Principal $p-$frequency estimates on non-compact manifolds with negative Ricci curvature
Xiaoshang Jin, Zhiwei Lü
TL;DR
The paper studies the principal $p$-frequency $\lambda_{1,p}(\Omega)$ on bounded domains within complete non-compact manifolds under a negative Ricci curvature lower bound $\operatorname{Ric} \ge (n-1)K$, $K<0$. It introduces a one-dimensional model with weight $e^{(n-1)\sqrt{-K}t}$ and defines $\bar{\lambda}_{D,K,n}$ as the first eigenvalue of this model; the main result is the sharp bound $\lambda_{1,p}(\Omega) > \bar{\lambda}_{D,K,n}$, with $D$ the domain diameter. The method combines a Busemann-function gradient framework, a Barta-type inequality, and a reduction to the 1D model via a test function built from the model solution $w$, yielding a quantitative link between eigenvalues, diameter, and curvature. Sharpness is established through a construction of warped-product manifolds with cusp-like ends, showing the bound is tight and illustrates how negative curvature can sharply reduce eigenvalues as the domain grows. The work extends known $K=0$ results to the negative-curvature setting and provides detailed asymptotics in the small- and large-diameter regimes.
Abstract
We establish a lower bound for the principal $p-$frequency $λ_{1,p}(Ω)$ on a bounded domain $Ω$ in a non-compact Riemannian manifold of dimension $n.$ Under the assumption that the Ricci curvature satisfies $\operatorname{Ric} \geq (n-1)K$ with $K<0,$ we prove that $λ_{1,p}(Ω) > \barλ_{D,K,n}$, where $D$ is the diameter of $Ω$ and $\barλ_{D,K,n}$ is explicitly defined as the first eigenvalue of an associated one-dimensional ordinary differential equation model that incorporates both $D$ and $K.$ Moreover, the estimate is sharp. This work extends previous results for the case $K=0$ to the geometrically more complex setting of negative Ricci curvature, and providing a new quantitative connection between the eigenvalue, the diameter of domains, and the curvature lower bound.