The anisotropic Weyl law and Grassmannian equidistribution

Abstract

We establish a Weyl law for the codimension-one anisotropic volume spectrum on compact Riemannian manifolds. We then use this law to obtain equidistribution in the full Grassmannian bundle while keeping the ambient metric fixed. More precisely, on every closed connected smooth Riemannian manifold of dimension three through six, we construct smooth, embedded hypersurfaces stationary for anisotropies smoothly converging to area. The cumulative sum of their area measures, when lifted to the Grassmannian bundle, converges to the uniform measure.