International Journal of Mathematics

In this paper, we consider the system of $p$-Laplacian equations with critical growth

We obtain sharp volume bound for a conic 2-sphere in terms of its Gaussian curvature bound. We also give the geometric models realizing the extremal volume. In particular, when the curvature is bounded in absolute value by $1$, we compute the minimal volume of a conic sphere in the sense of Gromov. In order to apply the level set analysis and isoperimetric inequality as in our previous works, we develop new analytical tools to treat regions with vanishing curvature.

We show the convergence properties of the eigenvalues of the Dirac operator on a spin manifold with a Riemannian flow when the metric is collapsed along the flow.

We show that the core of a weakly group-theoretical braided fusion category ${𝒞}$ is equivalent as a braided fusion category to a tensor product ${ℬ}⊠{𝒟}$, where ${𝒟}$ is a pointed weakly anisotropic braided fusion category, and ${ℬ}\cong Vect$ or ${ℬ}$ is an Ising braided category. In particular, if ${𝒞}$ is integral, then its core is a pointed weakly anisotropic braided fusion category. As an application we give a characterization of the solvability of a weakly group-theoretical braided fusion category. We also prove that an integral modular category all of whose simple objects have Frobenius–Perron dimension at most 2 is necessarily group-theoretical.

We compute the A-polynomial 2-tuple of twisted Whitehead links. As applications, we determine the canonical component of twisted Whitehead links and give a formula for the volume of twisted Whitehead link cone-manifolds.

Higher rank Brill–Noether theory for genus 6 is especially interesting as, even in the general case, some unexpected phenomena arise which are absent in lower genus. Moreover, it is the first case for which there exist curves of Clifford dimension greater than 1 (smooth plane quintics). In all cases, we obtain new upper bounds for non-emptiness of Brill–Noether loci and construct many examples which approach these upper bounds more closely than those that are well known. Some of our examples of non-empty Brill–Noether loci have negative Brill–Noether numbers.