During the **Spring 2022** semester, the CMSA will host a program on **General Relativity**. It will be a semester-long program, which includes four minicourses running in March, April, and May, and a conference April 4–8, 2022. There will also be a workshop May 2–5, 2022.

**Organizers: **

- Shing-Tung Yau (Harvard)
- Andrew Strominger (Harvard)
- Lydia Bieri (UMich)
- Lars Andersson (Max Planck Institute)
- Mu-Tao Wang (Columbia)
- Martin Lesourd (BHI)
- Achilleas Porfyriadis (BHI)

All in-person attendees must **register online**.

Schedule | Speaker | Title | Abstract |

March 1 – 3, 202210:00 am – 12:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10. Zoom link will be available. | Dr. Stefan Czimek | Title: Characteristic Gluing for the Einstein Equations | Abstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface. Then we turn to bifurcate characteristic gluing (i.e. gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay). |

March 22 – 25, 202222nd & 23rd, 10:00 am – 11:30am ET 24th & 25th, 11:00 am – 12:30pm ET Location: Hybrid. CMSA main seminar room, G-10. Zoom link will be available. | Prof. Lan-Hsuan Huang | Title: Existence of Static Metrics with Prescribed Bartnik Boundary Data | Abstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is1. The conjecture and an overview of the results 2. Static regular: a sufficient condition for existence and local uniqueness 3. Convex boundary, isometric embedding, and static regular 4. Perturbations of any hypersurface are static regular Lecture Notes Video on Youtube: March 22, 2022 |

March 29 – April 1, 2022 10:00am – 12:00pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10. Zoom link will be available. | Prof. Martin Taylor | Title: The nonlinear stability of the Schwarzschild family of black holes | Abstract: I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes. |

April 19 & 21, 202210 am – 12 pm ET, each dayZoom only | Prof. Håkan Andréasson | Title: Two topics for the Einstein-Vlasov system: Gravitational collapse and properties of static and stationary solutions. | Abstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole. |

May 16 – 17, 202210:00 am – 1:00 pm ET, each dayLocation: Hybrid. CMSA main seminar room, G-10. Zoom link will be available. | Prof. Marcelo Disconzi | Title: A brief overview of recent developments in relativistic fluids | Abstract: In this series of lectures, we will discuss some recent developments in the field of relativistic fluids, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary, a new formulation of the relativistic Euler equations tailored to applications to shock formation, and formulations of relativistic fluids with viscosity.1. Set-up, review of standard results, physical motivation. 2. The relativistic Euler equations: null structures and the problem of shocks. 3. The free-boundary relativistic Euler equations with a physical vacuum boundary. 4. Relativistic viscous fluids. |

This conference will be held virtually on Zoom. Registration is required. **Webinar Registration **

A few talks will be held in hybrid formats, with talks given from the CMSA seminar room, G-10. Advanced registration for in-person components is required. **In-Person Registration**

**Monday, April 4, 2022**

Time (ET) | Speaker | Title/Abstract |

9:30 am–10:30 am | Pieter Blue, University of Edinburgh, UK (virtual) | Title: Linear stability of the Kerr spacetime in the outgoing radiation gaugeAbstract: This talk will discuss a new gauge condition (i.e. coordinate condition) for the Einstein equation, the linearisation of the Einstein equation in this gauge, and the decay of solutions to the linearised Einstein equation around Kerr black holes in this gauge. The stability of the family of Kerr black holes under the evolution generated by the Einstein equation is a long-standing problem in mathematical relativity. In 1972, Teukolsky discovered equations governing certain components of the linearised curvature that are invariant under linearised gague transformations. In 1975, Chrzanowski introduced the “outgoing radiation gauge”, a condition on the linearised metric that allows for the construction of the linearised metric from the linearised curvature. In 2019, we proved decay for the metric constructed using Chrzanowski’s outgoing radiation gauge. Recently, using a flow along null geodesics, we have constructed a new gauge such that, in this gauge, the Einstein equation is well posed and such that the linearisation is Chrzanowski’s outgoing radiation gauge.This is joint work with Lars Andersson, Thomas Backdahl, and Siyuan Ma. |

10:30 am–11:30 am | Peter Hintz, MIT (virtual) | Title: Mode stability and shallow quasinormal modes of Kerr-de Sitterblack holes Abstract: The Kerr-de Sitter metric describes a rotating black hole with mass $m$ and specific angular momentum $a$ in a universe, such as our own, with cosmological constant $\Lambda>0$. I will explain a proof of mode stability for the scalar wave equation on Kerr-de Sitter spacetimes in the following setting: fixing $\Lambda$ and the ratio $|a/m|<1$ (related to the subextremality of the black hole in question), mode stability holds for sufficiently small black hole mass $m$. We also obtain estimates for the location of quasinormal modes (resonances) $\sigma$ in any fixed half space $\Im\sigma>-C$. Our results imply that solutions of the wave equation decay exponentially in time to constants, with an explicit exponential rate. The proof is based on careful uniform estimates for the spectral family in the singular limit $m\to 0$ in which, depending on the scaling, the Kerr-de Sitter spacetime limits to a Kerr or the de Sitter spacetime. |

11:30 am–12:30 pm | Lars Andersson, Albert Einstein Institute, Germany (virtual) | Title: Gravitational instantons and special geometryAbstract: Gravitational instantons are Ricci flat complete Riemannian 4-manifolds with at least quadratic curvature decay. In this talk, I will introduce some notions of special geometry, discuss known examples, and mention some open questions. The Chen-Teo gravitational instanton is an asymptotically flat, toric, Ricci flat family of metrics on $\mathrm{CP}^2 \setminus \mathrm{S}^1$, that provides a counterexample to the classical Euclidean Black Hole Uniqueness conjecture. I will sketch a proof that the Chen-Teo Instanton is Hermitian and non-Kähler. Thus, all known examples of gravitational instantons are Hermitian. This talks is based on joint work with Steffen Aksteiner, cf. https://arxiv.org/abs/2112.11863. |

12:30 pm–1:30 pm | break | |

1:30 pm–2:30 pm | Martin Taylor, Imperial College London (virtual) | Title: The nonlinear stability of the Schwarzschild family of black holesAbstract: I will present a theorem on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes. The proof employs a double null gauge, is expressed entirely in physical space, and utilises the analysis of Dafermos–Holzegel–Rodnianski on the linear stability of the Schwarzschild family. This is joint work with M. Dafermos, G. Holzegel and I. Rodnianski. |

2:30 pm–3:30 pm | Po-Ning Chen, University of California, Riverside (virtual) | Title: Angular momentum in general relativityThe definition of angular momentum in general relativity has been a subtle issue since the 1960s, due to the ‘supertranslation ambiguity’. In this talk, we will discuss how the mathematical theory of quasilocal mass and angular momentum leads to a new definition of angular momentum at null infinity that is free of any supertranslation ambiguity. Abstract: This is based on joint work with Jordan Keller, Mu-Tao Wang, Ye-Kai Wang, and Shing-Tung Yau. |

3:30 pm–4:00 pm | break | |

4:00 pm–5:00 pm | Dan Lee, Queens College (CUNY) (hybrid: in person & virtual) | Title: Stability of the positive mass theoremAbstract: We will discuss the problem of stability for the rigidity part of the Riemannian positive mass theorem, focusing on recent work with Kazaras and Khuri, in which we proved that if one assumes a lower Ricci curvature bound, then stability holds with respect to pointed Gromov-Hausdorff convergence. |

**Tuesday, April 5, 2022**

Time (ET) | Speaker | Title/Abstract |

9:30 am–10:30 am | Xinliang An, National University of Singapore (virtual) | Title: Anisotropic dynamical horizons arising in gravitational collapse Abstract: Black holes are predicted by Einstein’s theory of general relativity, and now we have ample observational evidence for their existence. However theoretically there are many unanswered questions about how black holes come into being and about the structures of their inner spacetime singularities. In this talk, we will present several results in these directions. First, in a joint work with Qing Han, with tools from scale-critical hyperbolic method and non-perturbative elliptic techniques, with anisotropic characteristic initial data we prove that: in the process of gravitational collapse, a smooth and spacelike apparent horizon (dynamical horizon) emerges from general (both isotropic and anisotropic) initial data. This result extends the 2008 Christodoulou’s monumental work and it connects to black hole thermodynamics along the apparent horizon. Second, in joint works with Dejan Gajic and Ruixiang Zhang, for the spherically symmetric Einstein-scalar field system, we derive precise blow-up rates for various geometric quantities along the inner spacelike singularities. These rates obey polynomial blow-up upper bounds; and when it is close to timelike infinity, these rates are not limited to discrete finite choices and they are related to the Price’s law along the event horizon. This indicates a new blow-up phenomenon, driven by a PDE mechanism, rather than an ODE mechanism. If time permits, some results on fluid dynamics will also be addressed. |

10:30 am–11:30 am | Sergiu Klainerman, Princeton (virtual) | Title: Nonlinear stability of slowly rotating Kerr solutionsAbstract: I will talk about the status of the stability of Kerr conjecture in General Relativity based on recent results obtained in collaboration with Jeremie Szeftel and Elena Giorgi. |

11:30 am–12:30 pm | Siyuan Ma, Sorbonne University (virtual) | Title: Sharp decay for Teukolsky master equation Abstract: I will talk about joint work with L. Zhang on deriving the late time dynamics of the spin $s$ components that satisfy the Teukolsky master equation in Kerr spacetimes. |

12:30 pm–1:30 pm | Break | |

1:30 pm–2:30 pm | Jonathan Luk, Stanford (virtual) | Title: A tale of two tailsAbstract: Motivated by the strong cosmic censorship conjecture, we introduce a general method for understanding the late-time tail for solutions to wave equations on asymptotically flat spacetimes in odd spatial dimensions. A particular consequence of the method is a re-proof of Price’s law-type results, which concern the sharp decay rate of the late-time tails on stationary spacetimes. Moreover, we show that the late-time tails are in general different from the stationary case in the presence of dynamical and/or nonlinear perturbations. This is a joint work with Sung-Jin Oh (Berkeley). |

2:30 pm–3:30 pm | Gary Horowitz, University of California Santa Barbara (virtual) | Title: A new type of extremal black holeAbstract: I describe a family of four-dimensional, asymptotically flat, charged black holes that develop (charged) scalar hair as one increases their charge at fixed mass. Surprisingly, the maximum charge for given mass is a nonsingular hairy black hole with a nondegenerate event horizon. Since the surface gravity is nonzero, if quantum matter is added, Hawking radiation does not appear to stop when this new extremal limit is reached. This raises the question of whether Hawking radiation will cause the black hole to turn into a naked singularity. I will argue that does not occur. |

3:30 pm–4:00 pm | Break | |

4:00 pm–5:00 pm | Lydia Bieri, University of Michigan (virtual) | Title: Gravitational radiation in general spacetimesAbstract: Studies of gravitational waves have been devoted mostly to sources such as binary black hole mergers or neutron star mergers, or generally sources that are stationary outside of a compact set. These systems are described by asymptotically-flat manifolds solving the Einstein equations with sufficiently fast decay of the gravitational field towards Minkowski spacetime far away from the source. Waves from such sources have been recorded by the LIGO/VIRGO collaboration since 2015. In this talk, I will present new results on gravitational radiation for sources that are not stationary outside of a compact set, but whose gravitational fields decay more slowly towards infinity. A panorama of new gravitational effects opens up when delving deeper into these more general spacetimes. In particular, whereas the former sources produce memory effects that are finite and of purely electric parity, the latter in addition generate memory of magnetic type, and both types grow. These new effects emerge naturally from the Einstein equations both in the Einstein vacuum case and for neutrino radiation. The latter results are important for sources with extended neutrino halos. |

**Wednesday, April 6, 2022**

Time (ET) | Speaker | Title/Abstract |

9:30 am–10:30 am | Gerhard Huisken, Mathematisches Forschungsinstitut Oberwolfach (virtual) | Title: Space-time versions of inverse mean curvature flowAbstract: In order to extend the Penrose inequality from a time-symmetric setting to general asymptotically flat initial data sets several anisotropic generalisations of inverse mean curvature flow have been suggested that take the full space-time geometry into account. The lecture describes the properties of such flows and reports on recent joint work with Markus Wolff on inverse flow along the space-time mean curvature. |

10:30 am–11:30 am | Carla Cederbaum, Universität Tübingen, Germany (virtual) | Title: Coordinates are messy Abstract: Asymptotically Euclidean initial data sets $(M,g,K)$ are characterized by the existence of asymptotic coordinates in which the Riemannian metric $g$ and second fundamental form $K$ decay to the Euclidean metric $\delta$ and to $0$ suitably fast, respectively. Provided their matter densities satisfy suitable integrability conditions, they have well-defined (ADM-)energy, (ADM-)linear momentum, and (ADM-)mass. This was proven by Bartnik using harmonic coordinates. To study their (ADM-)angular momentum and (BORT-)center of mass, one usually assumes the existence of Regge—Teitelboim coordinates on the initial data set $(M,g,K)$ in question. We will give examples of asymptotically Euclidean initial data sets which do not possess any Regge—Teitelboim coordinates We will also show that harmonic coordinates can be used as a tool in checking whether a given asymptotically Euclidean initial data set possesses Regge—Teitelboim coordinates. This is joint work with Melanie Graf and Jan Metzger. We will also explain the consequences these findings have for the definition of the center of mass, relying on joint work with Nerz and with Sakovich. |

11:30 am–12:30 pm | Stefanos Aretakis, University of Toronto (virtual) | Title: Observational signatures for extremal black holesAbstract: We will present results regarding the asymptotics of scalar perturbations on black hole backgrounds. We will then derive observational signatures for extremal black holes that are based on global or localized measurements on null infinity. This is based on joint work with Gajic-Angelopoulos and ongoing work with Khanna-Sabharwal. |

12:30 pm–1:30 pm | Break | |

1:30 pm–2:30 pm | Jared Speck, Vanderbilt University (virtual) | Title: The mathematical theory of shock waves in multi-dimensional relativistic and non-relativistic compressible Euler flowAbstract: In the last two decades, there have been dramatic advances in the rigorous mathematical theory of shock waves in solutions to the relativistic Euler equations and their non-relativistic analog, the compressible Euler equations. A lot of the progress has relied on techniques that were developed to study Einstein’s equations. In this talk, I will provide an overview of the field and highlight some recent progress on problems without symmetry or irrotationality assumptions. I will focus on results that reveal various aspects of the structure of the maximal development of the data and the corresponding implications for the shock development problem, which is the problem of continuing the solution weakly after a shock. I will also describe various open problems, some of which are tied to the Einstein–Euler equations. Various aspects of this program are joint with L. Abbrescia, M. Disconzi, and J. Luk. |

2:30 pm–3:30 pm | Lan-Hsuan Huang, University of Connecticut (virtual) | Title: Null perfect fluids, improvability of dominant energy scalar, and Bartnik mass minimizersAbstract: We introduce the concept of improvability of the dominant energy scalar, and we derive strong consequences of non-improvability. In particular, we prove that a non-improvable initial data set without local symmetries must sit inside a null perfect fluid spacetime carrying a global Killing vector field. We also show that the dominant energy scalar is always almost improvable in a precise sense. Using these main results, we provide a characterization of Bartnik mass minimizing initial data sets which makes substantial progress toward Bartnik’s stationary conjecture.Along the way we observe that in dimensions greater than eight there exist pp-wave counterexamples (without the optimal decay rate for asymptotically flatness) to the equality case of the spacetime positive mass theorem. As a consequence, we find counterexamples to Bartnik’s stationary and strict positivity conjectures in those dimensions. This talk is based on joint work with Dan A. Lee. |

3:30 pm–4:00 pm | Break | |

4:00 pm–5:00 pm | Demetre Kazaras, Duke University (virtual) | Title: Comparison geometry for scalar curvature and spacetime harmonic functionsAbstract: Comparison theorems are the basis for our geometric understanding of Riemannian manifolds satisfying a given curvature condition. A remarkable example is the Gromov-Lawson toric band inequality, which bounds the distance between the two sides of a Riemannian torus-cross-interval with positive scalar curvature by a sharp constant inversely proportional to the scalar curvature’s minimum. We will give a new qualitative version of this and similar band-type inequalities in dimension 3 using the notion of spacetime harmonic functions, which recently played the lead role in our recent proof of the positive mass theorem. This is joint work with Sven Hirsch, Marcus Khuri, and Yiyue Zhang. |

**Thursday, April 7, 2022**

Time (ET) | Speaker | Title/Abstract |

9:30 am–10:30 am | Piotr Chrusciel, Universitat Wien (virtual) | Title: Maskit gluing and hyperbolic massAbstract: “Maskit gluing” is a gluing construction for asymptotically locally hyperbolic (ALH) manifolds with negative cosmological constant. I will present a formula for the mass of Maskit-glued ALH manifolds and describe how it can be used to construct general relativistic initial data with negative mass. |

10:30 am–11:30 am | Greg Galloway, University of Miami (virtual) | Title: Initial data rigidity and applicationsAbstract: We present a result from our work with Michael Eichmair and Abraão Mendes concerning initial data rigidity results (CMP, 2021), and look at some consequences. In a note with Piotr Chruściel (CQG 2021), we showed how to use this result, together with arguments from Chruściel and Delay’s proof of the their hyperbolic PMT result, to obtain a hyperbolic PMT result with boundary. This will also be discussed. |

11:30 am–12:30 pm | Pengzi Miao, University of Miami (virtual) | Title: Some remarks on mass and quasi-local massAbstract: In the first part of this talk, I will describe how to detect the mass of asymptotically flat and asymptotically hyperbolic manifolds via large Riemannian polyhedra. In the second part, I will discuss an estimate of the Bartnik quasi-local mass and its geometric implications. This talk is based on several joint works with A. Piubello, and with H.C. Jang. |

12:30 pm–1:30 pm | Break | |

1:30 pm–2:30 pm | Yakov Shlapentokh Rothman, Princeton (hybrid: in person & virtual) | Title: Self-Similarity and Naked Singularities for the Einstein Vacuum EquationsAbstract: We will start with an introduction to the problem of constructing naked singularities for the Einstein vacuum equations, and then explain our discovery of a fundamentally new type of self-similarity and show how this allows us to construct solutions corresponding to a naked singularity. This is joint work with Igor Rodnianski. |

2:30 pm–3:30 pm | Marcelo Disconzi, Vanderbilt University (virtual) | Title: General-relativistic viscous fluids.Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues. |

3:30 pm–4:00 pm | Break | |

4:00 pm–5:00 pm | Maxime van de Moortel, Princeton (hybrid: in person & virtual) | Title: Black holes: the inside story of gravitational collapseAbstract: What is inside a dynamical black hole? While the local region near time-like infinity is understood for various models, the global structure of the black hole interior has largely remained unexplored. These questions are deeply connected to the nature of singularities in General Relativity and celebrated problems such as Penrose’s Strong Cosmic Censorship Conjecture. I will present my recent resolution of these problems in spherical gravitational collapse, based on the discovery of a novel phenomenon: the breakdown of weak singularities and the dynamical formation of a strong singularity. |

**Friday, April 8, 2022**

Time (ET) | Speaker | Title/Abstract |

9:30 am–10:30 am | Ye-Kai Wang, National Cheng Kun University, Taiwan (virtual) | Title: Supertranslation invariance of angular momentum at null infinity in double null gaugeAbstract: This talk accompanies Po-Ning Chen’s talk on Monday with the results described in the double null gauge rather than Bondi-Sachs coordinates. Besides discussing how Chen-Wang-Yau angular momentum resolves the supertranslation ambiguity, we also review the definition of angular momentum defined by A. Rizzi. The talk is based on the joint work with Po-Ning Chen, Jordan Keller, Mu-Tao Wang, and Shing-Tung Yau. |

10:30 am–11:30 am | Zoe Wyatt, King’s College London (virtual) | Title: Global Stability of Spacetimes with Supersymmetric Compactifications Abstract: Spacetimes with compact directions which have special holonomy, such as Calabi-Yau spaces, play an important role in supergravity and string theory. In this talk I will discuss a recent work with Lars Andersson, Pieter Blue and Shing-Tung Yau, where we show the global, nonlinear stability a spacetime which is a cartesian product of a high dimensional Minkowski space with a compact Ricci flat internal space with special holonomy. This stability result is related to a conjecture of Penrose concerning the validity of string theory. Our proof uses the intersection of methods for quasilinear wave and Klein-Gordon equations, and so towards the end of the talk I will also comment more generally on coupled wave–Klein-Gordon equations. |

11:30 am–12:30 pm | Elena Giorgi, Columbia University (hybrid: in person & virtual) | Title: The stability of charged black holes Abstract: Black hole solutions in General Relativity are parametrized by their mass, spin and charge. In this talk, I will motivate why the charge of black holes adds interesting dynamics to solutions of the Einstein equation thanks to the interaction between gravitational and electromagnetic radiation. Such radiations are solutions of a system of coupled wave equations with a symmetric structure which allows to define a combined energy-momentum tensor for the system. Finally, I will show how this physical-space approach is resolutive in the most general case of Kerr-Newman black hole, where the interaction between the radiations prevents the separability in modes. |

12:30 pm–1:30 pm | Break | |

1:30 pm–2:30 pm | Marcus Khuri, Stony Brook University (virtual) | Title: The mass-angular momentum inequality for multiple black holes : Consider a complete 3-dimensional initial data set for the Einstein equations which has multiple asymptotically flat or asymptotically cylindrical ends. If it is simply connected, axisymmetric, maximal, and satisfies the appropriate energy condition then the ADM mass of any of the asymptotically flat ends is bounded below by the square root of the total angular momentum. This generalizes previous work of Dain, Chrusciel-Li-Weinstein, and Schoen-Zhou which treated either the single black hole case or the multiple black hole case without an explicit lower bound. The proof relies on an analysis of the asymptotics of singular harmonic maps fromAbstract R^3 \ \Gamma –>H^2 where \Gamma is a coordinate axis. This is joint work with Q. Han, G. Weinstein, and J. Xiong. |

2:30 pm–3:30 pm | Martin Lesourd, Harvard (hybrid: in person & virtual) | Title: A Snippet on Mass and the Topology and Geometry of Positive Scalar CurvatureAbstract: I will talk about a small corner of the study of Positive Scalar Curvature (PSC) and questions which are most closely related to the Positive Mass Theorem. The classic questions are ”which topologies allow for PSC?” and ”what is the geometry of manifolds with PSC?”. This is based on joint work with Prof. S-T. Yau, Prof. D. A. Lee, and R. Unger. |

3:30 pm–4:00 pm | Break | |

4:00 pm–5:00 pm | Georgios Moschidis, Princeton (virtual) | Title: Weak turbulence for the Einstein–scalar field system.Abstract: In the presence of confinement, the Einstein field equations are expected to exhibit turbulent dynamics. In the presence of a negative cosmological constant, the AdS instability conjecture claims the existence of arbitrarily small perturbations to the initial data of Anti-de Sitter spacetime which, under evolution by the vacuum Einstein equations with reflecting boundary conditions at conformal infinity, lead to the formation of black holes after sufficiently long time.In this talk, I will present a rigorous proof of the AdS instability conjecture in the setting of the spherically symmetric Einstein-scalar field system. The construction of the unstable initial data will require carefully designing a family of initial configurations of localized matter beams and estimating the exchange of energy taking place between interacting beams over long periods of time, as well as estimating the decoherence rate of those beams. |

May 2–5, 2022**Workshop on scalar curvature, minimal surfaces, and initial data sets**

**Location**: Room G10, CMSA, 20 Garden Street, Cambridge MA 02138

Advanced registration for in-person components is required. **In-Person** **Registration is required. Register online.**

Zoom webinar: **Registration is required. ** **Webinar Registration **

Organizers: Dan Lee (CMSA/CUNY), Martin Lesourd (CMSA/BHI), and Lan-Hsuan Huang (University of Connecticut).

**Speakers:**

- Zhongshan An, University of Connecticut
- Paula Burkhardt-Guim, NYU
- Hyun Chul Jang, University of Miami
- Chao Li, NYU
- Christos Mantoulidis, Rice University
- Robin Neumayer, Carnegie Mellon University
- Andre Neves, University of Chicago
- Tristan Ozuch, MIT
- Annachiara Piubello, University of Miami
- Antoine Song, UC Berkeley
- Tin-Yau Tsang, UC Irvine
- Ryan Unger, Princeton
- Zhizhang Xie, Texas A & M
- Xin Zhou, Cornell University
- Jonathan Zhu, Princeton University

Download PDF

Monday, May 2, 2022

9:30–10:30 am | Hyun Chul Jang | Title: Mass rigidity for asymptotically locally hyperbolic manifolds with boundaryAbstract: Asymptotically locally hyperbolic (ALH) manifolds are a class of manifolds whose sectional curvature converges to −1 at infinity. If a given ALH manifold is asymptotic to a static reference manifold, the Wang-Chruściel-Herzlich mass integrals are well-defined, which is a geometric invariant that essentially measure the difference from the reference manifold. In this talk, I will present the result that an ALH manifold which minimize the mass integrals admits a static potential. To show this, we proved the scalar curvature map is locally surjective when it is defined on (1) the space of ALH metrics that coincide exponentially toward the boundary or (2) the space of ALH metrics with arbitrarily prescribed nearby Bartnik boundary data. And then, we establish the rigidity of the known positive mass theorems by studying the static uniqueness. This talk is based on joint work with L.-H. Huang. |

10:40–11:40 am | Annachiara Piubello | Title: Estimates on the Bartnik mass and their geometric implications.Abstract: In this talk, we will discuss some recent estimates on the Bartnik mass for data with non-negative Gauss curvature and positive mean curvature. In particular, if the metric is round the estimate reduces to an estimate found by Miao and if the total mean curvature approaches 0, the estimate tends to 1/2 the area radius, which is the bound found by Mantoulidis and Schoen in the blackhole horizon case. We will then discuss some geometric implications. This is joint work with Pengzi Miao. |

LUNCH BREAK | ||

1:30–2:30 pm | Ryan Unger | Title: Density and positive mass theorems for black holes and incomplete manifoldsAbstract: We generalize the density theorems for the Einstein constraint equations of Corvino-Schoen and Eichmair-Huang-Lee-Schoen to allow for marginally outer trapped boundaries (which correspond physically to apparent horizons). As an application, we resolve the spacetime positive mass theorem in the presence of MOTS boundary in the non-spin case. This also has a surprising application to the Riemannian setting, including a non-filling result for manifolds with negative mass. This is joint work with Martin Lesourd and Dan Lee. |

2:40–3:40 pm | Zhizhang Xie | Title: Gromov’s dihedral extremality/rigidity conjectures and their applications IAbstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu. |

TEA BREAK | ||

4:10–5:10 pm | Antoine Song (virtual) | Title: The spherical Plateau problemAbstract: For any closed oriented manifold with fundamental group G, or more generally any group homology class for a group G, I will discuss an infinite codimension Plateau problem in a Hilbert classifying space for G. For instance, for a closed oriented 3-manifold M, the intrinsic geometry of any Plateau solution is given by the hyperbolic part of M. |

Tuesday, May 3, 2022

9:30–10:30 am | Chao Li | Title: Stable minimal hypersurfaces in 4-manifoldsAbstract: There have been a classical theory for complete minimal surfaces in 3-manifolds, including the stable Bernstein conjecture in R^3 and rigidity results in 3-manifolds with positive Ricci curvature. In this talk, I will discuss how one may extend these results in four dimensions. This leads to new comparison theorems for positively curved 4-manifolds. |

10:40–11:40 am | Robin Neumayer | Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds IAbstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber. |

LUNCH BREAK | ||

1:30–2:30 pm | Zhongshan An | Title: Local existence and uniqueness of static vacuum extensions of Bartnik boundary dataAbstract: The study of static vacuum Riemannian metrics arises naturally in differential geometry and general relativity. It plays an important role in scalar curvature deformation, as well as in constructing Einstein spacetimes. Existence of static vacuum Riemannian metrics with prescribed Bartnik data — the induced metric and mean curvature of the boundary — is one of the most fundamental problems in Riemannian geometry related to general relativity. It is also a very interesting problem on the global solvability of a natural geometric boundary value problem. In this talk I will first discuss some basic properties of the nonlinear and linearized static vacuum equations and the geometric boundary conditions. Then I will present some recent progress towards the existence problem of static vacuum metrics based on joint works with Lan-Hsuan Huang. |

2:40–3:40 pm | Zhizhang Xie | Title: Gromov’s dihedral extremality/rigidity conjectures and their applications IIAbstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu. |

TEA BREAK | ||

4:10–5:10 pm | Tin-Yau Tsang | Title: Dihedral rigidity, fill-in and spacetime positive mass theorem Abstract: For compact manifolds with boundary, to characterise the relation between scalar curvature and boundary geometry, Gromov proposed dihedral rigidity conjecture and fill-in conjecture. In this talk, we will see the role of spacetime positive mass theorem in answering the corresponding questions for initial data sets. |

Speakers Banquet |

Wednesday, May 4, 2022

9:30–10:30 am | Tristan Ozuch | Title: Weighted versions of scalar curvature, mass and spin geometry for Ricci flowsAbstract: With A. Deruelle, we define a Perelman-like functional for ALE metrics which lets us study the (in)stability of Ricci-flat ALE metrics. With J. Baldauf, we extend some classical objects and formulas from the study of scalar curvature, spin geometry and general relativity to manifolds with densities. We surprisingly find that the extension of ADM mass is the opposite of the above functional introduced with A. Deruelle. Through a weighted Witten’s formula, this functional also equals a weighted spinorial Dirichlet energy on spin manifolds. Ricci flow is the gradient flow of all of these quantities. |

10:40–11:40 am | Robin Neumayer | Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds IIAbstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber. |

LUNCH BREAK | ||

1:30–2:30 pm | Christos Mantoulidis | Title: Metrics with lambda_1(-Delta+kR) > 0 and applications to the Riemannian Penrose InequalityAbstract: On a closed n-dimensional manifold, consider the space of all Riemannian metrics for which -Delta+kR is positive (nonnegative) definite, where k > 0 and R is the scalar curvature. This spectral generalization of positive (nonnegative) scalar curvature arises naturally, for different values of k, in the study of scalar curvature in dimension n + 1 via minimal surfaces, the Yamabe problem in dimension n, and Perelman’s surgery for Ricci flow in dimension n = 3. We study these spaces in unison and generalize, as appropriate, scalar curvature results that we eventually apply to k = 1/2, where the space above models apparent horizons in time-symmetric initial data sets to the Einstein equations and whose flexibility properties are intimately tied with the instability of the Riemannian Penrose Inequality. This is joint work with Chao Li. |

2:40–3:40 pm | Zhizhang Xie | Title: Gromov’s dihedral extremality/rigidity conjectures and their applications IIIAbstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu. |

TEA BREAK | ||

4:10–5:10 pm | Xin Zhou (Virtual) | Title: Min-max minimal hypersurfaces with higher multiplicityAbstract: It is well known that minimal hypersurfaces produced by the Almgren-Pitts min-max theory are counted with integer multiplicities. For bumpy metrics (which form a generic set), the multiplicities are one thanks to the resolution of the Marques-Neves Multiplicity One Conjecture. In this talk, we will exhibit a set of non-bumpy metrics on the standard (n+1)-sphere, in which the min-max varifold associated with the second volume spectrum is a multiplicity two n-sphere. Such non-bumpy metrics form the first set of examples where the min-max theory must produce higher multiplicity minimal hypersurfaces. The talk is based on a joint work with Zhichao Wang (UBC). |

May 5, 2022

9:00–10:00 am | Andre Neves | Title: Metrics on spheres where all the equators are minimalAbstract: I will talk about joint work with Lucas Ambrozio and Fernando Marques where we study the space of metrics where all the equators are minimal. |

10:10–11:10 am | Robin Neumayer | Title: An Introduction to $d_p$ Convergence of Riemannian Manifolds IIIAbstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge, and what the limiting objects look like. In this mini-course, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber. |

11:20–12:20 pm | Paula Burkhardt-Guim | Title: Lower scalar curvature bounds for C^0 metrics: a Ricci flow approachAbstract: We describe some recent work that has been done to generalize the notion of lower scalar curvature bounds to C^0 metrics, including a localized Ricci flow approach. In particular, we show the following: that there is a Ricci flow definition which is stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from C^0 initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from C^0 initial data. |

LUNCH BREAK | ||

1:30–2:30 pm | Jonathan Zhu | Title: Widths, minimal submanifolds and symplectic embeddingsAbstract: Width or waist inequalities measure the size of a manifold with respect to measures of families of submanifolds. We’ll discuss related area estimates for minimal submanifolds, as well as applications to quantitative symplectic camels. |

- Dan Lee, CMSA/CUNY, 01/24/22 – 05/20/22
- Stefan Czimek, Brown, 02/27/22 – 03/03/22
- Lan-Hsuan Huang, University of Connecticut, 03/13 – 03/19, 03/21 – 03/25, 04/17 – 04/23
- Mu-Tao Wang, Columbia, 03/21/22 – 03/25/22, 05/07/22 – 05/09/22
- Po-Ning Chen, University of California, Riverside, 03/21/22 – 03/25/22, 05/07/22 – 05/09/22
- Marnie Smith, Imperial College London, 03/27/22 – 04/11/22
- Christopher Stith, University of Michigan, 03/27/22 – 04/23/22
- Martin Taylor, Imperial College London, 03/27/22 – 04/11/22
- Marcelo Disconzi, Vanderbilt, 05/09/22 – 05/21/22
- Lydia Bieri, University of Michigan, 05/05/22 – 05/09/22