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

TL;DR

This paper refines a lower bound for the first Dirichlet eigenvalue of a Riemannian manifold by improving Ling's gradient-comparison method through variance-based averaging, resulting in a strictly stronger bound.

Contribution

It introduces a variance-based refinement to Ling's gradient-comparison method, enhancing the explicit in-diameter lower bound for the first Dirichlet eigenvalue.

Findings

The new bound is strictly stronger than Ling's original estimate for all positive curvature K.

The refinement retains more quantitative information by incorporating variance into the averaging process.

The method provides a closed-form explicit in-diameter bound applicable to compact manifolds with boundary.

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 $\lambda$ the first Dirichlet eigenvalue of the Laplacian. Ling's gradient-comparison method (Ling, 2006) provides an explicit lower bound for $\lambda$ 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-H\"older 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 $K>0$.