T. Grafke, S. Scholtes, A. Wagner, M. Westdickenberg

#### Abstract

We explore recent progress and open questions concerning local minima and saddle points of the Cahn-Hilliard energy in \(d\ge 2\) and the critical parameter regime of large system size and mean value close to \(-1\). We employ the String Method of E, Ren, and Vanden-Eijnden — a numerical algorithm for computing transition pathways in complex systems — in \(d=2\) to gain additional insight into the properties of the minima and saddle point. Motivated by the numerical observations, we adapt a method of Caffarelli and Spruck to study convexity of level sets in \(d\ge2\).