Extending the FEM holomorphic-differential and Laurent-series conformal parameterization pipeline from multiply connected genus-zero surfaces to high-genus surfaces

The 2014 paper of Kropf, Yin, Yau, and Gu (published online December 2013) introduced a two-step method for conformally parameterizing genus-zero surfaces with multiple boundary components: holomorphic differentials, computed by a finite element method based on Hodge decomposition, map the surface onto a planar annulus with concentric circular slits, and a Laurent series method then maps the slit domain to a canonical circle domain (a disk with circular holes). The open question is the generalization of this pipeline to high-genus surfaces, posed as the paper's future work; it requires extending both the holomorphic-differential framework and the Laurent series method to surfaces with handles. The general capability is not open — discrete Ricci flow parameterizes arbitrary topology, and Zhang, Li, Zeng, and Gu (2012) mapped high-genus surfaces with boundaries onto hyperbolic circle domains — but this specific essentially non-iterative, mesh-robust pipeline has not been extended beyond genus zero, and recent flattening work (Tan, Li, Lin, Yau, 2024) leaves higher-genus seamless conformal flattening as ongoing research. The open core is therefore the pipeline-specific extension, obstructed mainly by the planar character of the Laurent-series circle-domain step, which has no analogue for genus $g\ge1$ because uniformization forces a flat-torus (closed torus) or hyperbolic canonical domain.

Background

A conformal map between surfaces preserves angles and local shape; conformal parameterization maps a curved surface onto a planar domain with minimal distortion, and is a basic tool in computer graphics, shape analysis, and isogeometric analysis (a methodology that performs numerical simulation directly on computer-aided-design geometries by building spline spaces over a parameter domain, so it needs a high-quality planar parameterization of the surface). For a genus-zero surface (a surface with no handles) carrying several boundary components — a multiply connected domain — a classical theorem of Koebe guarantees a conformal map onto a circle domain, i.e. a disk with finitely many circular holes; the configuration of the inner circles, called the conformal module, is a complete invariant of the surface up to conformal equivalence and Möbius transformations.

Kropf, Yin, Yau, and Gu (2014) proposed an efficient, robust computational pipeline for this map, combining finite elements with classical complex analysis in two steps. Step one: a holomorphic differential (also called a holomorphic 1-form — a complex-valued differential $\omega=f\,dz$ with $f$ holomorphic, realized on a surface as a harmonic differential 1-form together with $i$ times its harmonic conjugate) is computed on the triangulated surface by a finite element method based on the Hodge decomposition, the representation of any differential 1-form as the sum of an exact part, a co-exact part, and a harmonic part; integrating the resulting closed differentials maps the surface conformally onto a planar annulus with concentric circular slits (a slit domain). Step two: the slit annulus is conformally mapped to the circle domain by a Laurent series method — expanding an analytic function as a power series together with principal parts around the inner circles and determining the coefficients from a linear collocation system that enforces constant modulus on circular slits and constant argument on radial slits — after which the map is inverted numerically by Newton iteration. Compared with the two prior alternatives, Koebe iteration (repeatedly filling all holes but one, mapping to an annulus, and reopening a hole until every boundary is a circle, which requires repeated mesh surgery) and surface Ricci flow (a metric-evolution process $\partial g/\partial t=-2Kg$ driving Gaussian curvature toward a constant, whose discrete versions are sensitive to triangle quality and can fail on raw scanned meshes), the authors report their method to be faster and more robust, with applications such as face and cortical-surface meshes taken directly from 3D scanners.

The pipeline is intrinsically genus-zero, because its target — a planar circle domain — exists only for planar multiply connected surfaces. By the uniformization theorem, the universal cover (the simply connected surface that unwraps a given surface onto itself infinitely many times) of a closed surface of genus $g\ge1$ is the Euclidean plane (genus 1, a flat torus) or the hyperbolic plane (genus $\ge2$), so no single global planar parameterization exists: handles force either cutting the surface along a homology basis (a maximal system of independent loops encoding the topology) with side identifications across the cuts, or a canonical hyperbolic domain. It is important for the status of this question that conformal parameterization of high-genus surfaces is not an open capability in general: Jin, Kim, Luo, and Gu (2008) gave a unified discrete Ricci flow framework designing conformal metrics on surfaces of arbitrary topology (spherical, Euclidean, and hyperbolic variants), and Zhang, Li, Zeng, and Gu (2012) — from the same research group — computed canonical conformal mappings of high-genus surfaces with boundaries onto circle domains in the hyperbolic plane, with a convergence proof. However, those methods use exactly the iterative, mesh-sensitive machinery (Ricci flow and Koebe-style iteration) whose cost and fragility the Kropf–Yin–Yau–Gu pipeline was designed to avoid. What the 2014 paper left open, and what remains open, is the extension of this specific essentially non-iterative, finite-element-based pipeline to surfaces with handles.

Problem Statement

Generalize the two-step conformal parameterization method of Kropf, Yin, Yau, and Gu (2014) from genus-zero surfaces with multiple boundary components to surfaces with one or more handles — topological type $(g,b)$, where $g\ge1$ is the genus (the number of handles) and $b\ge0$ is the number of boundary components, covering both closed surfaces and surfaces with boundary — the generalization posed as the paper's future work, requiring extension of both the holomorphic differential framework and the Laurent series method. Concretely, the question asks for an algorithm that, given a triangulated surface of genus $g\ge1$:

1. extends the finite-element holomorphic-differential step — computing, via Hodge decomposition, holomorphic differentials whose integration maps the surface to a canonical slit domain — so that the handles are correctly represented, i.e. the image is a canonical flat (closed torus, $g=1$ and $b=0$) or hyperbolic ($g\ge2$, or $b\ge1$) slit domain or fundamental domain carrying the side identifications mandated by the uniformization theorem; and

2. extends or replaces the planar Laurent-series step — which relies on Koebe's planar circle-domain theorem and therefore has no direct genus-$\ge1$ analogue — so that all boundary components become circles in the appropriate canonical geometry.

These two components serve one joint objective and are not independent deliverables: the extended pipeline must preserve the defining properties that motivate it, namely essentially non-iterative, linear-algebraic computation (solved by a fixed number of sparse linear solves together with at most local Newton inversion, with no global iteration to convergence), robustness on low-quality scanned triangulations (where discrete Ricci flow degenerates), and efficiency comparable to or better than Koebe iteration and Ricci flow. The extension is to be realized as a mathematically formulated algorithm, implemented and demonstrated numerically on genus-one and higher-genus surfaces, including surfaces with several boundary components and raw scanned meshes. Whether any extension of the Laurent-series-type formulation can exist while preserving its non-iterative character — including a rigorous obstruction or impossibility result — is part of the question.

References

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