Energy barrier and $Gamma$-convergence in the $d$-dimensional Cahn-Hilliard equation

We study the d-dimensional Cahn-Hilliard equation on the flat torus in a parameter regime in which the system size is large and the mean value is close---but not too close---to -1. We are particularly interested in a quantitative description of the energy landscape in the case in which the uniform state is a local but not global energy minimizer. In this setting, we derive a sharp leading order estimate of the size of the energy barrier surrounding the uniform state. A sharp interface version of the proof leads to a $Gamma$-limit of the rescaled energy gap between a given function and the uniform state.