Unbounded Invariant Distances

• Unbounded invariant distances are metric measures defined on moduli spaces that remain invariant under symmetry actions while being capable of reaching arbitrarily large values.
• They encode critical geometric and analytic information, revealing subtle boundary behaviors and directional divergences in complex, symplectic, and contact settings.
• Their construction employs boundary scaling, spectral invariants, and rescaling limits, with applications ranging from holomorphic mapping to spatial network analysis.

Unbounded invariant distances are metric-type quantities, defined on geometric or function-theoretic moduli spaces, that are both invariant under the symmetries of the problem (such as biholomorphic, Hamiltonian, or contactomorphism group actions) and—crucially—can attain arbitrarily large values on suitable pairs of points. Their paper arises across several branches of analysis, complex geometry, and symplectic/contact topology, and their structural properties are often dictated by sharp connections to boundary geometry, group orderability, or dynamical features.

1. Core Definitions and Geometric Framework

The notion of invariant distance appears in multiple contexts. In several complex variables, canonical examples include the Carathéodory, Kobayashi, and Bergman distances, all of which are invariant under biholomorphic maps. For a domain $D \subset \mathbb{C}^n$, precise lower bounds on these distances often reflect the subtle pseudoconvex geometry of $\partial D$. A significant structural insight is that—especially on domains with boundary degeneracies (e.g., pseudoconvex Levi corank one domains)—invariant distances may not merely diverge at the boundary, but their precise rate of divergence encodes complex analytic and geometric features of the boundary.

In symplectic and contact topology, invariant distances—such as the Hofer metric on Hamiltonian diffeomorphism groups, spectral invariants from Floer theory, or distances defined on isotopy classes (e.g., of Legendrians)—measure the "size" of isotopies or transformations, and unboundedness is tied to group-theoretic rigidity or the absence of certain symmetries.

An essential point is that "unbounded" here refers to the possibility that for arbitrarily large values (e.g., between submanifolds, Legendrian isotopy classes, or metric images), there is always a pair with distance exceeding any prescribed bound, reflecting either complexity, rigidity, or sharp geometric features.

2. Analytical and Algebraic Formulations

In complex geometry, the analysis of unbounded invariant distances centers on explicit lower and upper bounds. On smoothly bounded pseudoconvex Levi corank one domains, Balakumar, Mahajan, and Verma establish for the Kobayashi, Carathéodory, and Bergman distances between $z, w \in D$ the lower bound

$$d_D(z, w) \geq C \log\left(1 + \frac{Q(z, w)}{\delta_D(z)}\right) - C',$$

where $Q(z, w)$ aggregates weighted coordinate differences and $\delta_D(z)$ is the boundary distance. The weight structure reflects the directions in which the Levi form degenerates, inducing a slower divergence of the invariant metric in tangential directions. Further, the precise boundary expansions rely on normal forms for the defining function:

$$r(z) = r(z^0) + 2 \operatorname{Re}(z_n - S_n) + \sum_{j,k = 1}^{n-1} P_{j, k}(z - z^0) + R(z - z^0).$$

In symplectic topology, the unboundedness of the Hofer metric on spaces of Lagrangians and Hamiltonian diffeomorphisms reduces to the existence of functionals (such as spectral invariants or quasimorphisms) that grow without bound on sequences of isotopies. For example, in the work of Khanevsky, the Lagrangian Hofer distance on the space

$$\mathcal{L} = \{ \phi(L_0) : \phi \in \mathrm{Ham}(B^{2n}) \}$$

satisfies

$$\frac{\|f - g\|_\infty - C}{D} \leq d(\Psi(f), \Psi(g)) \leq \|f - g\|_\infty$$

where $\Psi$ is a map from compactly supported functions to Lagrangian submanifolds. The unboundedness follows as $\|f - g\|_\infty \to \infty$.

In contact topology, recent results show that invariant distances on Legendrian isotopy classes (on universal covers) can be defined via order structures and positive loops:

$$d(x, y) = \max\left\{ \ell_+(x,y), -\ell_-(x,y) \right\},$$

where $\ell_+$ and $\ell_-$ measure order-theoretic displacement via positive Legendrian loops, with the surprising consequence that such distances are necessarily discrete, and yet unbounded (Arlove, 24 Jul 2025).

3. Boundary Geometry, Unboundedness, and Sharpness

A central finding is that the unboundedness of invariant distances reflects subtle boundary geometry. In strongly pseudoconvex domains every boundary point enforces a comparable logarithmic growth for all invariant distances as points approach the boundary, while in Levi corank one settings degeneracy in the Levi form results in a logarithmic but directionally varying blow-up rate. For example, in complex ellipsoids $E(2m, 2, \dots, 2)$, the Kobayashi metric splits depending on tangential/normal components and can remain finite or diverge slowly depending on approach (1303.3439).

In symplectic topology, the construction of sets in $\mathbb{R}^d$ of slow-decaying density that avoid an unbounded collection of distances further demonstrates that one cannot strengthen certain density-implies-distance results: for any $f(R) \to 0$, it is possible to select $A \subset \mathbb{R}^d$ and $R_n \to \infty$ such that $\|x-y\| \neq R_n$ for all $x, y \in A$ and $\mu(A \cap B_{R_n}) \geq f(R_n) \mu(B_{R_n})$ for all $n$ (Rice, 2019).

In spatial networks, Aldous shows that in invariant tree networks on a rate-1 Poisson process in the plane, the mean distance between points at Euclidean distance $r \gg 1$ is infinite beyond some threshold (Aldous, 2021).

4. Construction Techniques and Model Examples

The realization of unbounded invariant distances often hinges on:

• Boundary scaling and normal forms: Invariant distances are extracted via holomorphic or contact-geometric scaling, leading to model domains (e.g., ellipsoids), explicit coordinate expansions, and rigorous control of asymptotics.
• Symplectic and contact orderability: For Legendrian isotopy classes and groups of contactomorphisms, orderability (absence of contractible positive loops) underpins the construction of nontrivial, unbounded invariant distances (Arlove, 24 Jul 2025).
• Spectral invariants and quasimorphisms: In the Hamiltonian setting, unboundedness of the Hofer metric is proved by constructing maps from function spaces to submanifolds (e.g., via spectral invariant-quasimorphism machinery) and leveraging rigidity of superheavy sets (Ishikawa, 2015).
• Rescaling limits in metric geometry: The notion of pretangent spaces at infinity, constructed by taking limits of rescaled metrics along sequences $r_n \to \infty$, reveals when unbounded metric spaces have equivalent asymptotic geometry (Bilet et al., 2021).

Model Example: A set $A \subset \mathbb{R}^d$ with slow-decaying density that avoids an unbounded sequence $\{R_n\}$ can be built by thickening lattice points in carefully separated shells and ensuring via the triangle inequality that distances in $A$ never fall in $\{R_n\}$ (Rice, 2019). This illustrates the sharpness of earlier density/distance theorems.

5. Applications, Rigidity, and Classification

Unbounded invariant distances yield applications in several domains:

• Function Theory and Complex Geometry: Sharp distance estimates inform the boundary regularity of holomorphic maps, rigidity phenomena (such as the nonexistence of holomorphic isometries between domains of different Levi-type), and explicit control of biholomorphic invariants (1303.3439).
• Symplectic Rigidity and Flexibility: The unbounded nature of the Lagrangian Hofer distance reflects symplectic rigidity, while the discreteness of invariant distances for Legendrian isotopy classes signals contact rigidity against the backdrop of smooth flexibility (Seyfaddini, 2013, Arlove, 24 Jul 2025).
• Network Theory and Spatial Models: The divergence of route-lengths in invariant networks constrains optimal design, demonstrating that, for trees with spatial invariance constraints, mean route lengths are inherently inefficient on large scales (Aldous, 2021).
• Operator Theory and Interpolation Problems: Generalized invariant functions (Minkowski functionals of Pick bodies) serve as tools for solving analytic interpolation problems and extend classical pseudodistances to unbounded multi-point scenarios (Biswas, 2023).

6. Comparative Analysis and Open Directions

A recurring theme is the interplay between geometry (boundary type, orderability), invariants (spectral, geometric, or analytic), and the behavior of distances. While much is now known for strongly pseudoconvex and Levi corank one domains about the scaling behavior and sharp divergence rates, several open problems persist regarding unbounded invariant distances in more degenerate or higher co-rank settings, or for broader classes of Legendrian or contactomorphism spaces without auxiliary group structures.

Recent work also compares multiple invariant distances (e.g., Lempert, Carathéodory, Kobayashi, Bergman) and their precise relationships even in fine asymptotic regimes near the boundary (Kosiński et al., 2023, Nikolov et al., 2021), generalizing classical results that only provided crude asymptotics for large separations.

The development of invariant distances that are both discrete and unbounded is a particularly striking phenomenon unique to the contact setting, contrasting with the smoother structure on diffeomorphism groups, and suggests new possibilities in the exploration of rigidity phenomena.

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Summary Table: Representative Unbounded Invariant Distances

Setting	Invariant Distance Type	Source of Unboundedness
Pseudoconvex Levi corank one domains	Kobayashi, Carathéodory, Bergman	Degeneracy in Levi geometry, logarithmic rate
Hamiltonian dynamics (Euclidean ball)	Lagrangian Hofer metric	Spectral invariants, function space mapping
Legendrian isotopy classes (universal cover)	Discrete invariant metric	Orderability, positive Legendrian loops
Spatial tree networks (random/Poisson)	Route length within tree	Lack of cycles, global invariance constraints
Metric spaces at infinity	Pretangent scaling limits	Infinite rescaling, mutual stability

These cases exemplify how unboundedness of invariant distances, far from being pathological, encodes essential structural information—about boundary geometry, topological order, algebraic rigidity, and asymptotic complexity. They unify geometric analysis, group actions, and rigidity theory in modern mathematics.