Universal self-gravitating skyrmions

The self-gravitating skyrmion is an exact solution of the Einstein $SU(2)$-Skyrme model describing a topological soliton with baryon number $B=1$, living in a $4$-dimensional space-time in the presence of a cosmological constant. Here we show that, using the maximal embedding Ansatz of $SU(2)$ into $SU(N)$ in the Euler angles parametrization, this solution can be generalized to include arbitrary values of the flavor number and, consequently, allowing higher values of the topological charge. Also, we show that higher-order corrections in the 't Hooft expansion can be considered while still preserving the analytical nature of the solutions. Finally we will show that from the gravitational solutions it is possible to construct skyrmions in flat space-time at a finite volume.