McKean Rigidity for Cocompact Negatively Curved Manifolds and the \(p\)-Laplacian

Abstract

Let \((M^m,g)\) be a closed Riemannian manifold with \(\sec_g\leq-1\). We prove that the bottom spectrum of its universal cover attains McKean's lower bound if and only if the universal cover is hyperbolic space of constant sectional curvature \(-1\). More generally, for every \(1<p<\infty\), the variational \(p\)-fundamental tone satisfies \[ \lambda_{1,p}(\wti M) \geq\left(\frac{m-1}{p}\right)^p, \] and equality for some \(p\in(1,\infty)\) holds if and only if \(\wti M\cong\bH^m(-1)\). In that case, equality holds for every \(p\in(1,\infty)\). The proof converts the two McKean defects of a minimizing sequence into a stationary probability measure on the compact horospherical suspension; heat-kernel positivity then forces its zero-defect support to contain a complete leaf.