  We consider the global attractor of the critical SQG semigroup $S(t)$ on the
scale-invariant space $H^1(\mathbb{T}^2)$. It was shown in~\cite{CTV13} that
this attractor is finite dimensional, and that it attracts uniformly bounded
sets in $H^{1+\delta}(\mathbb{T}^2)$ for any $\delta>0$, leaving open the
question of uniform attraction in $H^1(\mathbb{T}^2)$. In this paper we prove
the uniform attraction in $H^1(\mathbb{T}^2)$, by combining ideas from DeGiorgi
iteration and nonlinear maximum principles.
