Topics in Geometric Analysis: The Yamabe Problem (course 748F)

University of Maryland

Department of Mathematics

Spring 2013






Time: Tuesdays, Thursdays at 2pm.
Room: 0411 Mathematics Building.

Teacher: Y.A. Rubinstein. Office hours: By appointment.

Course plan:
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 proof.

Main references:

T. Aubin, 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.

Schedule:

  • January 29
    Overview. Curvature and conformal deformations. The Yamabe energy.

    HW1.

  • January 31
    The Yamabe invariant. The conformal Laplacian.

    HW2.

  • February 5
    The subcritical Yamabe problem in the negative case. (Presentation: Formula for the curvature under conformal deformation.)

  • February 7
    Moser iteration. (Presentation: Existence of smooth positive first eigenfunction for the conformal Laplacian, and the Strong Maximum Principle.)

  • February 12
    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.)

  • February 14
    Presentation: Linear stability for the scalar curvature equation.

  • February 19
    Kazdan-Warner identity. Eigenfuctions on the sphere.

    HW3.

  • February 21
    The sharp Sobolev inequality on the sphere and its extremal functions. An outline of the remaining steps for the resolution of the Yamabe Problem.

  • February 26
    Proof of Aubin's inequality in higher dimensions. Stereographic projection and the Green's function.

  • February 28
    Presentation: Obstacle and free boundary problems for the Laplacian.

  • March 5
    Review of previous HWs.

    HW4.

  • March 7
    Presentation: Schouten-Weil and Obata--Ferrand-Lelong Theorems.

  • March 12
    Asymptotic expansion of the Green function.

  • March 14
    The global test function and the mass. (Presentation: Conformal normal coordinates.)

  • March 28
    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).

  • April 4
    Presentation: An alternative approach to constructing the test function.

  • April 9
    The mass - well-defined for an asymptotically flat manifold of high enough order. Weighted Sobolev/Holder spaces and harmonic coordinates. curvature.

  • April 16, 11am, room 3206
    Informal Geometric Analysis Seminar: Kahler-Einstein geometry (Simon Donaldson).

  • April 18
    The identification of the mass with the coefficient from the expansion.

  • April 23
    Presentation: every compact manifold admits a metric of constant negative scalar curvature. Presentation: Weighted Schauder theorem and Sobolev embedding.

  • April 25
    Presentation: Witten's proof of the Positive Mass Theorem.

  • April 30
    Presentation: Witten's proof of the Positive Mass Theorem - continued.

  • May 2
    Presentation: Weighted Schauder estimates on asymptotically flat manifolds.

  • May 7
    Presentation: Schoen-Yau's work on locally conformally flat manifolds and the mass.

  • May 9
    Presentation: Schoen-Yau's work on locally conformally flat manifolds and the mass - continued.