Invariant Distances in Complex Geometry

Invariant distances lie at the heart of complex geometry, providing intrinsic measures of separation between points in complex manifolds that remain unchanged under biholomorphic mappings. Chief examples include the Carathéodory distance, which arises from extremal holomorphic discs, and the Kobayashi distance, defined through chains of holomorphic maps into the unit disc. The Bergman and Lempert metrics complement these by incorporating volume and extremal mapping considerations. Together, these distances capture both local and global features of a domain’s complex structure, reflecting properties such as convexity, pseudoconvexity and boundary regularity. They underpin fundamental results on holomorphic extendability, boundary behaviour of mappings and complex dynamical systems. Recent developments have further explored novel metrics—such as the minimal metric, motivated by variational problems—and connections with Gromov hyperbolicity, which links complex-analytic curvature to large-scale geometric behaviour. Practical applications span moduli of Riemann surfaces, iteration theory of holomorphic self‐maps and estimates on solution spaces of partial differential equations arising in complex analysis.

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Researchers have established a Gehring–Hayman inequality for strongly pseudoconvex domains, proving that the length of any Kobayashi–Royden geodesic between two interior points is bounded above by a uniform constant times their Euclidean distance. This result creates a precise link between the complex-analytic metric and the ambient Euclidean geometry, with implications for boundary regularity and extension phenomena.

In a study of worm domains—examples of smoothly bounded pseudoconvex regions with intricate boundary structure—it has been shown that these domains fail to be Gromov hyperbolic with respect to the Kobayashi distance. This negative result delineates limitations of hyperbolicity techniques in complex dynamics and informs classification efforts for pseudoconvex domains according to their large-scale metric geometry.

Advances in the theory of the minimal metric have demonstrated that every bounded strongly minimally convex domain is Gromov hyperbolic, and that its Gromov compactification coincides with the Euclidean closure. Further comparisons between the minimal metric and the Hilbert metric in convex settings illuminate how different intrinsic distances encode geometric convexity and boundary regularity in complex spaces.

Invariant Distances in Complex Geometry publication trend

The graph below shows the total number of articles in invariant distances in complex geometry across all publications each year (not limited to Nature Index journals).

Technical terms

Kobayashi distance: The largest pseudodistance on a complex manifold that does not exceed the Poincaré distance under every holomorphic map from the unit disc.

Carathéodory distance: A pseudodistance defined by the supremum of Poincaré distances between images of two points under all holomorphic functions into the unit disc.

Lempert function: A function giving the optimal holomorphic disc through two points in a complex domain, whose logarithmic modulus yields an important invariant metric.

Pseudoconvex domain: A domain in complex space characterised by the sublevel sets of a plurisubharmonic exhaustion function, generalising convexity to complex analysis.

Gromov hyperbolicity: A coarse geometric condition on metric spaces indicating that geodesic triangles are thin, reflecting negative-curvature behaviour at large scale.

References

1. A Gehring–Hayman Inequality for Strongly Pseudoconvex Domains. International Mathematics Research Notices (2024).
2. Worm Domains are not Gromov Hyperbolic. The Journal of Geometric Analysis (2023).
3. On the Gromov hyperbolicity of the minimal metric. Mathematische Zeitschrift (2024).