Variance-Refined In-Diameter Lower Bound for the First Dirichlet Eigenvalue
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908731782987776 |
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| author | Schürmann, Thomas |
| author_facet | Schürmann, Thomas |
| contents | 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 $K>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21517 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Variance-Refined In-Diameter Lower Bound for the First Dirichlet Eigenvalue Schürmann, Thomas Differential Geometry Analysis of PDEs Primary 58J50, 35P15, Secondary 53C21 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 $K>0$. |
| title | Variance-Refined In-Diameter Lower Bound for the First Dirichlet Eigenvalue |
| topic | Differential Geometry Analysis of PDEs Primary 58J50, 35P15, Secondary 53C21 |
| url | https://arxiv.org/abs/2512.21517 |