Branes, moduli spaces and smooth transition from big crunch to big bang

We consider branes $N$ in a Schwarzschild-${\text{AdS}}_{(n+2)}$ bulk, where the stress energy tensor is dominated by the energy density of a scalar fields map $\text{f}:N\text{ra}\text{mc}S$ with potential $V$, where $\text{mc}S$ is a semi-Riemannian moduli space. By transforming the field equation appropriately, we get an equivalent field equation that is smooth across the singularity $r=0$, and which has smooth and uniquely determined solutions which exist across the singularity in an interval $(-\text{e},\text{e})$. Restricting a solution to $(-\text{e},0)$ \resp $(0,\text{e})$, and assuming $n$ odd, we obtain branes $N$ \resp $\widehat{N}$ which together form a smooth hypersurface. Thus a smooth transition from big crunch to big bang is possible both geometrically as well as physically.