Uniqueness results for critical points of a non-local isoperimetric problem via curve shortening

Using area-preserving curve shortening flow, and a new inequality relating the potential generated by a set to its curvature, we study a non-local isoperimetric problem which arises in the study of di-block copolymer melts, also referred to as the Ohta-Kawasaki energy. We are able to show that the only connected critical point is the ball under mild assumptions on the boundary, in the small energy/mass regime. In particular this class includes all rectifiable, connected 1-manifolds in $\mathbb{R}^2$. We also classify the simply connected critical points on the torus in this regime, showing the only possibilities are the stripe pattern and the ball. In $\mathbb{R}^2$, this can be seen as a partial union of the well known result of Fraenkel \cite{Fraenkel} for uniqueness of critical points to the Newtonian Potential energy, and Alexandrov for the perimeter functional \cite{alexandrov}, however restricted to the plane. The proof of the result in $\mathbb{R}^2$ is analogous to the curve shortening result due to Gage \cite{Gage2}, but involving a non-local perimeter functional, as we show the energy of convex sets strictly decreases along the flow. Using the same techniques we obtain a stability result for minimizers in $\mathbb{R}^2$ and for the stripe pattern on the torus, the latter of which was recently shown to be the global minimizer to the energy when the non-locality is sufficiently small \cite{sternberg}.