ends and number of branch points (counting multiplicity). We will also discuss the 1-to-1 correspondence between Euclidean minimal and Bryant surfaces. This is based on joint work with Davi Maximo.
Teng Fei 费腾 (Rutgers University-Newark): The Type IIA flow and its geometric applications
The equations of flux compactifications of Type IIA superstrings were written down by Tomasiello and Tseng-Yau. To study these equations, we introduce a natural geometric flow called Type IIA flow on symplectic Calabi-Yau 6-manifolds. We prove the wellposedness of this flow and establish the basic estimates. We show that the Type IIA flow can be applied to find optimal almost complex structures on certain symplectic manifolds. We also prove its dynamical stability and apply it to prove a stability result about symplectic deformations of Calabi-Yau manifolds. This is based on joint work with Phong, Picard and Zhang.
Chung-Ming Pan 潘仲銘 (Université Paul Sabatier): Singular cscK metrics on smoothable varieties
Searching for canonical metrics in Kähler classes has been a central theme in Kähler geometry for decades. This talk aims to explain a method for investigating canonical metrics in families of singular varieties with a relative pluripotential theory and a variational approach in families. We shall start by revisiting fundamental concepts of canonical metrics and important properties within the variational picture for constant scalar curvature Kähler (cscK) metrics. I will then introduce notions of weak and strong topology of quasi-plurisubharmonic functions in families and explain several properties of entropy and Mabuchi functional extended to the family framework. Finally, I will present how these properties contribute to achieving uniform coercivity with almost optimal slopes of the Mabuchi functional and construct singular cscK potentials on smoothable varieties. This talk is based on joint work with T. D. Tô and A. Trusiani.
Tzu-Mo Kuo 郭子模 (University of California, Santa Cruz): conformal circles, local diffeomorphisms, and their holographic interpretation
In the context of a pseudo-Riemannian conformal manifold $(M^n, [g])$ with $n \geq 2$, a distinguished family of curves known as conformal circles, or conformal geodesics, emerges. These curves satisfy a third-order differential equation for non-null conformal circles. One classical problem is to determine whether conformal circles are essential for a conformal manifold by considering whether a diffeomorphism is a conformal diffeomorphism, provided it maps unparametrized conformal circles to unparametrized conformal circles. In this presentation, I will introduce the definition of a conformal circle and discuss its holographic interpretation. Additionally, I will present my results concerning both the classical problem and the holographic interpretation of the problem.
Chao-Ming Lin 林朝明 (Ohio State University): On the solvability of general inverse $\sigma_k$ equations
In this talk, first, I will introduce general inverse $\sigma_k$ equations in Kähler geometry. Some classical examples are the complex Monge–Ampère equation, the J-equation, the complex Hessian equation, and the deformed Hermitian–Yang–Mills equation. Second, by introducing some new real algebraic geometry techniques, we can consider more complicated general inverse $\sigma_k$ equations. Last, analytically, we study the solvability of these complicated general inverse $\sigma_k$ equations.
