arXiv subjects

Wai Yeung Lam

Publications and source records attributed to Wai Yeung Lam.

18 recordsLinked to original sources

arXiv2026

Discrete minimal surfaces: Old and New

We survey structure-preserving discretizations of minimal surfaces in Euclidean space. Our focus is on a discretization defined via parallel face offsets of polyhedral surfaces, which naturally leads to a notion of vanishing mean curvature and a corresponding variational characterization. All simply connected discrete minimal surfaces of this type can be constructed from circle patterns via a discrete Weierstrass representation formula. This representation links the space of discrete minimal surfaces to the deformation space of circle patterns, and thereby to classical Teichmüller theory. We also discuss variants of discrete minimal surfaces obtained by modifying the definition of mean curvature, restricting the variational criterion, or replacing circle pattern data with discrete conformal equivalence, Koebe-type circle packings, or quadrilateral meshes with factorized cross ratios. We conclude with open questions on discrete minimal surfaces.

math.DG↗

arXiv2026

Infinite circle patterns in the Weil-Petersson class

Analogous to Weil-Petersson quasicircles, we investigate infinite circle patterns in the Euclidean plane parameterized by discrete harmonic functions of finite Dirichlet energy. The space of such circle patterns forms an infinite-dimensional Hilbert manifold homeomorphic to the Sobolev space of half-differentiable functions on the unit circle. The Hilbert manifold is equipped with a Riemannian metric induced from the Hessian of a hyperbolic volume functional. We relate this Riemannian metric to the symplectic form on the Sobolev space of half-differentiable functions via an analogue of the Hilbert transform. Every such circle pattern induces a quasiconformal homeomorphism from the unit disk to itself, whose boundary extension belongs to the Weil-Petersson class of the universal Teichmüller space. Our results shed light on Jordan domains packed by infinite circle patterns of hyperbolic type, a subject highlighted by He and Schramm.

math.GT↗

arXiv2026

Period matrices and homological quasi-trees on discrete Riemann surfaces

We study discrete period matrices associated with graphs cellularly embedded on closed surfaces, resembling classical period matrices of Riemann surfaces. Defined via integrals of discrete harmonic 1-forms, these period matrices are known to encode discrete conformal structure in the sense of circle patterns. We obtain a combinatorial interpretation of the discrete period matrix, where its minors are expressed as weighted sums over certain spanning subgraphs, which we call homological quasi-trees. Furthermore, we relate the period matrix to the determinant of the Laplacian for a flat complex line bundle. We derive a combinatorial analogue of the Weil-Petersson potential on the Teichmüller space, expressed as a weighted sum over homological quasi-trees. Finally, we study the collection of homological quasi-trees from a (delta-)matroidal perspective. The discrete period matrix plays a role similar to that of the response matrix in circular planar networks, thereby addressing a question posed by Richard Kenyon.

math.CV↗

arXiv2025

A Structure-Preserving Numerical Method for Harmonic Maps Between High-genus Surfaces

Motivated by geometry processing for surfaces with non-trivial topology, we study discrete harmonic maps between closed surfaces of genus at least two. Harmonic maps provide a natural framework for comparing surfaces by minimizing distortion. Unlike conformal or isometric maps-which may not exist between surfaces with different geometries-harmonic maps always exist within a fixed homotopy class and yield optimal homeomorphisms when the target surface has negative curvature. We develop a structure-preserving algorithm to compute harmonic maps from a triangulated surface to a reference hyperbolic surface. The method minimizes Dirichlet energy over geodesic realizations of the surface graph into the target hyperbolic surface in the homotopy class of a homeomorphism. A central feature of our framework is the use of canonical edge weights derived from the hyperbolic metric, which generalize the classical cotangent weights from the Euclidean setting. These weights preserve injectivity and ensure that isometries remain harmonic in the discrete theory, reflecting their classical behavior.

math.NA↗

arXiv2025

Discrete Laplacians -- spherical and hyperbolic

The discrete Laplacian on Euclidean triangulated surfaces is a well-established notion. We introduce discrete Laplacians on spherical and hyperbolic triangulated surfaces. On the one hand, our definitions are close to the Euclidean one in that the edge weights contain the cotangents of certain combinations of angles and are non-negative if and only if the triangulation is Delaunay. On the other hand, these discretizations are structure-preserving in several respects. We prove that the area of a convex polyhedron can be written in terms of the discrete spherical Laplacian of the support function, whose expression is the same as the area of a smooth convex body in terms of the usual spherical Laplacian. We show that the conformal factors of discrete conformal vector fields on a triangulated surface of curvature $k \in \{-1,1\}$ are $-2k$-eigenfunctions of our discrete Laplacians, exactly as in the smooth setting. The discrete conformality can be understood here both in the sense of the vertex scaling and in the sense of circle patterns. Finally, we connect the $-2k$-eigenfunctions to infinitesimal isometric deformations of a polyhedron inscribed into corresponding quadrics.

math.MG↗

arXiv2024

Delaunay decompositions minimizing energy of weighted toroidal graphs

Given a weighted toroidal graph, each realization to a Euclidean torus is associated with the Dirichlet energy. By minimizing the energy over all possible Euclidean structures and over all realizations within a fixed homotopy class, one obtains a harmonic map into an optimal Euclidean torus. We show that only with this optimal Euclidean structure, the harmonic map and the edge weights are induced from a weighted Delaunay decomposition.

math.MG↗

arXiv2024

Space of circle patterns on tori and its symplectic form

We consider circle patterns on closed tori equipped with complex projective structures. There is an embedding of the space of circle patterns to the Teichmüller space of a punctured surface. Via the embedding, the Weil-Petersson symplectic form is pulled back to the space of circle patterns. We investigate its non-degeneracy. On the other hand, we also complete a conjecture that the space of circle patterns is homeomorphic to the Teichmüller space of the closed torus.

math.GT↗

arXiv2024

Discrete harmonic maps between hyperbolic surfaces

Given a topological cell decomposition of a closed surface equipped with edge weights, we consider the Dirichlet energy of any geodesic realization of the 1-skeleton graph to a hyperbolic surface. By minimizing the energy over all possible hyperbolic structures and over all realizations within a fixed homotopy class, one obtains a discrete harmonic map into an optimal hyperbolic surface. We characterize the extremum by showing that at the optimal hyperbolic structure, the discrete harmonic map and the edge weights are induced from a weighted Delaunay decomposition.

math.GT↗

arXiv2024

Pullback of symplectic forms to the space of circle patterns

We consider circle patterns on surfaces with complex projective structures. We investigate two symplectic forms pulled back to the deformation space of circle patterns. The first one is Goldman's symplectic form on the space of complex projective structures on closed surfaces. The other is the Weil-Petersson symplectic form on the Teichmüller space of punctured surfaces. We show that their pullbacks to the space of circle patterns coincide. It is applied to prove the smoothness of the deformation space, which is an essential step to the conjecture that the space of circle patterns is homeomorphic to the Teichmüller space of the closed surface. We further conjecture that the pullback of the symplectic forms is non-degenerate and defines a symplectic structure on the space of circle patterns.

math.GT↗

arXiv2023

CMC-1 surfaces via osculating Möbius transformations between circle patterns

Given two circle patterns of the same combinatorics in the plane, the Möbius transformations mapping circumdisks of one to the other induces a $PSL(2,\mathbb{C})$-valued function on the dual graph. Such a function plays the role of an osculating Möbius transformation and induces a realization of the dual graph in hyperbolic space. We characterize the realizations and obtain a one-to-one correspondence in the cases that the two circle patterns share the same shear coordinates or the same intersection angles. These correspondences are analogous to the Weierstrass representation for surfaces with constant mean curvature $H\equiv 1$ in hyperbolic space. We further establish convergence on triangular lattices.

math.GT↗

arXiv2020

Dimers and Circle patterns

We establish a correspondence between the dimer model on a bipartite graph and a circle pattern with the combinatorics of that graph, which holds for graphs that are either planar or embedded on the torus. The set of positive face weights on the graph gives a set of global coordinates on the space of circle patterns with embedded dual. Under this correspondence, which extends the previously known isoradial case, the urban renewal (local move for dimer models) is equivalent to the Miquel move (local move for circle patterns). As a consequence the Miquel dynamics on circle patterns is governed by the octahedron recurrence. As special cases of these circle pattern embeddings, we recover harmonic embeddings for resistor networks and s-embeddings for the Ising model.

math-ph↗

arXiv2020

Quadratic differentials and circle patterns on complex projective tori

Given a triangulation of a closed surface, we consider a cross ratio system that assigns a complex number to every edge satisfying certain polynomial equations per vertex. Every cross ratio system induces a complex projective structure together with a circle pattern on the closed surface. In particular, there is an associated conformal structure. We show that for any triangulated torus, the projection from the space of cross ratio systems with prescribed Delaunay angles to the Teichmüller space is a covering map with at most one branch point. Our approach is based on a notion of discrete holomorphic quadratic differentials.

math.GT↗

arXiv2019

Minimal surfaces from infinitesimal deformations of circle packings

We study circle packings with the combinatorics of a triangulated disk in the plane and parametrize deformations of circle packings in terms of vertex rotation and cross ratios. We show that there is a Weierstrass representation formula relating infinitesimal deformations of circle packings to discrete minimal surfaces of Koebe type. Furthermore, every minimal surface of Koebe type can be extended naturally to a discrete minimal surface of general type. In this way, discrete minimal surfaces via Steiner's formula are unified.

math.CV↗

arXiv2018

Infinitesimal conformal deformations of triangulated surfaces in space

We study infinitesimal conformal deformations of a triangulated surface in Euclidean space and investigate the change in its extrinsic geometry. A deformation of vertices is conformal if it preserves length cross-ratios. On one hand, conformal deformations generalize deformations preserving edge lengths. On the other hand, there is a one-to-one correspondence between infinitesimal conformal deformations in space and infinitesimal isometric deformations of the stereographic image on the sphere. The space of infinitesimal conformal deformations can be parametrized in terms of the change in dihedral angles, which is closely related to the Schläfli formula.

math.MG↗

arXiv2018

Holomorphic quadratic differentials on graphs and the chromatic polynomial

We study "holomorphic quadratic differentials" on graphs. We relate them to the reactive power in an LC circuit, and also to the chromatic polynomial of a graph. Specifically, we show that the chromatic polynomial $χ$ of a graph $G$, at negative integer values, can be evaluated as the degree of a certain rational mapping, arising from the defining equations for a holomorphic quadratic differential. This allows us to give an explicit integral expression for $χ(-k)$.

math.CO↗

arXiv2016

Discrete minimal surfaces: critical points of the area functional from integrable systems

We obtain a unified theory of discrete minimal surfaces based on discrete holomorphic quadratic differentials via a Weierstrass representation. Our discrete holomorphic quadratic differential are invariant under Möbius transformations. They can be obtained from discrete harmonic functions in the sense of the cotangent Laplacian and Schramm's orthogonal circle patterns. We show that the corresponding discrete minimal surfaces unify the earlier notions of discrete minimal surfaces: circular minimal surfaces via the integrable systems approach and conical minimal surfaces via the curvature approach. In fact they form conjugate pairs of minimal surfaces. Furthermore, discrete holomorphic quadratic differentials obtained from discrete integrable systems yield discrete minimal surfaces which are critical points of the total area.

math.DG↗

arXiv2016

Isothermic triangulated surfaces

We found a class of triangulated surfaces in Euclidean space which have similar properties as isothermic surfaces in Differential Geometry. We call a surface isothermic if it admits an infinitesimal isometric deformation preserving the mean curvature integrand locally. We show that this class is Möbius invariant. Isothermic triangulated surfaces can be characterized either in terms of circle patterns or based on conformal equivalence of triangle meshes. This definition generalizes isothermic quadrilateral meshes. A consequence is a discrete analog of minimal surfaces. Here the Weierstrass data needed to construct a discrete minimal surface consist of a triangulated plane domain and a discrete harmonic function.

math.DG↗

arXiv2015

Holomorphic vector fields and quadratic differentials on planar triangular meshes

Given a triangulated region in the complex plane, a discrete vector field $Y$ assigns a vector $Y_i\in \mathbb{C}$ to every vertex. We call such a vector field holomorphic if it defines an infinitesimal deformation of the triangulation that preserves length cross ratios. We show that each holomorphic vector field can be constructed based on a discrete harmonic function in the sense of the cotan Laplacian. Moreover, to each holomorphic vector field we associate in a Möbius invariant fashion a certain holomorphic quadratic differential. Here a quadratic differential is defined as an object that assigns a purely imaginary number to each interior edge. Then we derive a Weierstrass representation formula, which shows how a holomorphic quadratic differential can be used to construct a discrete minimal surface with prescribed Gauß map and prescribed Hopf differential.

math.CV↗