Semi-Riemannian Geometry
This course studies modern mathematical general relativity through differential geometry, geometric analysis, and nonlinear \textbf{hyperbolic} PDE. Beginning with the basic Semi-Riemannian geometry and Einstein equations, we develop the geometry of gravitational mass, scalar curvature, minimal and marginally trapped surfaces, and the Schoen-- Yau positive mass and Yau black-hole formation program. We then discuss quasilocal mass, with emphasis on the Brown--York, Liu--Yau, and Wang--Yau constructions, before turning to Einstein evolution, null focusing, rough solutions, and large-data black-hole formation. The final part connects these questions with critical regularity, profile decomposition, and concentration compactness, emphasizing how geometric rigidity and quasilocal quantities may enter a large-data theory of gravitational collapse.
讲师
日期
2026年09月16日 至 12月18日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周三,周五 | 13:30 - 15:05 | Shuimo | - | - | - |
修课要求
The course is intended for graduate students and advanced undergraduates in mathematics or physics at Quizhen College. Students should be familiar with smooth manifolds, basic Riemannian geometry, basic analysis. Familiarity with PDE is helpful but not required.
课程大纲
Week1 ( Introduction to Semi-Riemannian Geometry): Semi-Riemannian Manifolds, Lorentzian manifolds, timelike, spacelike, and null hypersurfaces, spacelike hypersurfaces and second fundamental forms; Spacelike initial data sets $(M,g,k)$; Gauss-Codazzi equations; Einstein equations, $n+1$ reduction of Einstein equations, Hamiltonian and momentum constraints; dominant energy condition; maximal and CMC data; asymptotic flatness. The time-symmetric reduction and the role of scalar curvature:
\[
R_g+(\tr_g k)^2-|k|_g^2=2\mu,
\qquad
\operatorname{div}_g k-d(\tr_g k)=J.
\]
Week 2 (Positive Mass and Scalar-Curvature Rigidity): ADM energy and momentum; statement and geometric meaning of the positive mass theorem; stable minimal surfaces and the Schoen--Yau method; rigidity in the zero-mass case; general initial data and comparison with Witten's spinorial approach.
\subsection*{Week 3: Minimal Surfaces, Trapped Surfaces, and Horizons}
Null normals and null expansions; trapped and marginally trapped surfaces; marginally outer trapped surfaces and apparent horizons; time-symmetric reduction to minimal surfaces; stability operators, outermost horizons.
\subsection*{Week 4: Yau-Type Geometric Criteria for Black Holes}
Condensation of pure gravity and matter; geometric notions of radius and size; lower bounds involving boundary mean curvature, $\mu-|J|$; apparent-horizon existence from interior concentration or boundary geometry; relation to the hoop-conjecture philosophy. A guiding principle is
\[
\text{geometric size}\times\text{energy concentration}
\quad\Longrightarrow\quad
\text{black-hole geometry}.
\]
\subsection*{Week 5: Quasilocal Mass}
Localization of gravitational energy; Brown--York, Liu--Yau, and Wang--Yau masses; reference isometric embeddings; observer dependence and the optimal embedding equation; positivity, rigidity, asymptotic limits, and relations with horizons.
\subsection*{Week 6: Einstein Evolution as a Geometric PDE}
Maximal globally hyperbolic developments; diffeomorphism invariance and gauge fixing; wave and CMC-type reductions; hyperbolic and hyperbolic--elliptic formulations; local well-posedness, constraint propagation, energy estimates, and continuation criteria. The focus is the PDE structure needed for large-data problems, not a full small-data stability theory.
\subsection*{Week 7: Null Geometry and Dynamical Focusing}
Optical functions and double-null foliations; null second fundamental forms; expansion, shear, and torsion; Raychaudhuri and null structure equations; curvature flux; quantitative focusing and trapped-surface formation in vacuum and Einstein--matter systems. Classical short-pulse results will be used selectively to illustrate the mechanism.
\subsection*{Week 8: Black-Hole Formation by Boundary Effect}
Boundary mean curvature and null expansions; geometric radius criteria; boundary contraction and interior MOTS formation; dynamical realization of Yau's black-hole criterion; comparison between boundary-induced collapse and concentration of incoming radiation. A typical mechanism is
\[
\Rad(M_t)\,c(t)>C_{\mathrm{crit}},
\qquad
c(t)=\inf_{\partial M_t}\bigl(H_t-|\tr_{\partial M_t}k_t|\bigr).
\]
\subsection*{Week 9: Rough Solutions and Curvature-Level Control}
Metric, connection, and curvature regularity; scaling and critical Sobolev spaces; derivative loss in geometric gauges; rough null geometry; curvature-flux and elliptic estimates; breakdown criteria; weak limits and persistence of mass, scalar-curvature inequalities, boundary geometry, and horizon conditions.
\subsection*{Week 10: Concentration Compactness for Nonlinear Waves}
Brief introduction to the main idea, LP theory, scaling and criticality; failure of compactness in Sobolev embeddings; linear and nonlinear profile decomposition; asymptotic orthogonality; minimal blow-up solutions; compactness modulo symmetries; the compactness--rigidity method, with energy-critical wave equations as the principal model.
\subsection*{Week 11/12: Concentration, Rigidity, and Black-Hole Geometry}
Analytic versus geometric concentration: critical norms, matter density, curvature flux, scalar curvature, and quasilocal mass. Geometric profiles for threshold sequences: scales, blow-up limits, mass splitting, necks, and trapped bubbles. Quasilocal mass as a concentration functional, with positive-mass and quasilocal rigidity as possible rigidity mechanisms.
\[
R_g+(\tr_g k)^2-|k|_g^2=2\mu,
\qquad
\operatorname{div}_g k-d(\tr_g k)=J.
\]
Week 2 (Positive Mass and Scalar-Curvature Rigidity): ADM energy and momentum; statement and geometric meaning of the positive mass theorem; stable minimal surfaces and the Schoen--Yau method; rigidity in the zero-mass case; general initial data and comparison with Witten's spinorial approach.
\subsection*{Week 3: Minimal Surfaces, Trapped Surfaces, and Horizons}
Null normals and null expansions; trapped and marginally trapped surfaces; marginally outer trapped surfaces and apparent horizons; time-symmetric reduction to minimal surfaces; stability operators, outermost horizons.
\subsection*{Week 4: Yau-Type Geometric Criteria for Black Holes}
Condensation of pure gravity and matter; geometric notions of radius and size; lower bounds involving boundary mean curvature, $\mu-|J|$; apparent-horizon existence from interior concentration or boundary geometry; relation to the hoop-conjecture philosophy. A guiding principle is
\[
\text{geometric size}\times\text{energy concentration}
\quad\Longrightarrow\quad
\text{black-hole geometry}.
\]
\subsection*{Week 5: Quasilocal Mass}
Localization of gravitational energy; Brown--York, Liu--Yau, and Wang--Yau masses; reference isometric embeddings; observer dependence and the optimal embedding equation; positivity, rigidity, asymptotic limits, and relations with horizons.
\subsection*{Week 6: Einstein Evolution as a Geometric PDE}
Maximal globally hyperbolic developments; diffeomorphism invariance and gauge fixing; wave and CMC-type reductions; hyperbolic and hyperbolic--elliptic formulations; local well-posedness, constraint propagation, energy estimates, and continuation criteria. The focus is the PDE structure needed for large-data problems, not a full small-data stability theory.
\subsection*{Week 7: Null Geometry and Dynamical Focusing}
Optical functions and double-null foliations; null second fundamental forms; expansion, shear, and torsion; Raychaudhuri and null structure equations; curvature flux; quantitative focusing and trapped-surface formation in vacuum and Einstein--matter systems. Classical short-pulse results will be used selectively to illustrate the mechanism.
\subsection*{Week 8: Black-Hole Formation by Boundary Effect}
Boundary mean curvature and null expansions; geometric radius criteria; boundary contraction and interior MOTS formation; dynamical realization of Yau's black-hole criterion; comparison between boundary-induced collapse and concentration of incoming radiation. A typical mechanism is
\[
\Rad(M_t)\,c(t)>C_{\mathrm{crit}},
\qquad
c(t)=\inf_{\partial M_t}\bigl(H_t-|\tr_{\partial M_t}k_t|\bigr).
\]
\subsection*{Week 9: Rough Solutions and Curvature-Level Control}
Metric, connection, and curvature regularity; scaling and critical Sobolev spaces; derivative loss in geometric gauges; rough null geometry; curvature-flux and elliptic estimates; breakdown criteria; weak limits and persistence of mass, scalar-curvature inequalities, boundary geometry, and horizon conditions.
\subsection*{Week 10: Concentration Compactness for Nonlinear Waves}
Brief introduction to the main idea, LP theory, scaling and criticality; failure of compactness in Sobolev embeddings; linear and nonlinear profile decomposition; asymptotic orthogonality; minimal blow-up solutions; compactness modulo symmetries; the compactness--rigidity method, with energy-critical wave equations as the principal model.
\subsection*{Week 11/12: Concentration, Rigidity, and Black-Hole Geometry}
Analytic versus geometric concentration: critical norms, matter density, curvature flux, scalar curvature, and quasilocal mass. Geometric profiles for threshold sequences: scales, blow-up limits, mass splitting, necks, and trapped bubbles. Quasilocal mass as a concentration functional, with positive-mass and quasilocal rigidity as possible rigidity mechanisms.
参考资料
Lecture Notes on Differential Geometry, R. Schoen, S-T Yau; General Relativity and Einstein equations, Yvone Choquet Bruhat, Proof of the positive mass theorem I,II, R.Schoen, S-T Yau, Geometry of Three Manifolds and Existence of Black holes due to boundary effect, S-T Yau, Existence of Black hole due to condensation of matter, R. Rchoen, S-T Yau, Dynamical Formation of Black Hole due to boundary effect in vacuum, P.Mondal, S-T Yau, Formation of Black holes in General Relativity, D. Christodoulou, Global well-posedness, scattering
and blow-up for the energy-critical
focusing non-linear wave equation, C. Kenig, F. Merle.
and blow-up for the energy-critical
focusing non-linear wave equation, C. Kenig, F. Merle.
听众
Advanced Undergraduate
, Graduate
视频公开
不公开
笔记公开
不公开
语言
英文
讲师介绍
Puskar Mondal is currently an assistant professor in BIMSA. He's been a fellow and lecturer at Harvard CMSA of Mathematics (his mentor is Prof. Shing-Tung Yau), and was a Ph.D. student at Yale University, under the supervision of Prof. Vincent Moncrief.