PreprintCC BY 4.0arXiv
Log-Concavity of First Dirichlet Eigenfunctions on ℂℙ2
Abstract
We study the log-concavity property of first Dirichlet eigenfunctions on domains in ℂℙ2. For every smooth $1$-convex domain Ω⊂ℂℙ2, we prove the quantitative estimate ∇2(−log u) > max{ψ(s),8/5}g, where ψ(|grad f|2) = s/(√(1+s))−log(1+s), for its first Dirichlet eigenfunction u. In particular, u is strictly log-concave. As consequences, we obtain a uniform convexity estimate for the regular level sets of u and the fundamental gap bound λ2−λ1>46/5.
Related
- Strict Log-Concavity of First Dirichlet Eigenfunctions under Symmetry in ℂℙn
- Agmon estimates for the difference of exact and approximate Dirichlet eigenfunctions for difference operators
- Generic boundary behavior of Dirichlet eigenfunctions
- Minimal immersions by first eigenfunctions of Riemann surfaces with large conformal automorphism groups. The Klein quartic
- Neumann--Dirichlet eigenvalue comparison at the first Dirichlet threshold in the plane
BibTeX
@misc{xia2026logconcavity,
title = {{Log-Concavity of First Dirichlet Eigenfunctions on $\mathbbCP^2$}},
author = {Yusen Xia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22659},
eprint = {2609.22659},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22659},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Log-Concavity of First Dirichlet Eigenfunctions on ℂℙ^2
AU - Yusen Xia
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22659
UR - https://arxiv.org/abs/2609.22659
AB - We study the log-concavity property of first Dirichlet eigenfunctions on domains in ℂℙ^2. For every smooth $1$-convex domain Ω⊂ℂℙ^2, we prove the quantitative estimate ∇^2(−log u) > max{ψ(s),8/5}g, where ψ(|grad f|^2) = s/(√(1+s))−log(1+s), for its first Dirichlet eigenfunction u. In particular, u is strictly log-concave. As consequences, we obtain a uniform convexity estimate for the regular level sets of u and the fundamental gap bound λ_2−λ_1>46/5.
ER -