Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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}.

30
The categories of this paper are: q-alg
25
The categories of this paper are: patt-sol
17
The categories of this paper are: q-fin
5
The categories of this paper are: gr-qc
26
The categories of this paper are: bayes-an
23
The categories of this paper are: solv-int
20
The categories of this paper are: acc-phys
12
The categories of this paper are: nucl-th
15
The categories of this paper are: nucl-ex
21
The categories of this paper are: adap-org
10
The categories of this paper are: q-bio
32
The categories of this paper are: atom-ph
29
The categories of this paper are: funct-an
35
The categories of this paper are: mtrl-th
24
The categories of this paper are: dg-ga
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The categories of this paper are: cmp-lg