MATH 668--TOPICS IN DIFFERENTIAL GEOMETRY--SPRING 2023, A. FREIRE

Tu+Th 2:30--3:45, Ayres G013

Office hours: Th 4:00--4:45 (Ayres 325), or by appointment.

Texts: 1) Dan A. Lee, Geometric Relativity, AMS/GSM 201

2) Demetrios Christodoulou, Mathematical Problems of General Relativity I, EMS/Zurich Lectures in Advsnced Math
(2.3 and Chapter 3)

Course outline and Reading List
(Classical and recent research papers and surveys)

Goal: Introduction to problems in Riemannian geometry (and asymptotic invariants) motivated by General Relativity, from the 1980s to
the current research literature

Prerequisites: some knowledge of Riemannian geometry, and the main results on elliptic PDE in Sobolev spaces.

Grading; based on attendance and presentation of a topic in the course.

TOPICS FOR STUDENT PRESENTATIONS


COURSE LOG

1/24 Tu  Riemann curvature tensor: irreducible decomposition, conformal change of metric
Ref: [Lee], ch 1; Lee-Parker survey, Schoen 1987 Montecatini survey (see the reading list)

Conformal change formulas

1/26 Th First variation of the scalar curvature and applications/critical points of EInstein-Hilbert functional
Linearization of scalar curvature

1/31 Tu Spatial Schwarzschild family in dimension n/motivation for ADM mass/definition of AF manifold

2/2 Th Independence of ADM mass from the exhaustion [Bartnik]/, indep of chart at infinity ([Bartnik, statement only]/
survey of scalar curvature results ([Lee, ch. 1]/ non-PSC manifolds with nonnegative R are Ricci-flat (start)

2/7 Tu Intro to the Yamabe problem ([Lee-Parker, Schoen-Montecatini]

2/9 Th Yamabe problem: best Sobolev constant, variational approach
Notes on the Yamabe problem
(Guide to Lee&Parker's 1987 survey--essentially complete, 16pp.)

2/14 Tu Yamabe problem: existence for the subcritical problem

2/16 Th Yamabe problem: convergence criterion for subcritical solutions

2/21 Tu Yamabe problem:  Alternative expression for ADM mass/ Reduction  of PMT to conf flat, harm flat case

2/23 Th Yamabe problem: Conformal blowup, estimate for a test function

2/28 Tu: Solution of the Yamabe problem using the PMT (conclusion)

3/2 Th: First and second variations of hypersurface volume

First and second variation for hypersurfaces: derivation

3/7 Tu 2nd variation, application: scalar curvature of rot sym metrics, DeSitterSchw, AdSS metrics

3/9 Th class canceled

3/14, 3/16: SPRING BREAK

Plan for the second half

3/21 to 5/9: 16 lectures planned (inc. 1 to replace 3/9)

PMT proof: 4 lectures

3/21 Tu: PMT in special cases/Weighted Sobolev and Hoelder spaces/Reduction to scalar-flat (start)

3/23 Th: reduction to harmonic-flat, Lohkamp's reduction

3/28 Tu Continuity of mass, preparation theorem, geometric heredity

3/30 Th topological heredity principle/ uniqueness proof.

4/4 Tu Two theorems on mass and Ricci curvature (Tariq, Nathan)--papers by Bartnik and Herzlich

4/6 Th: "no class day" (UTK)

4/11, 4/13, 4/18, 4/20: Riem Penrose ineq. and IMCF

4/25, 4/27, 5/2: Bray's approach to RPI

5/2Tu  (supplemental class) Student presentations: Ben, Steve

5/4 Th Student presentations: Ivy, Sathya

5/9 Tu Student presentations: Bryan, George