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

TL;DR

This work strengthens Ling's in-diameter bound for the first Dirichlet eigenvalue $\lambda$ of the Laplacian on a compact Riemannian manifold with boundary by incorporating a variance term into the one-dimensional comparison. By exploiting the uniform strong convexity of $x^{-1/2}$, the authors derive a variance-refined integral inequality and compute the variance of the auxiliary function $\xi$, obtaining an explicit closed-form bound $\lambda \ge \frac{(\alpha+D) + \sqrt{(\alpha+D)^2 + V\alpha^2}}{2}$ with $\alpha=\frac{(n-1)K}{2}$, $D=\frac{\pi^2}{\tilde d^2}$, and $V=4\zeta(3)-\frac{1}{3}(\pi^2+4)>0$. The refinement yields a strict improvement over Ling's bound $\lambda \ge \alpha+D$ for $K>0$, with a universal relative gain around 4.5% and greater when curvature dominates diameter. The approach highlights that the oscillation of the one-dimensional comparison function can be quantitatively retained, offering a robust, closed-form enhancement with implications for spectral geometry under positive Ricci curvature.

Abstract

Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold with nonempty boundary and $n\geq 2$. 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 $λ$ the first Dirichlet eigenvalue of the Laplacian. Ling's gradient-comparison method (Ling, 2006) provides an explicit lower bound for $λ$ 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 $K>0$.

Theorems & Definitions (22)

• Theorem 1.1: Variance-refined in-diameter bound
• Remark 1.2
• Lemma 2.1: The auxiliary function $\xi$
• proof
• Lemma 2.2: Ling's integral inequality
• proof
• Remark 2.3
• Proposition 3.1: Variance improvement for $x^{-1/2}$
• proof
• Lemma 3.2: Mean and variance of $z(t)=1+\delta\xi(t)$