Large-Scale Regularity Meets the Weyl Law on Ricci Shrinkers

We establish a large-scale regularity property for complete gradient Ricci shrinkers and apply it to the spectral asymptotics of the weighted Laplacian. Although uniform bounded geometry is not known for general Ricci shrinkers, we prove that, inside large geodesic balls, the region where the curvature radius degenerates at the reciprocal scale occupies an asymptotically negligible proportion of the volume. The proof uses the Ricci flow induced by the shrinker, together with the compactness and regularity theories developed by Bamler and Li-Wang. As an application, we prove that the weighted Laplacian, or equivalently its conjugate Schrödinger operator, on every complete gradient Ricci shrinker satisfies the classical Weyl law.