3 jun 2022 -- 11:00   [open in google calendar]

Aula Volterra

The seminar is in presence. Please write to andrea.malchiodi@sns.it if you want to attend online via Teams

Abstract.

Let $X^4$ be a differenntial 4-manifold with boundary $M^3 = \partial X^4$. Given the conformal class $(M, [h])$ of a Riemannian metric $h$ on $M$, we try to find ”conformal filling in” a asymptot- ically hyperbolic Einstein $g_+$ on $X$ such that $r^2 g_{+}\mid_{M} = h$ for some defining function $r$ on $X$. The study of complete AH Einstein manifolds has become very active due to the $AdS/CFT$ correspondence in string theory.

In this talk, instead of addressing the existence problem of a conformal filling in, we discuss the compactness problem, that is, how the compactness of the sequence of conformal infinity metrics leads to the compactness result of the compactified filling in AHE manifolds under the suitable assumptions on the topology of $X$ and some conformal invariants. We briefly survey some known results then report recent joint work in progress with Alice Chang. Some applications will be discussed.