Poly-Annulus Conformal Mapping

• Poly-annulus conformal mapping is a bijective transformation that maps multiply connected domains onto a standard unit disk with multiple circular holes, preserving local angles.
• It relies on uniformization theory and sophisticated numerical methods such as geodesic Voronoi tessellation, partial welding, and integral-equation formulations to achieve accurate mappings.
• Applications include geometric control in multi-agent systems on poriferous surfaces, simplifying domain partitioning and enabling robust obstacle avoidance through metric design.

A poly-annulus conformal mapping is a conformal diffeomorphism from a compact, orientable, two-dimensional Riemannian manifold with boundary—topologically equivalent to a planar domain with multiple holes—onto the canonical domain of the unit disk in the complex plane with $m$ interior, non-overlapping circular holes. Such mappings generalize the classical Riemann mapping theorem to multiply connected domains and are essential for the geometric analysis and control of systems (e.g., multi-agent systems) on domains with multiple disconnected obstacles or topological complexities (Feng et al., 20 Jan 2026, Choi, 2020, Trefethen, 20 Jul 2025).

1. Formal Definition and Theoretical Foundation

Let $S$ be a compact, orientable, 2D Riemannian manifold with boundary in $\mathbb{R}^3$, where the boundary $\partial S$ consists of an "outer" curve $\partial S_0$ and $m$ "inner" curves $\partial O_1,\dots,\partial O_m$, each enclosing an obstacle. $S$ is, therefore, an $(m+1)$-connected planar domain, or a poly-annulus. The canonical target is the multi-holed unit disk:

$$\mathbb{D}_m = \{ w \in \mathbb{C} : |w| \leq 1,\, |w-c_k| \geq r_k,\;k=1,\dots,m \}$$

where centers $c_k$ and radii $r_k$ parametrize the location and size of each hole.

A poly-annulus conformal mapping $\tau: S \rightarrow \mathbb{D}_m$ is a diffeomorphism that is conformal in the interior, maps the outer boundary $\partial S_0$ precisely to the unit circle $\{|w|=1\}$, and each inner boundary $\partial O_k$ to the corresponding circle $\{|w-c_k|=r_k\}$. Uniqueness (up to three real normalization conditions) is guaranteed by the Koebe–Poincaré Uniformization Theorem, the generalization of the Riemann Mapping Theorem for multiply connected domains (Feng et al., 20 Jan 2026, Trefethen, 20 Jul 2025).

2. Existence, Uniqueness, and Cohomological Structure

The existence of poly-annulus conformal mappings relies on the uniformization theorem for planar domains of finite connectivity. Formally, given a bounded domain $U \subset \mathbb{C}$ with boundary comprised of $m+1$ disjoint Jordan curves, there exists a unique conformal diffeomorphism to a disk with $m$ circular holes, up to the normalization of three real degrees of freedom (Feng et al., 20 Jan 2026, Trefethen, 20 Jul 2025).

If $S$ is homeomorphic to such a planar domain, one constructs a conformal atlas reducing $S$ to the multiply connected planar case. The Hodge decomposition and de Rham cohomology imply the global existence of $m$ linearly independent harmonic 1-forms, providing the necessary structure to parametrize the annular moduli—logarithmic ratios of radii distinguishing the conformal types of the multiple holes (Choi, 2020). Any two such conformal maps differ by a Möbius transformation preserving the boundary correspondence (Feng et al., 20 Jan 2026).

3. Constructive Algorithms and Numerical Methods

Several approaches are established to construct these mappings:

• Domain Decomposition: Employ geodesic Voronoi tessellation (GVT) to decompose $S$ into simply-connected (type I) and annular (type II) patches, facilitating parallel computation (Feng et al., 20 Jan 2026).
• Local Riemann Mapping: For simply-connected patches, compute the Riemann map to the unit disk by solving the Laplace equation with appropriate boundary data, using schemes such as the Zipper algorithm or disk harmonic maps with cotangent-weight Laplacian on meshes (Choi, 2020).
• Partial Welding: At interfaces between patches, align local parameterizations using Möbius transformations minimizing $L^2$ interface mismatch. This "partial welding" is composed along chains to assemble topological annuli for each obstacle (Feng et al., 20 Jan 2026).
• Annulus Conformal Maps: For each annular cluster, solve for the conformal map to an annulus $A_k = \{z: r_k < |z| < 1\}$, with the modulus determined by solving an integral equation or using methods from quasi-conformal theory, such as Choi's approach (Choi, 2020).
• Global Assembly and Rectification: Position the obtained annuli in the plane via similarity transforms that minimize global interface mismatch. A final geometric rectification projects numerical boundary images onto exact circles and optionally applies a Möbius transformation to minimize area distortion (Feng et al., 20 Jan 2026, Choi, 2020).
• Integral-Equation Formulation: Boundary integral methods, such as the single-layer (generalized Neumann kernel) approach, set up a linear system for unknown density functions and normalization constants, incorporating neutrality and period-zero conditions to determine the radii of the annular target domain (Trefethen, 20 Jul 2025).
• Harmonic Extension and Correction: After geometric rectification, solve the Laplace equation on $S$ with Dirichlet boundary data to compute the final conformal map. Quasi-conformal correction guarantees numerical conformality and bijectivity. The Linear Beltrami Solver is used for efficiency, and convergence is superlinear under adequate regularity (Choi, 2020, Feng et al., 20 Jan 2026).

Algorithmic Component	Purpose	Paper Source
Geodesic Voronoi Tessellation (GVT)	Decomposition into tractable subdomains	(Feng et al., 20 Jan 2026)
Partial welding (local/global)	Seamless assembly of local parameterizations	(Feng et al., 20 Jan 2026)
Sparse Laplace/Beltrami solves	Disk/annulus parameterizations, quasi-conformal correction	(Choi, 2020)
Integral-equation/Nyström methods	High-precision mapping for analytic boundaries	(Trefethen, 20 Jul 2025)

4. Analytic Structure, Properties, and Series Expansions

For annular pieces, the conformal map admits a Laurent series representation:

$$f_k(z) = e^{i\theta_0} z \cdot \exp\left(\sum_{n \neq 0} a_n z^n \right)$$

where coefficients $a_n$ enforce the modulus and circularity of boundaries. The modulus $\log(1/r_k)$ is intrinsic to the conformal class of each annulus (Feng et al., 20 Jan 2026, Choi, 2020). The final poly-annulus mapping $τ$ is globally conformal ($\partial τ / \partial \bar z = 0$ in the interior) and univalent, with boundary correspondence (each boundary maps to its prescribed target circle), smoothness up to the piecewise boundary, and normalization fixing global rotation and translation.

Discretized integral-equation approaches use the periodic trapezoidal rule for smooth curves, corrected for logarithmic singularities by techniques such as Kress' rule, and yield spectral convergence for analytic boundaries (Trefethen, 20 Jul 2025).

5. Practical Implementation and Computational Considerations

Efficient computation employs the following strategies:

• Parallelization: Per-hole subproblems (annulus computations) are independent and suitable for parallel or distributed architectures.
• Sparse Linear Solvers: The Laplace equation and Beltrami corrections are formulated as sparse symmetric positive definite systems (cotangent-weight Laplacian or Linear Beltrami Solver) for mesh-based representations (Choi, 2020).
• Integral-Equation Discretization: Nyström discretization for boundary integrals, which is particularly effective for smooth (analytic) boundaries, with convergence rates depending on boundary regularity (Trefethen, 20 Jul 2025).
• Complexity: For mesh-based approaches, computational cost is approximately $O(N^{1.5}/M)$ per subdomain, where $N$ is the number of boundary vertices and $M$ the number of subdomains (Feng et al., 20 Jan 2026). The method scales linearly in mesh size and number of holes for iterative (per-annulus) approaches (Choi, 2020).
• Rectification: Ensures that the boundary of the target domain is exactly circular to machine precision after assembly and Möbius correction, essential for physical fidelity and subsequent use in geometric control (Feng et al., 20 Jan 2026).

6. Applications and Geometric Control on Poriferous Surfaces

The canonical application of poly-annulus conformal mapping in recent literature is distributed coverage control for multi-agent systems operating on poriferous surfaces—surfaces with multiple disconnected forbidden regions (holes). The conformal map transforms the complex domain to a normalized disk with holes, where partition, planning, and control tasks are greatly simplified.

Key elements include:

• Sectorial Partitioning: Enables collision-free, strictly connected subregion assignment and workload balance, critical in robotics and distributed MAS contexts (Feng et al., 20 Jan 2026).
• Pull-back Riemannian Metric: The complex derivative of the conformal map defines $g_S(z) = |\tau'(z)|^2 |dz|^2$; after equipping the canonical domain with a safety-aware metric

$$\hat{\eta}_0(w) = (1+\sigma_0(w))^2 \,\mathrm{diag}(1,|w|^2),\quad \sigma_0(w) = \sum_{k=1}^m (e^{\mu\,\mathrm{dist}(w, \partial O_k)} - 1)^{-1}$$

the pull-back metric encodes strict obstacle avoidance into any variational formulation on the original surface (Feng et al., 20 Jan 2026).

• Coverage Control Law: A distributed gradient-based law leverages this metric for provable input-to-state stability and asymptotic convergence of the closed-loop multi-agent system.
• Numerical Robustness: Simulations validate reachability and robustness on complex surfaces with multiple holes, and confirm the efficiency and partition accuracy afforded by the mapping procedure.

7. Further Directions and Extensions

Extensions of the poly-annulus conformal mapping framework include:

• Surface Genus: Surfaces of genus $g>0$ may be handled by cutting along canonical spanning trees, reducing the problem to poly-annulus mapping (Choi, 2020).
• Optimal-Mass-Transport and Area Equalization: Integration of these methods can enforce area preservation or density-equalization constraints in the mapping, important for physical simulations or texture mapping (Choi, 2020).
• Higher-order Numerical Methods: Use of composite Gauss–Legendre quadrature for corners or adaptive mesh refinement near boundaries to address domains with singularities or sharp features (Trefethen, 20 Jul 2025).
• Parallel and GPU Implementation: For extremely large or real-time applications, multigrid preconditioning and GPU acceleration may be adopted (Choi, 2020).

Poly-annulus conformal mapping thus provides a mathematically rigorous, computationally efficient foundation for the geometric control and parameterization of multiply connected surfaces, with direct implications for robotics, differential geometry, and numerical analysis (Feng et al., 20 Jan 2026, Choi, 2020, Trefethen, 20 Jul 2025).