to the new Penrose limit of $\mbox{AdS}_5\times S^5$ we also explore Penrose limits of $\mbox{AdS}_4 \times {\mathbb{C}}P^3$. Here two different classes of Penrose limits have been found, one in which there are no explicit space-like isometries [@Nishioka:2008gz; @Gaiotto:2008cg] and another in which there are two explicit space-like isometries [@Grignani:2008is; @Astolfi:2009qh] which makes it suitable for studying the $SU(2)\times SU(2)$ sector of ABJM theory. We find in this paper a new Penrose limit of the $\mbox{AdS}_4 \times {\mathbb{C}}P^3$ background giving a pp-wave background with one explicit space-like isometry.

The new Penrose limit of $\mbox{AdS}_5\times S^5$ found in this paper is also relevant for studying the finite temperature behavior of AdS/CFT. It is conjectured that the confinement/deconfinement transition temperature of planar $\mathcal{N}=4$ SYM on $R\times S^3$ is dual to the Hagedorn temperature of type IIB string theory on $\mbox{AdS}_5 \times S^5$ [@Witten:1998zw; @Sundborg:1999ue; @Polyakov:2001af; @Aharony:2003sx]. Using the Penrose limit [@Bertolini:2002nr] this was shown quantitatively to be true [@Harmark:2006ta] by matching the confiment/deconfinement temperature of planar $\mathcal{N}=4$ SYM on $R\times S^3$ in a limit with R-charge chemical potentials to the Hagedorn temperature of type IIB string on the pp-wave background of [@Bertolini:2002nr][^3]. We furthermore expect that our results could help in understanding more 