Discrete Real Riemann Surfaces

• Discrete real Riemann surfaces are combinatorial and metric models that replicate the geometric, analytic, and topological features of classical real Riemann surfaces through quad-graph cellular decompositions.
• They incorporate discrete holomorphic functions, antiholomorphic involutions, and period matrices via linear discretizations of the Cauchy–Riemann equations, ensuring rigorous numerical convergence.
• The framework supports practical applications such as computing period matrices, uniformizing minimal surfaces, and providing combinatorial interpretations that mirror classical conformal theories.

A discrete real Riemann surface is a combinatorial and metric object designed to model classical (smooth) real Riemann surfaces within a discrete, algorithmically tractable framework. Discrete analogues replicate the key geometric, analytic, and topological structures of their classical counterparts—including antiholomorphic involutions, real structures, period matrices, and the classification of real forms—by encoding them in quad-graph cellular decompositions endowed with real-weight (orthodiagonal) complex structures and linear discretizations of the Cauchy–Riemann equations. This approach not only mirrors the smooth theory at the level of conformal and period data but recovers the full topological classification of real algebraic curves while enabling rigorous numerical algorithms and convergence guarantees.

1. Discrete Riemann Surfaces and Linear Discretization

A discrete Riemann surface is defined by a quad-graph cellular decomposition $\Lambda=(V,E,F)$ of a closed oriented surface $\Sigma$, with each face $Q\in F$ a quadrilateral and the 1-skeleton a bipartite graph admitting a black–white coloring. The discrete complex structure is specified by assigning to each $Q$ a complex weight

$$\rho_Q = -i \frac{w_+ - w_-}{b_+ - b_-} \in \{\zeta\in \mathbb{C}: \Re \zeta > 0\},$$

where $(b_-, w_-, b_+, w_+)$ are the vertices of $Q$ in counterclockwise order. For real discrete Riemann surfaces, one requires $\rho_Q\in\mathbb{R}_{>0}$, ensuring orthogonality of the diagonals and geometric realization by rhombic quad-surfaces.

Discrete holomorphic functions $f:V(\Lambda)\rightarrow\mathbb{C}$ satisfy the linear discrete Cauchy–Riemann equation on each face $Q$: $f(w_+) - f(w_-) = i\rho_Q [f(b_+) - f(b_-)].$ The medial graph $X$ and its associated discrete exterior calculus (differential forms, Hodge star, wedge product, codifferential, and Laplacian) enable the definition of discrete holomorphic and harmonic 1-forms, whose properties closely parallel the analytic structure of smooth surfaces. The canonical discrete period matrix is constructed from cycles in the homology of $X$ and the solution spaces of discrete holomorphic differentials (Bobenko et al., 2015, Düntsch et al., 31 Dec 2025, Günther, 2023).

2. Discrete Antiholomorphic Involution and Real Structure

A discrete real Riemann surface $(\Sigma,\Lambda,\tau)$ is a discrete Riemann surface together with an antiholomorphic involution

$\tau:\Lambda\rightarrow\Lambda,$

which either preserves (type 1) or swaps (type 2) the black/white coloring, and acts on the complex weights by

• Type 1: $\rho_{\tau(Q)} = \overline{\rho_Q}$,
• Type 2: $\rho_{\tau(Q)} = 1/\overline{\rho_Q}$.

The fixed-point set $\operatorname{Fix}(\tau)$ consists of disjoint closed cycles ("discrete real ovals") on $\Lambda$ or $X$. The number of real ovals $k$ and their distribution determine the topological type, perfectly paralleling classical results:

• $0 \leq k \leq g+1$,
• $k=g+1$ (discrete M-curve): surface is dividing,
• $k=0$: surface is non-dividing.

This classification replicates Harnack's and Weichold's bounds combinatorially via polyhedral realization and Euler characteristic arguments (Düntsch et al., 31 Dec 2025).

3. Canonical Homology Basis and Discrete Period Matrix

A canonical symplectic homology basis $\{a_i, b_i\}_{i=1}^g$ is constructed on $H_1(X,\mathbb{Z})$ and adapted to $\tau$ such that

$$\tau(a_i) = a_i, \qquad \tau(b_i) = \sum_j h_{ji} a_j - b_i,$$

for a symmetric integer matrix $H=(h_{ij})$. The topology (dividing/non-dividing, number of ovals) is encoded in the diagonal and $\mathbb{Z}_2$-rank of $H$:

• Dividing: $\operatorname{diag}(H)=0$, $\operatorname{rank}_{\mathbb{Z}_2}H=g+1-k$.
• Non-dividing, $k>0$: $\operatorname{diag}(H)=1$, $\operatorname{rank}_{\mathbb{Z}_2}H=g+1-k$.
• Non-dividing, $k=0$: formula depends on the parity of $g$.

A canonical basis of discrete holomorphic differentials can be specified by normalization on black and white $a$-cycles, producing associated $b$-period blocks and the complete discrete period matrix $\widetilde{\Pi}$ (Düntsch et al., 31 Dec 2025, Bobenko et al., 2015).

The discrete period matrix is defined by averaging: $$\Pi = \frac{1}{2}(\Pi^{B,B} + \Pi^{B,W} + \Pi^{W,B} + \Pi^{W,W}).$$ The principal structural result is the canonical decomposition (Düntsch et al., 31 Dec 2025): $\boxed{ \Pi = \tfrac{1}{2}H + iT },$ where $H$ encodes the real structure, and $T$ is a real symmetric positive definite matrix (the "discrete Torelli matrix").

For discrete M-curves ($k=g+1$), $H=0$, so $\Pi$ is purely imaginary, matching the smooth case.

4. Discrete Beltrami and Quasiconformal Theory

Discrete quasiconformal maps are formulated on triangulated surfaces via prescribed discrete Beltrami differentials $\mu:V\rightarrow\mathbb{C}$, assigning data on local charts compatible under overlaps. The auxiliary metric is defined (Zeng et al., 2010): $$\widetilde{\ell}_{ij} = \ell_{ij} \cdot \frac{ |dz_{ij} + \mu_{ij} \overline{dz_{ij}}| }{ |dz_{ij}| },$$ where $dz_{ij} = z(v_j) - z(v_i)$ is the difference in a global conformal parameter $z$. The algorithm proceeds via the discrete Yamabe flow for a conformal factor $u:V\to\mathbb{R}$, flowing to the prescribed curvature. The flow is gradient descent of a convex energy and converges exponentially to the desired (uniformization) metric.

This scheme is topologically general, applying for surfaces of all genera:

• genus 0: planar or annular flattening,
• genus 1: parallelogram fundamental domain with periodicity,
• genus $g\geq2$: hyperbolic 4$g$-gon with prescribed curvature and appropriate $\mu$-compatibility.

Exponential convergence, first-order $L_2$ and sup-norm convergence as mesh size vanishes, and numerical stability are guaranteed (Zeng et al., 2010).

5. Discrete Holomorphic Differentials, Period Matrices, and Convergence

A discrete holomorphic 1-form is defined as a type $(1,0)$ form on the double graph (vertices, edges of primal and dual), satisfying linear relations derived from the discrete Hodge star and Cauchy–Riemann equations (0909.1305, Bobenko et al., 2012, Bobenko et al., 2015, Günther, 2023). The construction proceeds via Dirichlet energy minimization under period constraints, yielding basis forms and the period matrix.

Convergence results are robust:

• As mesh size $h\to0$ and mesh regularity is maintained, discrete period matrices $\Pi(h)$ converge to their continuous analogues with error $O(h^\alpha)$, or $O(h)$ with suitable adaptation near cone or branch points (Bobenko et al., 2018, Günther, 2023, Bobenko et al., 2012).
• Abelian integrals of the first kind (discrete holomorphic integrals with prescribed periods) converge uniformly on compacts.
• All spectral invariants, such as extremal length, can be computed combinatorially and converge to smooth conformal invariants (Wood, 2010).

These results extend to branched coverings, using adapted triangulations near ramification points to control singularity behavior. The discrete Riemann–Roch and Riemann–Hurwitz theorems are established, structurally matching the smooth theory (Bobenko et al., 2015).

6. Combinatorial Theory: Quasi-Trees, Delta-Matroids, and Spectral Invariants

The discrete period matrix admits a purely combinatorial expansion in terms of homological quasi-trees—spanning subgraphs whose minors in the period matrix index contributions weighted by their conductance and algebraic homology class (Lam et al., 2 Jun 2025). This combinatorial interpretation directly links discrete conformal structure with network-theoretic and matroidal concepts:

• Minors of the period matrix correspond to weighted sums over $k$-quasi-trees determined by their intersection pairing with a symplectic basis.
• The collection of quasi-trees forms a delta-matroid, supporting exchange axioms and encoding the richness of cycle-rooted spanning forests.
• The discrete period matrix is related to the Laplacian of a flat $\mathbb{C}^*$-bundle, with the combinatorial version of the Weil–Petersson potential expressible as a sum over quasi-trees, paralleling the Kähler geometry of Teichmüller space.

This framework enables the precise enumeration and manipulation of discrete period matrices for spectral, geometric, and moduli-theoretic applications (Lam et al., 2 Jun 2025).

7. Examples, Applications, and Convergence to Smooth Theory

Discrete real Riemann surfaces have been used to model and numerically compute period matrices for surfaces such as the Lawson genus-2 minimal surface, hyperelliptic curves, and discrete uniformizations via hyper-ideal circle patterns for branched covers of the sphere (0909.1305, Bobenko et al., 2015, Düntsch et al., 31 Dec 2025). Practical algorithms exploit convexity, variational principles, and combinatorial flows with provable convergence.

As sequences of quad-graph (or triangulation) refinements tend toward the smooth surface, all discrete structures—period matrices, real structures, homology bases, Abel–Jacobi maps—converge quantitatively to their smooth counterparts. The canonical decomposition $\Pi = \frac{1}{2}H + iT$ is preserved in the limit, faithfully bridging combinatorial and classical theories (Düntsch et al., 31 Dec 2025, Bobenko et al., 2012, Günther, 2023, Bobenko et al., 2018).

The discrete theory thus provides a robust, rigorous, and computationally explicit framework for real Riemann surfaces, maintaining structural, analytic, and topological fidelity to the classical theory at every stage.