What secret lets a hyperbolic domain self‑extend into the space of horofunctions?

Imagine a magic map that turns a hyperbolic domain into a larger picture without losing its shape. Researchers have found a condition that lets the identity map extend into the space of horofunctions. This condition also works for domains that are not Gromov‑hyperbolic but remain hyperbolic with respect to the Kobayashi distance. In practice, the map can self‑extend even if the domain is infinite and not hyperbolic in the Kobayashi sense. For planar domains, a large class of examples meets the condition, making the result more accessible. This opens new links between complex geometry and functional analysis, without relying on restrictive assumptions. In short, the study shows a hyperbolic domain can expand into a larger space while keeping its original shape. The finding is important because it connects hyperbolic geometry with function analysis. Horofunctions act like shadows describing a set’s shape, even when it is very large. This approach lets us study domains without artificially compressing or deforming them. The authors proved the condition holds for unlimited domains, expanding application possibilities. Moreover, the class of planar domains satisfying the condition is rich and varied, including curvilinear and polygonal shapes.

The paper investigates the Horofunction compactification of complete hyperbolic domains in ℂⁿ equipped with the Kobayashi distance k_Ω. Introduced by Gromov, the Horofunction compactification maps a set into a space of affine functions that, in the limit, capture its asymptotic shapes. The author proposes a sufficient condition for the identity map id: Ω → Ω to extend as an embedding \(\overline{Ω} ↪ \widehat{Ω}\), where \(\widehat{Ω}\) is the Horofunction compactification of (Ω,k_Ω). This condition requires that for every point x∈Ω and every horofunction h∈\widehat{Ω} there exists a sequence (y_n)⊂Ω with h(y_n)→h(x) and the Kobayashi distance between y_n and x remains bounded. Consequently, the identity map becomes an immersion at the boundary, preserving the domain’s geometric structure. The condition is expressed via controlled growth of horofunctions along Kobayashi‑limited curves and Busemann limits. This result is significant because it shows the Horofunction compactification can house domains that are not Gromov‑hyperbolic, thus widening the class of metric spaces under study. Moreover, the work provides a large family of planar hyperbolic domains—such as disk‑type or star‑type domains—that satisfy the condition, making the theorem usable in concrete settings. By linking hyperbolic geometry with functional analysis, the study opens new avenues for completing non‑compact metric spaces.