Planar-Conformal Regularization

• Planar-conformal regularization is a method that maps irregular, fractal subsets of the Riemann sphere to highly structured round disk packings.
• It generalizes classical uniformization by using packing-conformal maps under the minimal square-summability condition for disk diameters.
• This approach unifies tools from geometric function theory, complex dynamics, and probability, offering new insights into fractal and random structures.

Planar-conformal regularization is a powerful methodology for "geometric regularization" of highly irregular, possibly fractal, subsets of the Riemann sphere by constructing canonical mappings to round disk packings. This concept generalizes classical conformal uniformization—such as the Riemann mapping theorem and Koebe’s Circle Domain theorem—beyond open, simply- or finitely-connected domains to residual sets like Sierpiński carpets, Sierpiński packings, and random structures (e.g., CLE carpets) that arise in geometric function theory, complex dynamics, and probability.

1. Definition and Explanation: Conformal Uniformization in Planar Packings

A Sierpiński packing is a planar (or spherical) set obtained by removing a countable collection of disjoint, non-separating continua (closed connected sets) from the Riemann sphere, with diameters tending to zero. When each continuum is a geometric disk, the resulting set is a round Sierpiński packing or disk packing.

Conformal uniformization in this context seeks a mapping from a wildly irregular packing $Y$ to a highly regular round disk packing $X$, using a "packing-conformal map". This map generalizes the notion of a conformal map to cases where the domain is not open and may be fractal or disconnected in a strong sense.

The significance of this viewpoint is the extension of conformal geometry tools to the regularization of sets well beyond classical domain theory, equipping fractal and randomly generated sets with canonical geometric "coordinates".

2. Mathematical Formulation

The core result asserts that for any Sierpiński packing $Y = \widehat{\mathbb{C}}\setminus \bigcup_{i} q_i$ whose peripheral continua $q_i$ have square-summable diameters ($\sum_{i} \operatorname{diam}(q_i)^2 < \infty$), there exists:

• A disjoint collection of closed disks $\{p_i\}$ (forming a round Sierpiński packing $X$),
• A continuous, surjective, monotone mapping $H: \widehat{\mathbb{C}} \to \widehat{\mathbb{C}}$ with $$H^{-1}(\operatorname{int}(q_i)) = \operatorname{int}(p_i)$$ for each $i$,
• A Borel function $\rho_H \in L^2(\widehat{\mathbb{C}})$ such that for all curves $\gamma$ outside a modulus-zero exceptional family,

$$\sigma(H(\gamma(a)), H(\gamma(b))) \leq \int_{\gamma} \rho_H\, ds + \sum_{i: p_i \cap |\gamma| \neq \emptyset} \operatorname{diam}(q_i)$$

where $\sigma$ is the spherical metric,

• For all Borel sets $E$,

$$\int_{H^{-1}(E)} \rho_H^2\, d\Sigma \leq \Sigma(E \cap Y)$$

with $\Sigma$ the spherical measure.

The maps $H$ are termed "packing-conformal"; they are monotone (point preimages are continua) and are the uniform limit of homeomorphisms. If the removed sets are uniformly "fat" (cofat), the mapping is a homeomorphism. The associated modulus and upper gradient inequalities codify the analytic regularity replacing the traditional differential notion of conformality.

Transboundary Modulus: For a curve family $\Gamma$ in the packing, the modulus is defined via

$$\operatorname{Mod}_X \Gamma = \inf_\rho \left\{ \int_X (\rho \circ \pi_X)^2 d\Sigma + \sum_i \rho(\hat{p}_i)^2 \right\}$$

where $\pi_X$ collapses each peripheral continuum $p_i$ to a point.

3. Applications and Implications

Sierpiński Carpets: Every Sierpiński carpet (a subset where the peripheral continua are topological disks, with interiors dense) with square-summable diameters can be mapped by a packing-conformal map to a round Sierpiński carpet, regularizing its geometric structure without loss of topological information. No geometric condition (such as uniform size, separation, or roundness of the holes) is required.

Conformal Loop Ensemble (CLE) Carpets: CLE carpets, constructed as the complement of a random collection of disjoint loops (from conformal loop ensemble processes) in the unit disk, almost surely fall into this class. The paper demonstrates that they too can be uniformized to round disk packings via packing-conformal maps, thereby canonically encoding these random fractals in geometric packings and addressing a question posed by Rohde-Werness.

The ability to regularize such objects has implications for geometric measure theory, stochastic analysis, and the transfer of probabilistic information to canonical geometric models.

4. Comparison with Other Regularization Methods in Planar Geometry

Method	Map Type	Applicability / Target	Geometric Assumptions
Riemann Mapping / Koebe's	Conformal, Homeo.	Open, (finitely/countably) conn.	Open set, boundary properties
Quasisymmetric (Bonk, etc.)	Quasisymmetric	Carpets/quasicircles/disks	Uniform separation, "quasidisks"
Packing-conformal (this)	Monotone, modulus	Any Sierpiński packing, $L^2$ diam.	None beyond square-summability

Highlights:

• The present results do not require geometric regularity or separation of the peripheral sets—regularization is possible under the minimal $L^2$ condition.
• For sets failing to be cofat, the packing-conformal map may not be injective, but is always a monotone surjection.
• This generalizes, unifies, and extends much previous work on uniformization and regularization of planar sets.

5. Future Directions

The paper identifies directions for further research and unresolved problems:

• Uniqueness: Under what geometric (or probabilistic) conditions is the uniformizing map unique up to Möbius automorphism?
• Rigidity and Removability: Can one classify the regularized packings according to conformal removability or rigidity properties, particularly in random settings?
• Higher Regularity: What additional geometric properties (e.g., uniform fatness, relative separation) ensure the mapping is a homeomorphism or even quasisymmetric?
• Extension to More General Sets: Can the methods be adapted to packings with peripheral sets merely in $\ell^p$ for $p > 2$ or even outside $L^2$?
• Analysis and Probability on Regularized Packings: Developing extremal length, capacity, or potential theory (including Laplacians and random walks) for these new canonical models of previously wild sets.

Summary: Planar-Conformal Regularization in Planar Geometry

Planar-conformal regularization, via uniformization of Sierpiński packings to round disk packings using packing-conformal maps, provides a flexible and universal method to geometrically encode and regularize highly irregular planar sets—including those arising deterministically (wild carpets) or randomly (CLE carpets). The method is robust, depending only on the mild analytic condition of square-summable diameters, and does not demand geometric niceness, connectivity, or separation. It effectively extends the reach of conformal mapping theory to an expansive new universe of fractal structures, allowing the application of classical geometric and analytic techniques in settings where they were previously inapplicable or undefined. This regularization framework opens new ground for research in analytic, geometric, and probabilistic aspects of planar and spherical topology.