Time: Tuesdays, Thursdays at 2pm.
Rubinstein. Office hours: By appointment.
Room: 0411 Mathematics Building.
The goal will be to give an introduction to the Yamabe Problem.
In 1960 Yamabe described a proof of the following fact:
Given a compact Riemannian manifold (M,g) there exists
a smooth positive function u on M such that the metric ug
has constant scalar curvature.
His proof was flawed, since it assumed that a certain critical
Sobolev embedding was compact. This gap was only fixed 25 years
later, through the work of Trudinger, Aubin, Schoen, and others.
This work provides a beautiful overview of key techniques in
Geometric Analysis. Time permitting, we will also discuss more
recent developements and open problems concerning the Yamabe flow,
Morse theory of the Yamabe energy, and compactness of solutions.
The course should be suitable for students with some background
in PDE or/and Differential Geometry. However, key elementary facts
will be recalled as we go along, and the detailled proof of some of
these facts will be given in the introductory course
MATH742 (Geometric Analysis) that will be offered in Autumn 2013.
Thus, MATH748F might be interesting also for rather beginning students.
At the end of the lectures (in addition to the usual lecture time),
students will give short presentations on a
regular basis about results that have been stated in class without full
Some nonlinear problems in Riemannian geometry, Springer, 1998.
D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of
Second Order, Springer, 2001.
J.M. Lee, T.H. Parker, The Yamabe Problem, Bull. AMS 17 (1987), 37-91.
P. Li, Geometric Analysis, Cambridge University Press, 2012.
R. Schoen, S.-T. Yau, Lectures on Differential Geometry, Int. Press, 1994.
Additional references: TBA as we go along.
Overview. Curvature and conformal deformations. The Yamabe energy.
The Yamabe invariant. The conformal Laplacian.
The subcritical Yamabe problem in the negative case.
(Presentation: Formula for the curvature
under conformal deformation.)
Moser iteration. (Presentation: Existence of smooth positive first
eigenfunction for the conformal Laplacian, and the Strong Maximum
Completion of the subcritical Yamabe problem in the zero and positive
case. The Yamabe invariant of the standard conformal class on the sphere,
and Aubin's inequality.
(Presentation: Hardy-Littlewood-Sobolev Lemma.)
Presentation: Linear stability for the scalar curvature equation.
Kazdan-Warner identity. Eigenfuctions on the sphere.
The sharp Sobolev inequality on the sphere and its extremal functions.
An outline of the remaining steps for the resolution of the Yamabe Problem.
Proof of Aubin's inequality in higher dimensions. Stereographic projection
and the Green's function.
Presentation: Obstacle and free boundary problems for the Laplacian.
Review of previous HWs.
Presentation: Schouten-Weil and Obata--Ferrand-Lelong Theorems.
Asymptotic expansion of the Green function.
The global test function and the mass.
(Presentation: Conformal normal coordinates.)
The mass - relation to the spherical density function and the
asymptotic expansions developed earlier.
April 2, Room 3206
Informal Geometric Analysis Seminar: Isoperimetric inequality and Q-curvature (Yi Wang).
Presentation: An alternative approach to constructing the test function.
The mass - well-defined for an asymptotically flat manifold of high enough
order. Weighted Sobolev/Holder spaces and harmonic coordinates.
April 16, 11am, room 3206
Informal Geometric Analysis Seminar: Kahler-Einstein geometry (Simon
The identification of the mass with the coefficient from the expansion.
Presentation: every compact manifold admits a metric of constant negative scalar curvature.
Presentation: Weighted Schauder theorem and Sobolev embedding.
Presentation: Witten's proof of the Positive Mass Theorem.
Presentation: Witten's proof of the Positive Mass Theorem - continued.
Presentation: Weighted Schauder estimates on asymptotically flat manifolds.
Presentation: Schoen-Yau's work on locally conformally flat manifolds and the mass.
Presentation: Schoen-Yau's work on locally conformally flat manifolds and the mass - continued.