Log-Concavity and Fundamental Gaps on Complex Projective Domains
Abstract
We prove quantitative log-concavity and fundamental-gap estimates for Dirichlet Schrödinger operators on complex projective domains. With holomorphic sectional curvature four, the boundary condition II≥√(44/5) g suffices in every complex dimension under Hess V≥2|∇ V|g and oscV≤4n. It yields Hess(−logϕ1)>g and λ2−λ1>max{n+2+√(n2+5n+11), 2n+14/3}. The main difficulty is to control the distinguished Hopf direction together with a transverse Hessian block whose dimension grows with n. We extend modified-Hessian first-contact arguments by retaining the Hopf term and using longitudinal and transverse rotational tests. A trace minimization reduces all gradient orientations to three extreme directions. A dimension-scaled logarithmic barrier then compares their contact costs with a single limiting profile. A radial spectral estimate supplies the required quadratic-in-dimension spectral threshold from a dimension-independent boundary curvature bound. For n≥3 these are, to our knowledge, the first log-concavity estimates for domains in ℂℙn without symmetry assumptions. We also prove a sharp diameter bound and Hessian levels linear in n under larger dimension-free curvature bounds.
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BibTeX
@misc{cho2026logconcavity,
title = {{Log-Concavity and Fundamental Gaps on Complex Projective Domains}},
author = {Gunhee Cho},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35431},
eprint = {2610.35431},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35431},
}
RIS
TY - UNPB
TI - Log-Concavity and Fundamental Gaps on Complex Projective Domains
AU - Gunhee Cho
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35431
UR - https://arxiv.org/abs/2610.35431
AB - We prove quantitative log-concavity and fundamental-gap estimates for Dirichlet Schrödinger operators on complex projective domains. With holomorphic sectional curvature four, the boundary condition II≥√(44/5) g suffices in every complex dimension under Hess V≥2|∇ V|g and oscV≤4n. It yields Hess(−logϕ_1)>g and λ_2−λ_1>max{n+2+√(n^2+5n+11), 2n+14/3}. The main difficulty is to control the distinguished Hopf direction together with a transverse Hessian block whose dimension grows with n. We extend modified-Hessian first-contact arguments by retaining the Hopf term and using longitudinal and transverse rotational tests. A trace minimization reduces all gradient orientations to three extreme directions. A dimension-scaled logarithmic barrier then compares their contact costs with a single limiting profile. A radial spectral estimate supplies the required quadratic-in-dimension spectral threshold from a dimension-independent boundary curvature bound. For n≥3 these are, to our knowledge, the first log-concavity estimates for domains in ℂℙ^n without symmetry assumptions. We also prove a sharp diameter bound and Hessian levels linear in n under larger dimension-free curvature bounds.
ER -