Convergence theory for computing quasiconformal maps via the auxiliary metric and discrete Yamabe flow beyond disk topology

The paper 'Computing quasiconformal maps using an auxiliary metric and discrete curvature flow' (Zeng, Lui, Luo, Chan, Yau, Gu; Numerische Mathematik 121:671-703, 2012) computes quasiconformal maps on triangulated surfaces with a prescribed Beltrami differential by encoding the differential in an auxiliary metric, under which the target map becomes conformal, and then computing the conformal flattening with a discrete Yamabe flow. Its theory is incomplete in three linked respects: (i) the exponential convergence of the flow is proven only under the hypothesis that no singularity develops after finitely many mesh-surgery (edge-flip) operations, while the surgery mechanism itself is not formalized; (ii) the admissible set of discrete conformal factors is nonconvex, and the Newton iteration used in practice relies on heuristic step-length adjustment and edge swaps, with no globalization proof on irregular meshes; (iii) convergence of the computed discrete quasiconformal maps to the continuous solution as the mesh is refined is proven only for disk-type surfaces with regular triangulations. The conformal core of these gaps is now resolved: discrete-uniformization convergence is proven for Jordan domains (planar domains bounded by a single simple closed curve), genus-zero surfaces, and closed surfaces of genus at least one, and the edge-flip/surgery question is settled for closed surfaces. However, no published work proves mesh-refinement convergence, or solver-level global convergence, for the quasiconformal pipeline with prescribed nonzero Beltrami data beyond disk topology. The open problem is to complete the convergence theory of the auxiliary-metric quasiconformal computation to the generality at which it is used - arbitrary surface topology and general triangulations - including an analysis of the discretization error of the auxiliary metric itself.

Background

Setting and objects. A Riemann surface is a one-complex-dimensional manifold whose coordinate charts are related by holomorphic (angle-preserving) maps; a triangulated surface is a surface obtained by gluing Euclidean triangles along edges. Quasiconformal (QC) maps generalize conformal (angle-preserving) maps by allowing bounded angle distortion: an orientation-preserving homeomorphism $f$ is $K$-quasiconformal if infinitesimal circles are mapped to infinitesimal ellipses whose ratio of major to minor axis is at most $K$. In local conformal coordinates $z=x+iy$, writing $\partial_{\bar z}=\frac12(\partial_x+i\partial_y)$, the distortion is measured by the Beltrami coefficient $\mu_f=f_{\bar z}/f_z$; $f$ is quasiconformal precisely when $\|\mu_f\|_\infty<1$. Given a prescribed function $\mu$ with $\|\mu\|_\infty<1$ on a surface, the Beltrami equation $f_{\bar z}=\mu f_z$ has quasiconformal solutions, unique up to post-composition with conformal maps; prescribing $\mu$ together with normalization or boundary data pins down the map. Computing QC maps with prescribed distortion is a basic operation in surface parameterization, registration, and medical image analysis.

The auxiliary metric idea. The key identity behind the source paper: a QC map with coefficient $\mu$ becomes conformal if the metric on the source surface is replaced by the auxiliary metric

$$\tilde g_1=e^{2\lambda_1(z)}\,|dz+\mu(z)\,d\bar z|^2,$$

where the scalar function $\lambda_1$ is chosen (via a prescribed-curvature equation) so that the target map is conformal from the source with metric $\tilde g_1$ to the target with its original metric. Thus computing a QC map reduces to a conformal computation on a modified metric.

Discrete conformal geometry. On a triangulated surface, the standard discrete analogue of a conformal change of metric $g\mapsto e^{2u}g$ is vertex scaling: edge lengths are transformed by $\tilde l_{ij}=e^{(u_i+u_j)/2}l_{ij}$, with the discrete conformal factors $u=(u_i)$ assigned to vertices (Luo 2004; Springborn-Schröder-Pinkall 2008; Bobenko-Pinkall-Springborn 2015 supplied the hyperbolic-polyhedron interpretation and Springborn 2020 the variational picture). A vector $u$ is admissible only if every rescaled face still satisfies the triangle inequality, and this admissible set is nonconvex. The discrete Yamabe flow is the negative gradient flow of a (on each admissible cell, convex) energy that drives the discrete Gaussian curvatures at vertices - the angle deficits, $2\pi$ minus the sum of the angles of the triangles meeting at the vertex - toward prescribed target values; on a simply connected surface with zero targets it flattens the mesh to the plane. An edge flip (also called mesh surgery) replaces the shared edge of two adjacent triangles by the other diagonal; for Delaunay flips (flips that restore the condition that the two angles opposite an edge sum to at most $\pi$) the discrete conformal class is preserved (Gu-Luo-Sun-Wu 2018).

The Quasi-Yamabe pipeline. Zeng-Lui-Luo-Chan-Yau-Gu (2012) discretize the prescribed Beltrami differential edge-wise, $\mu_{ij}=\frac{\mu(v_i)+\mu(v_j)}{2}$, and define the discrete auxiliary edge lengths

$$\tilde l_{ij}=l_{ij}\,\frac{|dz_{ij}+\mu_{ij}\,d\bar z_{ij}|}{|dz_{ij}|},$$

then run the discrete Yamabe flow ('Quasi-Yamabe flow') on the mesh with these lengths - performing Newton steps with step-length adjustment and edge swaps whenever triangle inequalities fail - and reconstruct from the resulting factors a discrete QC map. On surfaces with boundary the computation is carried out under prescribed boundary data; on closed surfaces it is carried out on a fundamental domain embedded in the universal cover, a finite portion of which is tiled by copies of a fundamental polygon (preprint Sections VII-C-VII-D). The numerical experiments report empirical convergence orders near one. The paper proves exponential convergence of the flow only under the hypothesis that no singularity develops after finitely many surgeries (preprint Theorem 4.9), and proves mesh-refinement convergence of the discrete QC maps to the continuous QC map (published Theorem 6.3) only when source and target are disks (disk topology) and the triangulations are regular, meaning the triangles' aspect ratios stay uniformly bounded under refinement. The paper supplies no discretization-error analysis of the auxiliary metric, and its Newton step-length adjustment is heuristic.

What later work settled (the conformal case). For $\mu\equiv 0$ the corresponding questions are now theorems: every closed polyhedral surface admits a discretely conformally equivalent constant-curvature metric computable by a discrete Yamabe flow with the flip structure controlled (Gu-Luo-Sun-Wu 2018; Gu-Guo-Luo-Sun-Wu 2018); the vertex-scaling Yamabe flow extends without surgery and converges exponentially (Ge-Jiang 2016); only finitely many flips ever occur (Wu, preprint 2023); and mesh-refinement convergence of discrete uniformization (the discrete computation of the constant-curvature uniformizing metric) holds for Jordan domains (planar domains bounded by a single simple closed curve) (Luo-Sun-Wu 2022), genus-zero closed surfaces (Luo-Wu-Zhu 2024), and closed surfaces of genus at least one (Wu-Zhu 2024). For QC maps on the disk, convergence of a different discretization - circle packing, a configuration of mutually tangent circles whose tangency pattern encodes the triangulation - was already known (He 1990). No published work provides the corresponding theory for the QC pipeline with prescribed nonzero $\mu$ beyond disk topology, or a solver-level convergence guarantee for that pipeline.

Problem Statement

Complete the convergence theory of the auxiliary-metric method for computing quasiconformal maps with prescribed Beltrami differentials on triangulated surfaces - the Quasi-Yamabe flow pipeline of Zeng-Lui-Luo-Chan-Yau-Gu (2012), or a discretization demonstrably equivalent to it - at the generality at which the method is used. The common scientific objective is a provable end-to-end guarantee: the maps the method computes must be shown to converge, under mesh refinement, to the continuous quasiconformal map determined by the prescribed Beltrami data on surfaces beyond disk topology, and the computational procedure must be shown to reach them. Two linked components jointly discharge this objective.

(a) Topology-general refinement convergence. Prove that, for at least one class of surfaces beyond disk topology - closed surfaces of genus at least one (compact boundaryless surfaces with one or more handles), and/or multiply connected planar domains (domains with holes) - the discrete quasiconformal maps produced by the pipeline (or the equivalent discretization) converge, as the mesh is refined, to the continuous quasiconformal map solving the Beltrami equation with the prescribed differential, under explicitly stated hypotheses on the differential's regularity and on the mesh sequence. The analysis must account for the discretization error of the auxiliary metric itself (the vertex sampling and edge-wise averaging of the differential and the resulting edge lengths), which the source paper leaves unanalyzed even for disk topology. The theorem may rely on strong hypotheses on mesh regularity, provided these are stated explicitly, and must hold beyond disk-type domains; no convergence rate is required.

(b) Solver-level guarantee. Prove that the computational procedure realizing (a) converges, with a guarantee that is not conditional on the unproven 'no singularity after finitely many surgeries' hypothesis. The surgery-finiteness and no-surgery flow theorems for the conformal case are taken as established.

The two components are scientifically linked parts of one objective - the method's claimed applicability to general surfaces is only as sound as both: (a) without (b) describes a discretization that no proven algorithm is known to compute, and (b) without (a) proves convergence of a solver to a map whose relation to the continuous quasiconformal map is unestablished. Both components concern the auxiliary-metric construction - the prescribed Beltrami data encoded in a discrete metric on the source surface, with the desired map computed as a discrete conformal map of that metric - and apply to any discretization that faithfully realizes it, not only the exact formulas of the source paper. The conformal sub-case (zero prescribed differential) is settled by the discrete uniformization literature; the open content is the quasiconformal case.

References

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