# A non-compactness result on the fractional Yamabe problem in large dimensions

##
Monica Musso

created by daniele on 27 Nov 2015

2 dec 2015
-- 15:00

Aula Mancini, SNS, Pisa

Seminario di Matematica, Scuola Normale Superiore di Pisa

**Abstract.**

Let $(X^{n+1}, g^+)$ be an $(n+1)$-dimensional asymptotically hyperbolic
manifold with a conformal infinity $(M^n, [h])$. The fractional Yamabe
problem addresses to solve \[P^{\gamma}[g^+,h] (u) = cu^{n+2\gamma \over
n-2\gamma}, \quad u > 0 \quad \text{on } M\] where $c \in {\mathbb{R}} $ and
$P^{\gamma}[g^+,h]$ is the fractional conformal Laplacian whose principal
symbol is $(-\Delta)^{\gamma}$. We construct a metric on the half space $X
= {\mathbb{R}}^{n+1}_+$, which is conformally equivalent to the unit ball,
for which the solution set of the fractional Yamabe equation is non-compact
provided that $n \ge 24$ for $\gamma \in (0, \gamma^*)$ and $n \ge 25$ for
$\gamma \in [\gamma^*,1)$ where $\gamma^* \in (0, 1)$ is a certain
transition exponent. The value of $\gamma^*$ turns out to be approximately
0.940197. This is a joint work with Seunghyeok Kim and Juncheng Wei.