How the p-Bergman metric approaches the Carathéodory metric
Abstract
We study the rate at which the p-Bergman metric approaches the Carathéodory metric on convex domains with measures comparable to δαdV, α>−1. On smooth convex domains of finite type the logarithmic coefficient is bounded above by the harmonic-measure average of the tangential multitype dimension plus 1+α, with equality when almost every endpoint of the extremal disc is coercive. For a class of convex domains in ℂ2 with C4 boundary and rank-one real tangential contact along an axis disc, a winding number distinguishes the coefficient 2+α from an upper bound 7/4+α; quartic contact makes the latter bound exact. The exact value holds along the entire axis of explicit real-algebraic strongly pseudoconvex domains, including perturbations without the weighted circle action, although their pointwise boundary models agree up to unitary equivalence with those of a domain having coefficient 2+α. A C2 example has negative winding but coefficient 2+α, with a −1/2loglog p correction, so boundary regularity is essential to the gap theorem.
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BibTeX
@misc{cho2026pbergman,
title = {{How the $p$-Bergman metric approaches the Carathéodory metric}},
author = {Gunhee Cho},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08616},
eprint = {2611.08616},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08616},
}
RIS
TY - UNPB
TI - How the p-Bergman metric approaches the Carathéodory metric
AU - Gunhee Cho
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08616
UR - https://arxiv.org/abs/2611.08616
AB - We study the rate at which the p-Bergman metric approaches the Carathéodory metric on convex domains with measures comparable to δ^αdV, α>−1. On smooth convex domains of finite type the logarithmic coefficient is bounded above by the harmonic-measure average of the tangential multitype dimension plus 1+α, with equality when almost every endpoint of the extremal disc is coercive. For a class of convex domains in ℂ^2 with C^4 boundary and rank-one real tangential contact along an axis disc, a winding number distinguishes the coefficient 2+α from an upper bound 7/4+α; quartic contact makes the latter bound exact. The exact value holds along the entire axis of explicit real-algebraic strongly pseudoconvex domains, including perturbations without the weighted circle action, although their pointwise boundary models agree up to unitary equivalence with those of a domain having coefficient 2+α. A C^2 example has negative winding but coefficient 2+α, with a −1/2loglog p correction, so boundary regularity is essential to the gap theorem.
ER -