Boundary Regularity for Complete Conformal Metrics

Definitions and mathematical statement

Let $\Omega\subseteq{\mathbb{C}}$ be a domain, let $\Gamma\subseteq\partial\Omega$, and fix $z_0\in\Omega$. Under the standard regular-metric convention, let $\lambda\in C^2(\Omega)$ be strictly positive and let $\lambda(z)|{\,\mathrm{d}} z|$ be its conformal length element. Its intrinsic distance is \[ d_\lambda(z,w)=\inf_\gamma\int_\gamma\lambda(\zeta)|{\,\mathrm{d}}\zeta|, \] and its Gaussian curvature is \[ K_\lambda(z)=-\lambda(z)^{-2}\Delta\log\lambda(z). \] Assume the source's bound $K_\lambda\geq-4$. This is a lower curvature bound and is not replaced by the opposite inequality from another Ahlfors-lemma convention.

The two conditions to compare are \[ \forall\xi\in\Gamma,\qquad \lim_{\Omega\ni z\to\xi}d_\lambda(z,z_0)=\infty \] and \[ \forall\xi\in\Gamma,\qquad \lim_{\Omega\ni z\to\xi}\lambda(z)=\infty. \] Approach is Euclidean approach from the domain, with no nontangential restriction. These are pointwise conditions at every boundary point, not a separately stated uniform limit over $\Gamma$. The distance condition is the source's specified local completeness near $\Gamma$, not an assertion of completeness at the whole boundary or at infinity. Its validity is independent of the chosen interior basepoint by the triangle inequality.

The problem asks which hypotheses on the boundary set and its relation to $\Omega$ make these conditions equivalent for the indicated metrics, and which such regularity assumptions are sharp. The source does not identify a scale of regularity classes, local one-sided boundary geometry, endpoint conventions, or a precise least regularity index. Smoothness of an abstract subset $\Gamma$ and smoothness of the actual local boundary of $\Omega$ must be distinguished. The standard $C^2$ positive-density meaning of regular also requires source confirmation; singular or vanishing conformal metrics would need a different definition.

The supplied assessment checked 2026-08-21 reports the $C^2$-subarc case proved by the cited boundary Ahlfors lemma, while the minimal boundary regularity remains a stated problem in the checked record. The exact subarc and metric hypotheses of that result are not reproduced, so it is not extended to arbitrary $C^2$ subsets or rough boundaries.

Short English statement

Under Gaussian curvature at least $-4$, determine when infinite intrinsic distance to each boundary point is equivalent to blow-up of the conformal density, and how little boundary regularity suffices. The exact approach, metric, and boundary-arc conventions are stated explicitly.

Sources

• Local UnsolvedMath registry, permanent ID 30000704, catalogue entry, Boundary Regularity for Complete Conformal Metrics.
• Normal Families and Complex Dynamics (2007), Oberwolfach Reports, pp. 487--548, DOI https://doi.org/10.4171/owr/2007/09.
• Supplied reference "boundary Ahlfors lemma": https://doi.org/10.1007/s11854-007-0009-x.
• Supplied literature triage checked 2026-08-21 reports the $C^2$-subarc case and an unresolved sharp threshold, without full result hypotheses.