Integrally closed $\mathfrak{m}$-primary ideals have extremal resolutions

We show that every integrally closed $\mathfrak{m}$-primary ideal $I$ in a commutative Noetherian local ring $(R,\mathfrak{m},k)$ has maximal complexity and curvature, i.e., ${\mathrm{c}\mathrm{x}}_R(I)={\mathrm{c}\mathrm{x}}_R(k)$ and ${\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{v}}_R(I)={\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{v}}_R(k)$. As a consequence, we characterize complete intersection local rings in terms of complexity, curvature and complete intersection dimension of such ideals. The analogous results on projective, injective and Gorenstein dimensions are known. However, we provide short proofs of these results as well.