Talks and presentations

Axially Symmetric Solution of the Teukolsky System in Very Slowly Rotating and Strongly Charged Sub-Extremal Kerr-Newman Spacetime

July 16, 2024

Talk, [Oberseminar Topics in General Relativity](https://www.uni-muenster.de/FB10/Service/show_article.shtml?id=10138&brettid=48), Muenster Univerity

We establish boundedness and polynomial decay results for the Teukolsky system in the exterior spacetime of very slowly rotating and strongly charged sub-extremal Kerr-Newman black holes, with a focus on axially symmetric solutions. The key step in achieving these results is deriving a physical-space Morawetz estimate for the associated generalized Regge-Wheeler system, without relying on spherical harmonic decomposition. This is based on a joint work with Elena Giorgi. References: [(Joint with Elena Giorgi) Boundedness and Decay for the Teukolsky System in Kerr-Newman Spacetime II: The Case |a| « M, |Q| < M in Axial Symmetry, arXiv preprint arXiv:2407.10750 (2024)]

Rigidity of Area Non-Increasing Maps

April 29, 2024

Talk, [Westlake geometric analysis seminar](https://its.westlake.edu.cn/info/1133/2032.htm), Westlake University, online

In this talk, we discuss the approach of Mean Curvature Flow to demonstrate that area non-increasing maps between certain positively curved closed manifolds are rigid. Specifically, this implies that an area non-increasing self-map of $\mathbb{CP}^n$, is either homotopically trivial or is an isometry, answering a question by Tsai-Tsui-Wang. Moreover, by coupling the Mean Curvature Flow for the graph of a map with Ricci Flows for the domain and the target, we can also study the rigidity of area non-increasing maps from closed manifolds with positive 1-isotropic curvature (PIC1) to closed Einstein manifolds, where Prof. Brendle’s PIC1 Sphere Theorem is applied. The key to studying the rigidity of area non-increasing maps under various curvature conditions lies in the application of the Strong Maximum Principle along the MCF/MCF-RF. We will focus our attention on one particular case to illustrate the SMP argument. This is a joint work with Professor Man-Chun Lee and Professor Luen-Fai Tam from CUHK. References: [(Joint with Man-Chun Lee and Luen-Fai Tam) Rigidity of area non-increasing maps., arXiv preprint arXiv:2312.10940 (2023)]

The Dirichlet Problem for Concave Equations

March 19, 2024

Talk, [Student Geometric & Analysis Seminar (Spring 2024)](https://math.columbia.edu/~ypharry/seminar/ma), Columbia University

In this talk, we continue our discussion of the concave equation $F(D^2u)=0$, focusing on its Dirichlet Problem. We first note that the standard Schauder estimate enables us to demonstrate that interior/global $C^{2,\alpha}$ solutions are indeed interior/global smooth. (No concavity is necessary in this “bootstrap” process.) Next, we establish the $C^{2,\alpha}$ estimate up to the boundary, progressing step by step ($C^0$, $C^1$, $C^2$, $C^{2,\alpha}$). The main technique at each step is selecting appropriate test functions and applying the Maximum Principle. Finally, we use everything we have discussed (uniqueness and regularity) along with the method of continuity to solve the Dirichlet problem. References: [Caffarelli-Cabre Fully Nonlinear Elliptic Equations]

Rigidity of Area Non-Increasing Maps

March 06, 2024

Talk, [BIMCR geometric analysis seminar](https://bicmr.pku.edu.cn/cn/content/show/43-3239.html), BIMCR, online

In this talk, we discuss the approach of Mean Curvature Flow to demonstrate that area non-increasing maps between certain positively curved closed manifolds are rigid. Specifically, this implies that an area non-increasing self-map of $\mathbb{CP}^n$, is either homotopically trivial or is an isometry, answering a question by Tsai-Tsui-Wang. Moreover, by coupling the Mean Curvature Flow for the graph of a map with Ricci Flows for the domain and the target, we can also study the rigidity of area non-increasing maps from closed manifolds with positive 1-isotropic curvature (PIC1) to closed Einstein manifolds, where Prof. Brendle’s PIC1 Sphere Theorem is applied. The key to studying the rigidity of area non-increasing maps under various curvature conditions lies in the application of the Strong Maximum Principle along the MCF/MCF-RF. We will focus our attention on one particular case to illustrate the SMP argument. This is a joint work with Professor Man-Chun Lee and Professor Luen-Fai Tam from CUHK. References: [(Joint with Man-Chun Lee and Luen-Fai Tam) Rigidity of area non-increasing maps., arXiv preprint arXiv:2312.10940 (2023)]

Evans-Krylov and interior $C^{2,alpha}$ regularity for concave equationss

February 27, 2024

Talk, [Student Geometric & Analysis Seminar (Spring 2024)](https://math.columbia.edu/~ypharry/seminar/ma), Columbia University

In this talk, we restrict ourselves to the fully nonlinear uniformly elliptic equation of the form $F(D^2u)=0$ with the additional condition that $F$ is concave as a function on symmetric matrices. In particular, we will make use of weak Harnack inequality and local maximum principle in chapter 4 of Caffarelli-Cabre as ingredients, demonstrating their application in proving the Evans-Krylov theorem and interior $C^{2,\alpha}$ regularity for concave equations. During the proof, some results implied by Jensen approximation we discussed last time (Chapter 5 of Caffarelli-Cabre) would also be used, leveraging translation invariance of the PDE we are interested in. References: [Caffarelli-Cabre Fully Nonlinear Elliptic Equations]

Jenson Approximation and uniqueness of solutions

February 13, 2024

Talk, [Student Geometric & Analysis Seminar (Spring 2024)](https://math.columbia.edu/~ypharry/seminar/ma), Columbia University

Within this talk, we restrict ourselves to the fully nonlinear uniformly elliptic equation of the form $F(D^2u)=0$, which is invariant under translation. We will introduce a useful approximation of the viscosity solution, called Jenson Approximation, which effectively captures the translation invariance of our PDE. Using Jenson Approximation, we establish a variation of Jenson’s uniqueness theorem for Dirichlet problem, which concerns the difference of two viscosity solutions. And again, due to the translation invariance of the PDE, we can apply the same idea to study the difference quotient of a viscosity solution, leading to interior $C^{1,\alpha}$ estimate. If time permits, we will also cover the application to concave $F$, in which we basically apply the same idea to study second order difference quotient (hence second derivatives) of a solution. (Notice: the additional concavity condition is crucial for discussing second order difference quotient.) References: [Caffarelli-Cabre Fully Nonlinear Elliptic Equations]

Bray’s conformal flow and Riemannian Penrose Inequality

December 11, 2023

Talk, [Student Geometric & Analysis Seminar (Fall 2023)](http://math.columbia.edu/~ypharry/seminar/mcf), Columbia University

This talk explores an alternative method for establishing the Riemannian Penrose inequality through Bray’s conformal flow. The conformal flow preserves the non-negativity of scalar curvature and the outermost minimizing property of the inner boundary. By using the positive mass theorem, one can demonstrate that the ADM mass is non-increasing under Bray’s conformal flow, eventually converging to half of Schwarzschild space. This convergence implies the desired Riemannian Penrose inequality. Notably, Bray’s approach offers advantages over IMCF approach, as it can be generalized to dimensions less than eight due to the absence of Gauss-Bonnet and enhances the inequality by replacing the area of the largest component of the horizon with the total area of the horizon, owing to fewer topology restrictions on the flow.

Inverse Mean Curvature Flow and Riemannian Penrose Inequality: Part II

December 04, 2023

Talk, [Student Geometric & Analysis Seminar (Fall 2023)](http://math.columbia.edu/~ypharry/seminar/mcf), Columbia University

Last time, we discussed the Geroch monotonicity of Hawking mass under smooth inverse mean curvature flow that can lead to Riemannian Penrose Inequality. We also gave a brief overview of Huisken-Ilmanen’s argument. In this talk, we aim to delve deeper into the intricacies of Huisken-Ilmanen’s weak formalism concerning the inverse mean curvature flow. To provide a more comprehensive understanding, we will focus on the freezing and variation approach, essential components in defining a weak solution for the level set inverse mean curvature flow. This approach not only grants us analytical advantages but also plays a crucial role in demonstrating the existence of weak solutions. To enhance our geometric understanding of weak IMCF, we will also cover an equivalence formulation utilizing sublevel sets. With both the analytic and geometric understanding of the weak IMCF, we will briefly talk about the heuristic reason for Geroch monotonicity along weak IMCF.

Inverse Mean Curvature Flow and Riemannian Penrose Inequality: Part I

November 27, 2023

Talk, [Student Geometric & Analysis Seminar (Fall 2023)](http://math.columbia.edu/~ypharry/seminar/mcf), Columbia University

In mathematical general relativity, the Penrose inequality, first conjectured by Sir Roger Penrose, estimates the mass of a spacetime in terms of the total area of its black holes and is a generalization of the positive mass theorem. The Riemannian Penrose inequality is an important special case, which was first proved by Huisken-Ilmanen using the Inverse mean curvature flow in dimension three. As the first part of this series of talks, we are going to discuss some basic notions and the Geroch monotonicity of Hawking mass under smooth inverse mean curvature flow. After this brief introduction, we will focus on the Huisken-Ilmanen’s weak formalism of inverse mean curvature flow, which is the essential ingredient in the proof of Riemannian Penrose inequality.

Lagrangian Mean Curvature Flow in the Gibbon-Hawking Ansatz II

October 16, 2023

Talk, [Complex Geometry Student Learning Seminar (Fall 2023)](https://www.math.columbia.edu/~sliang/seminar), Columbia University

We continue the discussion on LMCF in Hyperkahler 4-manifolds with circle symmetry. In this talk, we will discuss the hyperkahler moment map reduction of the LMCF where special Lagrangians will be mapped to certain line segments in the image and the LMCF will be reduced to a modified curve shortening flow. With all these understanding, we will give a heuristic outline of the Thomas conjecture and Thomas-Yau conjecture in this setting.

Inverse Mean Curvature Flow and a Minkowski inequality in AdS-Schwarzschild manifold, Part II

October 16, 2023

Talk, [Student Geometric & Analysis Seminar (Fall 2023)](http://math.columbia.edu/~ypharry/seminar/mcf), Columbia University

We continue the discussion about IMCF and a Minkowksi inequality in AdS-Schwarzschild manifold. Last time, we went over the basic a priori estimates for IMCF using both the parametric and non-parametric versions of the flow. In particular, short time existence and long time existence were clarified. In this talk, we will delve into an improved roundness estimate for the flow and the monotonicity formula along the flow. It’s worth noting that IMCF doesn’t fully converge in asymptotically hyperbolic manifolds. To overcome this, we will rely on Beckner’s sharp Sobolev inequality on standard spheres and a geometric inequality by Professor Brendle to estimate the lower bound of the monotone quantity as we approach the limit. References: A Minkowski Inequality for Hypersurfaces in the Anti‐de Sitter‐Schwarzschild Manifold

Lagrangian Mean Curvature Flow in the Gibbon-Hawking Ansatz I

October 09, 2023

Talk, [Complex Geometry Student Learning Seminar (Fall 2023)](https://www.math.columbia.edu/~sliang/seminar), Columbia University

Lagrangian Mean Curvature Flow (LMCF) provides a natural way to find special Lagrangian. In this talk, we discuss papers by Lotay-Oliveira that investigates the LMCF in very special ambient spaces, which are HyperKahler 4-manifolds generated by Gibbon-Hawking Ansatz. Notably, these spaces include the Multi-Eguchi-Hanson (ALE) and Multi-Taub-NUT (ALF). Moreover, HyperKahler 4-manifold in Gibbon-Hawking Ansatz admits a U(1)-symmetry which makes studying the U(1)-invariant Lagrangian surfaces and U(1)-invariant special Lagrangian surfaces very accessible. In particular, a U(1)-invariant LMCF in Gibbon-Hawking Ansatz will reduce to modified curve shortening flow on a plane by the HyperKahler moment map. This reduction provides us opportunities to understand various statements in the field of LMCF such as the notion of stability in Thomas-Yau conjecture and neck-pinch phenomenon in Joyce conjecture.

Inverse Mean Curvature Flow and a Minkowski inequality in AdS-Schwarzschild manifold

October 09, 2023

Talk, [Student Geometric & Analysis Seminar (Fall 2023)](http://math.columbia.edu/~ypharry/seminar/mcf), Columbia University

In this talk, we’ll discuss a paper by Brendle-Hung-Wang that applies a monotonicity formula of inverse mean curvature flow to prove a Minkowski inequality in AdS-Schwarzschild manifolds. Despite the lack of full convergence in this context (IMCF on Asymptotically hyperbolic manifold), we establish a roundedness estimate that helps estimate the lower bound of a monotone quantity along the flow. Besides IMCF, Professor Brendle’s geometric inequality and a sharp Sobolev inequality on the standard sphere are key components of the proof.

Rigidity of contracting map using harmonic map heat flow

September 28, 2023

Talk, [CUNY Geometric Analysis Seminar](https://www.lehman.edu/faculty/rbettiol/seminar.html), CUNY Graduate Center

In this talk, we are going to consider the rigidity of map between possibly curved closed manifolds, which is motivated by the recent work of Tsai-Tsui-Wang. We show that distance non-increasing map between complex projective spaces is either an isometry or homotopically trivial. The rigidity result also holds on a wider class of manifolds with positive curvature and weaker contracting property on the map in between distance non-increasing and area non-increasing. This is based on the harmonic map heat flow and it partially solves a Conjecture of Tsai-Tsui-Wang.

Rigidity of contracting map using harmonic map heat flow

September 25, 2023

Talk, [Student Geometric & Analysis Seminar (Fall 2023)](http://math.columbia.edu/~ypharry/seminar/mcf), Columbia University

In this talk, we are going to consider the rigidity of map between possibly curved closed manifolds, which is motivated by the recent work of Tsai-Tsui-Wang. We show that distance non-increasing map between complex projective spaces is either an isometry or homotopically trivial. The rigidity result also holds on a wider class of manifolds with positive curvature and weaker contracting property on the map in between distance non-increasing and area non-increasing. This is based on the harmonic map heat flow and it partially solves a Conjecture of Tsai-Tsui-Wang.

Rigidity of contracting map using harmonic map heat flow

August 24, 2023

Talk, [HKUST SEMINAR ON PURE MATHEMATICS](https://www.math.hkust.edu.hk/intranet/file/?c=seminar_abstract&f=20230817172736_22-Jingbo%20WAN_0824.pdf), THE HONG KONG UNIVERSITY OF SCIENCE & TECHNOLOGY

In this talk, we are going to consider the rigidity of map between possibly curved closed manifolds, which is motivated by the recent work of Tsai-Tsui-Wang. We show that distance non-increasing map between complex projective spaces is either an isometry or homotopically trivial. The rigidity result also holds on a wider class of manifolds with positive curvature and weaker contracting property on the map in between distance non-increasing and area non-increasing. This is based on the harmonic map heat flow and it partially solves a Conjecture of Tsai-Tsui-Wang.

Short course in semi-linear wave equation in Minkowski and Schwarzschild background

August 11, 2023

Talk, [CUHK seminar in geometric analysis], 501a, IMS

Lecture series on semi-linear wave equation in Minkowski and Schwarzschild background:

• On 11/8, we discuss semi-linear wave equations in Minkowski - Energy type estimate
• On 18/8, we discuss semi-linear wave equations in Minkowski - Dynamic estimate & Global decay
• On 25/8, we discuss semi-linear wave equations in Schwarzschild - Morawetz estimate
• On 1/9, we disucss the recent development of Dawei Shen on Exterior stability of Minkowski using Dafermos-Rodnianski’s r^p estimate. (cancelled due to Typhoon)

Rigidity of contracting map using harmonic map heat flow

August 05, 2023

Talk, [MIST 2023 Workshop on Geometry](http://www.ims.cuhk.edu.hk/activities/conferences/mist/2023/mist_workshop_3-6aug2023/), The Chinese University of Hong Kong

In this talk, we are going to consider the rigidity of map between possibly curved closed manifolds, which is motivated by the recent work of Tsai-Tsui-Wang. We show that distance non-increasing map between complex projective spaces is either an isometry or homotopically trivial. The rigidity result also holds on a wider class of manifolds with positive curvature and weaker contracting property on the map in between distance non-increasing and area non-increasing. This is based on the harmonic map heat flow and it partially solves a Conjecture of Tsai-Tsui-Wang.

Wave equations on Blackholes

June 02, 2023

Talk, [Joint Geometric Analysis Seminar](https://www.math.cuhk.edu.hk/seminars/wave-equations-blackholes), The Chinese University of Hong Kong

In this talk, I will review the vector field method/Energy method towards scalar wave equations on Schwarzschild or Kerr spacetime. To do so, we will emphasize some geometric/physical phenomena of Schwarzschild and Kerr involving the ergoregion, the photon sphere and also the Red-shift effect. Remarkably, these phenomena represent either obstructions or stabilizing mechanism for decay estimates of the wave equations. During the Energy estimate of wave equations, I intend to explain which phenomenon causes what obstruction and which phenomenon can help us fix some of the obstructions. At the end, I will also try to explain why the study of scalar wave equations is an essential step towards the (non-)linear stability of Blackholes.

Isoperimetric surface technique & Penrose Inequality

April 19, 2023

Talk, [Student Geometric & Analysis Seminar (Spring 2023): Scalar Curvature and Dihedral Rigidity](http://math.columbia.edu/~ypharry/seminar/dihedral-rigidity.html), Columbia University

In this talk, we will continue to discuss the isoperimetric surface technique that was developed by Professor Hubert Bray in his PhD thesis. This time, we are going to study the effect of non-negativity of scalar curvature on a complete asymptotically flat three-manifold. In particular, we will use the isoperimetric surface technique to prove two conditional Penrose inequality for single/multiple horizons cases. In contrast to last time, we are dealing with a minimizing process in non-compact manifolds, so we need to address the existence of area minimizers with its volume constraint.

Isoperimetric surface technique & Volumne Comparison theorems

April 12, 2023

Talk, [Student Geometric & Analysis Seminar (Spring 2023): Scalar Curvature and Dihedral Rigidity](http://math.columbia.edu/~ypharry/seminar/dihedral-rigidity.html), Columbia University

In this talk, we will discuss the isoperimetric surface technique that was developed by Professor Hubert Bray in his PhD thesis. In particular, we are going to use such technique to study volume comparison theorems for positively curved manifolds, which includes a new proof for the Bishop’ volume comparison theorem and also the “Bray’s football theorem”- a volume comparison theorem for scalar curvature in the case that the metric is close to the standard S^3 metric. We remark that the rigid case of Bray’s football theorem can be found in a beautiful survey paper written by Professor S.Brendle. (But we might not have the time cover it)

Elliptic Boundary Value Problem for Dirac-Type Operators: Part 3

March 22, 2023

Talk, [Student Geometric & Analysis Seminar (Spring 2023): Scalar Curvature and Dihedral Rigidity](http://math.columbia.edu/~ypharry/seminar/dihedral-rigidity.html), Columbia University

Last time, we studied the hybrid Sobolev spaces using the spectrum of a boundary adapted operator. In particular, the hybrid Sobolev space $\check{H}(A)$ is the image of (extended) trace map on the Dirac-maximal domain, and any boundary condition we will consider is a closed subspace of $\check{H}(A)$. Among all the boundary conditions, a special class called D-elliptic boundary conditions will be the main subject we are discussing. To understand this D-elliptic boundary condition, we will start with the famous APS condition, and try to explore based on that. Under these D-elliptic boundary conditions, nice boundary regularity result is obtained. And we will also discuss many other examples which belong to the class of D-elliptic boundary conditions.

Elliptic Boundary Value Problem for Dirac-Type Operators: Part 2

March 01, 2023

Talk, [Student Geometric & Analysis Seminar (Spring 2023): Scalar Curvature and Dihedral Rigidity](http://math.columbia.edu/~ypharry/seminar/dihedral-rigidity.html), Columbia University

Last time, we defined the differential operator on the boundary associated to the given Dirac type operator using the information of the symbols. Recall that such associated boundary operator is a self-adjoint elliptic first order operator defined on a closed manifold (the boundary), so we can make use of its spectrum (L^2 decomposition of sections defined on the boundary) to investigate what kind of boundary condition is natural for the Dirac type operator. Statements will be given and the ideas of the proof will be sketched, and some important examples will be discussed.

Elliptic Boundary Value Problem for Dirac-Type Operators: Part 1

February 15, 2023

Talk, [Student Geometric & Analysis Seminar (Spring 2023): Scalar Curvature and Dihedral Rigidity](http://math.columbia.edu/~ypharry/seminar/dihedral-rigidity.html), Columbia University

Dirac Operator is a powerful tool to study positive scalar curvature. Concerning positive scalar curvature on a manifold with boundary, it’s natural to ask how to formulate a valid boundary value problem for Dirac operator (1st order elliptic). In fact, Dirichlet boundary condition, which is natural for Laplacian operator (2nd order elliptic), turns out to be too strong for 1st order elliptic operators. In this talk, we focus on Dirac-type operators, with principal symbols capturing the Clifford relation just like the usual Dirac operator, and discuss some basic materials to get ready for elliptic boundary value problems.

Maximum Principles and applications

December 12, 2022

Talk, [Scalar Curvature and Topology Student Learning Seminar](https://math.columbia.edu/~axu/seminars/scalar-curvature-seminar-post/), Columbia University

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.

A Generalization of Hawking’s Black Hole Topology Theorem to Higher Dimensions

December 05, 2022

Talk, [Scalar Curvature and Topology Student Learning Seminar](https://math.columbia.edu/~axu/seminars/scalar-curvature-seminar-post/), Columbia University

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.

The Existence of a Black Hole Due to Condensation of Matter

October 14, 2022

Talk, [Columbia General Relativity & Geometric Analysis seminar](https://www.math.columbia.edu/~egiorgi/seminar-c.html), Columbia University

For an asymptotically flat initial data set (three dimensional), with the mass density large on a large region, Schoen and Yau showed that there is an apparent horizon in the initial data. (SY83’) Their proof is based on a contradiction argument where assuming no apparent horizon in the region will give a global solution to Jang’s equation over the region (SY81’). Furthermore, positivity of mass density and this global solvability of Jang’s equation will give rise to positivity of a certain operator on this region. Finally, the argument is completed by showing such positivity of certain operator on the region will assert that the region can not be too large in a certain sense. Aaron Chow and the speaker used a slicing technique that was introduced in a recent paper of S. Brendle, S. Hirsch, and F. Johne (BHJ22’) to show that in a $n+1$-dimensional torical band $T^n \times [0,1]$ where $n+1\leq 7$, positivity of a similar operator will assert that the band cannot be too long. If time permits, we will also discuss the possibility of generalising Schoen-Yau existence result of Black hole to higher dimensions, with such torical band width estimate.

Stern’s Bochner formula on compact three-manifolds

September 19, 2022

Talk, [Scalar Curvature and Topology Student Learning Seminar](https://math.columbia.edu/~axu/seminars/scalar-curvature-seminar-post/), Columbia University

We give the basic ingredients for Stern’s Bochner formula and apply this formula to harmonic maps $M^3\rightarrow \mathbb{S}^1$ 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 $M$.