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Discrete minimal surfaces: Old and New

Wai Yeung Lam, Masashi Yasumoto

TL;DR

This survey presents a structure-preserving discretization of minimal surfaces in $\,\mathbb{R}^3$ based on parallel-face offsets, introducing a discrete mean curvature via Steiner's formula and a discrete Weierstrass representation built from circle patterns. It proves that simply connected discrete minimal surfaces arise from circle-pattern data, linking the geometric theory to the deformation space of circle patterns and to Teichmüller theory through complex cross ratios. The work then surveys numerous variants, including alternative mean-curvature notions, different discrete conformal data (circle packings, discrete conformal equivalence), and integrable-systems perspectives, highlighting both the breadth and limits of current discretizations. It concludes with open questions on local embeddedness, topology and periodicity, generalized cell decompositions, and Bernstein-type results for discrete minimal surfaces, pointing to rich avenues for future research.

Abstract

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.

Discrete minimal surfaces: Old and New

TL;DR

This survey presents a structure-preserving discretization of minimal surfaces in based on parallel-face offsets, introducing a discrete mean curvature via Steiner's formula and a discrete Weierstrass representation built from circle patterns. It proves that simply connected discrete minimal surfaces arise from circle-pattern data, linking the geometric theory to the deformation space of circle patterns and to Teichmüller theory through complex cross ratios. The work then surveys numerous variants, including alternative mean-curvature notions, different discrete conformal data (circle packings, discrete conformal equivalence), and integrable-systems perspectives, highlighting both the breadth and limits of current discretizations. It concludes with open questions on local embeddedness, topology and periodicity, generalized cell decompositions, and Bernstein-type results for discrete minimal surfaces, pointing to rich avenues for future research.

Abstract

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.

Paper Structure

This paper contains 23 sections, 10 theorems, 32 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Classical minimal surfaces
  3. Mean curvature via Steiner's formula
  4. Weierstrass representation
  5. Further comments on Weierstrass-type representations
  6. A discretization of minimal surfaces
  7. Mean curvature
  8. Weierstrass representation via circle patterns
  9. Deformation space of circle patterns
  10. Miscellaneous results
  11. Infinitesimal deformations of circumscribed polyhedral surfaces
  12. Discrete spherical Laplacian
  13. Delaunay circle patterns and locally convex discrete Gauss maps
  14. Convergence
  15. Variants of discretization
  16. ...and 8 more sections

Key Result

Proposition 2.1

Every conformal minimal immersion $f$ can be written in the form where $(g,\omega)$ is a pair of a meromorphic function $g$ and a holomorphic 1-form $\omega=\hat{\omega}dz \ (z=x+\sqrt{-1}y)$ with $g^2\hat{\omega}$ being holomorphic, and $q=\omega\,dg$ is a holomorphic quadratic differential. Furthermore, the Gauss map $n$ of $f$ is given by which is precisely the inverse image under the stereog

Figures (7)

  • Figure 1: A classical minimal surface in $\mathbb{R}^3$. This is called Enneper's minimal surface.
  • Figure 2: Two discrete minimal surfaces satisfying Definition \ref{['def:mean']}, both reminiscent of the classical Enneper minimal surface.
  • Figure 3: A circle pattern with constant intersection angles $\Theta=\pi/3$.
  • Figure 4: A circle pattern with complex cross ratio $X_{ij} = -\frac{(g_k - g_i)(g_l -g_j)}{(g_i - g_l)(g_j - g_k)}$. The intersection angle of the circumcircles of triangles $\{ijk\}$ and $\{jil\}$ is given by $\Theta_{ij} = \arg(X_{ij})$.
  • Figure 5: A discrete minimal surface which is a realization of a topological sphere and each face has self-intersection. Its Gauss map fails to be locally convex. Indeed, the Gauss map is Jessen's orthogonal icosahedron, which is known to possess a non-trivial infinitesimal isometric deformation.
  • ...and 2 more figures

Theorems & Definitions (14)

  • Proposition 2.1
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Corollary 3.3: Karpenkov-Wallner Karpenkov2014
  • Theorem 3.4
  • proof
  • Definition 3.5
  • Theorem 3.6: Lam Lam2015b
  • Proposition 3.7
  • ...and 4 more