An Explicit Bound for the First Dirichlet Eigenvalue via Two Moments
Abstract
Let (Mn,g), n≥2, be a compact connected Riemannian manifold with smooth nonempty boundary, RicM≥(n−1)Kg for K>0, and nonnegative outward boundary mean curvature. We derive an explicit algebraic lower bound Bη for the first Dirichlet eigenvalue from Kasue's mixed inradius model. With R the inradius and s=4KR2/π2, the bound involves one square root and approximates the model eigenvalue with relative error On(s4) as s↓0 for fixed n. The construction uses the first two moments of a positive representation of the averaged model potential and a sharp pointwise corrector estimate. For the chosen spectral transfer, the first two moments determine an optimal bound. We also show that the model eigenvalue is the geometric infimum at fixed (n,K,R). For 0<s<1, it is approached by smooth collapsing balls with sectional curvature at least K and totally geodesic boundary. At the maximal inradius, the model value is attained by the round hemisphere. The bound strictly improves the variance refinement throughout 0<s≤1. We also give a direct algebraic comparison with an optimized power-test benchmark.
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BibTeX
@misc{schurmann2026explicit,
title = {{An Explicit Bound for the First Dirichlet Eigenvalue via Two Moments}},
author = {Thomas Schürmann},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14069},
eprint = {2610.14069},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14069},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Explicit Bound for the First Dirichlet Eigenvalue via Two Moments
AU - Thomas Schürmann
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.14069
UR - https://arxiv.org/abs/2610.14069
AB - Let (M^n,g), n≥2, be a compact connected Riemannian manifold with smooth nonempty boundary, Ric_M≥(n−1)Kg for K>0, and nonnegative outward boundary mean curvature. We derive an explicit algebraic lower bound B_η for the first Dirichlet eigenvalue from Kasue's mixed inradius model. With R the inradius and s=4KR^2/π^2, the bound involves one square root and approximates the model eigenvalue with relative error O_n(s^4) as s↓0 for fixed n. The construction uses the first two moments of a positive representation of the averaged model potential and a sharp pointwise corrector estimate. For the chosen spectral transfer, the first two moments determine an optimal bound. We also show that the model eigenvalue is the geometric infimum at fixed (n,K,R). For 0<s<1, it is approached by smooth collapsing balls with sectional curvature at least K and totally geodesic boundary. At the maximal inradius, the model value is attained by the round hemisphere. The bound strictly improves the variance refinement throughout 0<s≤1. We also give a direct algebraic comparison with an optimized power-test benchmark.
ER -