We introduce an asymptotic quantity that counts area-minimizing surfaces in negatively curved closed 3-manifolds and show that quantity to only be minimized, among all metrics of sectional curvature $\le -1$, by the hyperbolic metric.

We construct motives over the rational numbers associated with symmetric power moments of Kloosterman sums, and prove that their L-functions extend meromorphically to the complex plane and satisfy a functional equation conjectured by Broadhurst and Roberts. Although the motives in question turn out to be “classical,” we compute their Hodge numbers by means of the irregular Hodge filtration on their realizations as exponential mixed Hodge structures. We show that all Hodge numbers are either zero or one, which implies potential automorphy thanks to recent results of Patrikis and Taylor.

If f is a function supported on the truncated paraboloid in ${\mathbb{R}}^3$ and E is the corresponding extension operator, then we prove that for all $p>3+3∕13$, $\Vert Ef{\Vert }_{L^p({\mathbb{R}}^3)}\le C\Vert f{\Vert }_{L^{\mathrm{\infty }}}$. The proof combines Wolff’s two ends argument with polynomial partitioning techniques. We also observe some geometric structures in the wave packets.