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 $δ^αdV$, $α>-1$. 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 $\mathbb{C}^2$ 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 $-\frac12\log\log p$ correction, so boundary regularity is essential to the gap theorem.