Variance-Refined In-Diameter Lower Bound for the First Dirichlet Eigenvalue

Abstract

Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold with nonempty boundary and $n\ge 2$. Assume that $\mathrm{Ric}(M)\ge (n-1)K$ for some $K>0$ and that $\partial M$ has nonnegative mean curvature with respect to the outward unit normal. Denote by $\hat{I}»$ the first Dirichlet eigenvalue of the Laplacian. Ling's gradient-comparison method (Ling, 2006) provides an explicit lower bound for $\hat{I}»$ in terms of $K$ and the in-diameter $\tilde{d}$ (twice the maximal distance from a point of $M$ to $\partial M$). We isolate the only step in Ling's argument that loses quantitative information: a Jensen-Hölder averaging that replaces a nonconstant one-dimensional comparison function by its mean. Using the uniform strong convexity of $x\to x^{-1/2}$ on $(0,1]$, we refine this averaging by a variance term and thereby retain part of the discarded oscillation. This yields an explicit closed-form in-diameter bound that is strictly stronger than Ling's estimate for every $K>0$.

10 pages