## 特别数学讲座第14期

**Time: ** 08:00-18:00 June 13, 2011 - July 13, 2011

**Venue: ** 北京国际数学研究中心 资源大厦1218教室

时间： 2011年6月13日 - 2011年7月13日

地点：北京国际数学研究中心 资源大厦1218教室

1. Current Progress on Stability of Black Holes

Speaker: Pieter Blue

Abstract: General relativity is a geometric theory of gravity, and a theory that has been tested to an incredible accuracy. In general relativity, the universe is described by a four dimensional manifold of space-time points with a Lorentz metric. The metric satisfies a hyperbolic partial differential equation, known as Einstein's equation. A class of particular important solutions is the family of Kerr solutions, which includes the Schwarzschild subfamily. It is generally accepted by physicists that all stationary, asymptotically flat solutions that contain a single black hole belong to the Kerr family of solutions, and that all black holes will asymptotically approach a Kerr solution under the evolution given by the Einstein equation. In particular, it is expected that the Kerr black holes are asymptotically stable. In these lectures, we will discuss the background for, results related to, and current progress on this problem.

2. SOLUTIONS OF THE EINSTEIN INITIAL VALUE CONSTRAINT EQUATIONS

James Isenberg <isenberg@uoregon.edu>

An initial data set for Einstein's gravitational field equations cannot be chosen freely; it must satisfy the Einstein constraint equations. In this series of lectures we discuss what we know about the solutions of these constraint equations. We focus in particular on what the use of the conformal method and the use of gluing techniques has taught us about the parametrization and the construction of initial data sets satisfying the constraints. We review the well-known results regarding constant mean curvature and near constant mean curvature solutions, and we report on recent progress that has been made in studying solutions which fit in neither of these categories.

3. The Penrose Inequality, Quasi-Local Mass, and the Hoop Conjecture

Speaker: Marcus Khuri

Abstract: We will begin by reviewing the basic mathematical formulation of General Relativity, with the aim of understanding the consequences and proofs of the positive mass theorem. Next we will study a refinement of the positive mass theorem to the case in which spacetime contains black holes. This is known as the Penrose Inequality, and provides a lower bound for the mass in terms of the area of the black hole horizons. It has been proven in the time symmetric case, but remains open in general. We will study the known proofs and also the proposals for establishing the general theorem. The second part of the course will focus on the existence problem for black holes, known as the hoop conjecture. The goal is to provide a rigorous mathematical theorem for the heuristic physical intuition that black holes form when too much matter/energy is enclosed in a sufficiently small region. Directly related to this problem is the question of quasi-local mass, which asks for a geometric quantity describing the total mass (gravitation plus matter) content of a bounded region. Various proposals for such a definition will be surveyed.

4. Topics in Geometric Analysis

Speaker: Xiaodong Wang

Abstract: We plan to focus on the scalar curvature in Riemannian geometry. In the first part we will discuss some conformal geometry involving the scalar curvature. In particular we will study the Yamabe problem and its solution. In the second part, we will discuss some topological results involving scalar curvature. There are two approaches: one using spinors and Dirac operators and the other using minimal hypersurfaces and the second variation formula. We will discuss both approaches. In the third part we will discuss the positive mass theorem and its geometric applications.

5. The Yamabe Problem: existence and compactness questions

Speaker: Fernando Coda Marques

Abstract: The Yamabe Problem is a natural generalization of the Uniformization Theorem to dimensions greater than 2. The goal is to study the set of constant scalar curvature metrics on a compact manifold M, in a given conformal class. This is equivalent to studying positive solutions to a certain nonlinear partial dierential equation, of variational nature, and therefore the study of the Yamabe Problem requires deep geometric ideas and analytic techniques.

The existence problem was solved around 27 years ago, after the com-bined works of Yamabe (1960), Trudinger (1968), Aubin (1976), and Schoen (1984). In 1988, Richard Schoen started studying the qualitative properties of the solutions and conjectured that the set of solutions of volume one should be always compact, unless the manifold M is conformally dffeomor-phic to the round sphere (Schoen was assuming the validity of the Positive Mass Conjecture, and for this reason only we restrict to the case in which M is spin). The conjecture has been recently solved in a series of three papers. In [4], M. Khuri, F.C. Marques and R. Schoen proved that the Compactness Conjecture is true if the dimension is less than or equal to 24. On the other hand S. Brendle [1] discovered smooth counterexamples to the conjecture in dimensions greater than or equal to 52, which were later extended by Brendle and Marques [2] to the remaining dimensions (25<= n<=51).

In this minicourse we plan to start with the basics of the Yamabe Problem, then address the proof of the existence of solutions, and finally discuss the more recent works on the compactness and noncompactness questions. The references [3] and [5] are surveys on the material to be covered in class.

13JuneMonday

Morning Session

08:30-09:30 Xiaodong Wang (Michigan State University) Mini-course: Topics in geometric analysis

09:40-10:40 Marcus Khuri (Stony Brook) Mini-course: The Penrose inequality, quasi-local mass, and the Hoop conjecture

10:50-11:50 Pieter Blue (University of Edinburgh) Mini-course: The current progress on stability of black holes

Afternoon Session

14:00-15:00 Romain Gicquaud (Universite de Tours) Seminar: Conformal compactification of asymptotically locally hyperbolic metrics

15:30-16:30 Xin Zhou (Stanford University) Seminar: Axial symmetry and Mass Angular momentum inequality

14JuneTuesday

Morning Session

08:30-09:30 Xiaodong Wang (Michigan State University) Mini-course: Topics in geometric analysis

09:40-10:40 Marcus Khuri (Stony Brook) Mini-course: The Penrose inequality, quasi-local mass, and the Hoop conjecture

10:50-11:50 Pieter Blue (University of Edinburgh) Mini-course: The current progress on stability of black holes

Afternoon Session

14:00-15:00 Romain Gicquaud (Universite de Tours) Seminar: Conformal compactification of asymptotically locally hyperbolic metrics

15:30-16:30 Xin Zhou (Stanford University) Seminar: Axial symmetry and Mass Angular momentum inequality

15JuneWednesday

Morning Session

08:30-09:30 Xiaodong Wang (Michigan State University) Mini-course: Topics in geometric analysis

09:40-10:40 Marcus Khuri (Stony Brook) Mini-course: The Penrose inequality, quasi-local mass, and the Hoop conjecture

10:50-11:50 Pieter Blue (University of Edinburgh) Mini-course: The current progress on stability of black holes

Afternoon Session

14:00-15:00 James Isenberg (University of Oregon) Solutions of Einstein initial value constraint equations

15:30-16:30 Peng Lu (University of Oregon) Seminar: Conformal Ricci flow and Shi’s type curvature estimates

16JuneThursday

Morning Session

08:30-09:30 Xiaodong Wang (Michigan State University) Mini-course: Topics in geometric analysis[

Afternoon Session

14:00-15:00 James Isenberg (University of Oregon) Solutions of Einstein initial value constraint equations

15:30-16:30 Jose Espinar (Stanford University) Seminar: Compactness Theorem for marginally outer trapped surfaces

17JuneFriday

Morning Session

08:30-09:30 Xiaodong Wang (Michigan State University) Mini-course: Topics in geometric analysis

Afternoon Session

14:00-15:00 Xiaodong Wang (Michigan State University) Mini-course: Topics in geometric analysis

15:30-16:30 Wei Yuan (UC Santa Cruz) Seminar: Scalar curvature and deformations of metrics

20JuneMonday

Morning Session

08:30-09:30 Xiaodong Wang (Michigan State University) Mini-course: Topics in geometric analysis

10:50-11:50 Marcus Khuri (Stony Brook) Mini-course: The Penrose inequality, quasi-local mass, and the Hoop conjecture

Afternoon Session

14:00-15:00 Xiaodong Wang (Michigan State University) Mini-course: Topics in geometric analysis

15:30-16:30 Xue Hu (Peking University) Seminar: On static flows

21JuneTuesday

Morning Session

08:30-09:30 Marcus Khuri (Stony Brook) Mini-course: The Penrose inequality, quasi-local mass, and the Hoop conjecture

Afternoon Session

14:00-15:00 Lau Loi So (National Central University) Seminar: The physical meaning of the completely traceless property of the tensors B and V

15:30-16:30 Jie Wu (Peking University) Seminar: Ricci flows on asymptotically hyperbolic manifolds

22JuneWednesday

Morning Session

08:30-09:30 Xiaodong Wang (Michigan State University) Mini-course: Topics in geometric analysis

Afternoon Session

14:00-15:00 Xiaodong Wang (Michigan State University) Mini-course: Topics in geometric analysis

15:30-16:30 Dandan Ji (Peking Universty) Seminar: on the topology of conformally compact einstein 4 manifolds

23JuneThursday

Morning Session

08:30-09:30 James Isenberg (University of Oregon) Mini-course: Solutions of Einstein initial value constraint equations

09:40-10:40 Pieter Blue (University of Edinburgh) Mini-course: The current progress on stability of black holes

Afternoon Session

14:00-15:00 Vincent Bonini (Cal Poly CSU) Seminar: Hypersurfaces in Hyperbolic space and conformal metric on subdomains in the sphere

15:30-16:30 Nishanth Abu Gudapati (AEI) Seminar: Critical self-gravitating wave maps with symmetry

24JuneFriday

Morning Session

08:30-09:30 James Isenberg (University of Oregon) Mini-course: Solutions of Einstein initial value constraint equations

09:40-10:40 James Isenberg (University of Oregon) Mini-course: Solutions of Einstein initial value constraint equations

Afternoon Session

14:00-15:00 Lifeng Zhao (UTSC, Hefei) Seminar: Wellposedness of energy-critical Schrodinger Equations on torus

15:30-16:30 Ellery B. Ames (University of Oregon) Seminar: Fuchsian techniques in general relativity

Mini Courses by Fernando Coda Marques (IMPA)

4JulyMonday

09:00-11:00 Fernando Coda Marques (IMPA) Mini-course: The Yamabe Problem: existence and compactness questions

6JulyWednesday

09:00-11:00 Fernando Coda Marques (IMPA) Mini-course: The Yamabe Problem: existence and compactness questions

8JulyFriday

09:00-11:00 Fernando Coda Marques (IMPA) Mini-course: The Yamabe Problem: existence and compactness questions

11JulyMonday

13JulyWednesday