Principal $p-$frequency estimates on non-compact manifolds with negative Ricci curvature

We establish a lower bound for the principal $p-$frequency ${\lambda }_{1,p}(\mathrm{\Omega })$ on a bounded domain $\mathrm{\Omega }$ in a non-compact Riemannian manifold of dimension $n\mathrm{.}$ Under the assumption that the Ricci curvature satisfies $\mathrm{Ric}\ge (n-1)K$ with $K<0,$ we prove that ${\lambda }_{1,p}(\mathrm{\Omega })>{\bar{\lambda }}_{D,K,n}$, where $D$ is the diameter of $\mathrm{\Omega }$ and ${\bar{\lambda }}_{D,K,n}$ is explicitly defined as the first eigenvalue of an associated one-dimensional ordinary differential equation model that incorporates both $D$ and $K\mathrm{.}$ 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.