Compactness of conformally compact Einstein manifolds
speaker: Sun-Yung Alice Chang (Princeton University)abstract:
Given a manifold (Mn; h), when is it the boundary of a conformally compact Einstein manifold (Xn+1; g+) with r2g+
M = h for some defining function r on Xn+1? This problem of finding ”conformal filling in” is motivated by problems in the AdSCFT 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.In this mini-course, instead of addressing the existence problem of a conformal filling in, we will discuss the compactness problem. That is, given a sequence of compactification metrics of conformally compact Einstein (CCE) manifolds with boundary, we will study the compactness of the sequence under the assumption of the compactness of their restrictions on the boundary.In the first lecture, I will give a brief survey of selected results in this research area. We then discuss some main techniques, including the some works of R. Graham which lay the foundation of the subsequent research, the control of of interior Yamabe invariants in terms of boundary Yamabe invariants for CCE manifolds, Lee’s metric, Poisson equations of the scattering matrix. In the second lecture, I will discuss some earlier joint works of C-Yuxin Ge, C-Ge-Jie Qing, on a perturbation compactness Theorem, and as a consequence the uniqueness of conformal filling metrics for the class of boundary metrics on the sphere which are close to the standard canonical metric on the sphere, the existence of such conformal filling metrics
were established by Graham-Lee in 1991.
In the third lecture, I will describe a recent joint work of C-Ge, in which we establish
a general compactness theorem for 4-dimensional CCE manifolds under the conformally
invariant assumptions: the compactness of the boundary metrics, L
2 bound of their Weyl curvature, and some topological assumptions.
timetable:
Mon 6 Jun, 14:00 - 14:45, Aula Dini
Mon 6 Jun, 14:45 - 15:30, Aula Dini
Tue 7 Jun, 9:00 - 9:45, Aula Dini
Tue 7 Jun, 9:45 - 10:30, Aula Dini
Wed 8 Jun, 9:00 - 9:45, Aula Dini
Wed 8 Jun, 9:45 - 10:30, Aula Dini
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