How the $p$-Bergman metric approaches the Carath\'eodory metric

arXiv:2611.08616v1 Announce Type: new Abstract: We study the rate at which the $p$-Bergman metric approaches the Carath\'eodory metric on convex domains with measures comparable to $\delta^\alpha dV$, $\alpha>-1$. On smooth convex domains of finite type the logarithmic coefficient is bounded above by the harmonic-measure average of the tangential multitype dimension plus $1+\alpha$, 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 $2+\alpha$ from an upper bound $7/4+\alpha$; 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 $2+\alpha$. A $C^2$ example has negative winding but coefficient $2+\alpha$, with a $-\frac12\log\log p$ correction, so boundary regularity is essential to the gap theorem.

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