一、課程背景與目的：

This mini-course will focus on conformal and CR geometry. Conformal and CR geometry has been a rich topic of study going back to the work of Cartan in the early 20th century. It continues to be the source of interesting new problems as well as to spawn new developments in related areas. It is characterized as a meeting ground of researchers from a wide variety of backgrounds in geometry, analysis, and algebra. 

 

二、課程之大綱與講者：

Lecturer: Professor Sun-Yung Alice Chang; Princeton University

Lecture 1 and Lecture 2 (On Dec. 29-30, 2019)

Title: Compactness of Conformally Compact Einstein Manifolds in Dimension 4.

Abstract: Given a manifold (Mⁿ,[h]), when is it the boundary of a conformally compact Einstein manifold (Xⁿ⁺¹,g⁺) with r²g⁺|M=h? This problem of finding “conformal filling in” is motivated by problems in the AdS/CFT correspondence in quantum gravity (proposed by Maldacena in 1998) and from the geometric considerations to study the structure of non- compact asymptotically hyperbolic Einstein manifolds. The problem is largely open, but recently there has been substantial progress made in this research area. I will present the background and some recent progress concerning the aspects of the existence and non-existence, the uniqueness and compactness results of this problem. 

Lecture 3 (on Jan. 2, 2020)

Title: Improved Moser-Trudinger-Onofri Inequality under Constraints.

Abstract: I will report some recent joint work with Fengbo Hang. A classical result of Aubin states that the constant in Moser- Trudinger-Onofri inequality on S² can be imporved for furnctions with zero first order moments of the area element. We generalize it to higher order moments case. These new inequalities bear similarity to a sequence of Lebedev- Milin type inequalities on S¹ coming from the work of Grenander-Szego on Toeplitz determinants (as pointed out by Widom). We also discuss the related sharp inequality by the method of perturbation.

 

Lecturer: Professor Paul Yang; Princeton University

Lecture 4  (On Dec. 29, 2019)

Title: Theory of Surfaces in the Heisenberg.

Abstract: The classification of singular points, application to the isoperimetric problem.
The analogue of the codazzi equation.
CR analogues of the Willmore energy and examples.

Lecture 5  (On Dec. 30, 2019)

Title: The CR Yamabe Equation.

Abstract: The positivity criteria for the CR Paneitz operator, application to the embedding problem.
The positive mass theorem and the Sobolev quotient.
Counter examples.

Lecture 6  (on Jan. 2, 2020)

Title: The Q-prime Curvature.

Abstract: The analogue of the Q-curvature equation, extension of the rigidity theorem of Gursky.
The total Q-prime curvature as a global invariant.

** Professor Sun-Yung Alice Chang and Paul Yang will each give one lecture in conference from Jan. 4-9.