Combinatorial Calabi Flow Fundamentals

• Combinatorial Calabi flow is a discrete curvature flow on triangulated surfaces that evolves conformal parameters using the negative gradient of a curvature-squared energy.
• It encompasses various geometric models—including circle packing and vertex scaling—utilizing discrete Laplacians to drive metrics toward uniform or prescribed curvature.
• Advanced methods incorporate surgery techniques such as edge flips to maintain Delaunay conditions across Euclidean, hyperbolic, and ideal circle pattern settings, ensuring global convergence.

Searching arXiv for recent and foundational papers on combinatorial Calabi flow to ground the article in the current literature. arXiv search results found relevant papers including:

• "Combinatorial Calabi flow with surgery on surfaces" (Zhu et al., 2018)
• "Combinatorial Calabi flows on surfaces" (Ge, 2012)
• "Fractional combinatorial Calabi flow on surfaces" (Wu et al., 2021)
• "Combinatorial curvature flows with surgery for inversive distance circle packings on surfaces" (Xu et al., 2023)
• "Combinatorial Calabi flows on surfaces with boundary" (Luo et al., 2021)
• "Combinatorial Calabi flow for ideal circle pattern" (Li et al., 3 Jan 2025)
• "Combinatorial Calabi flows with ideal circle patterns" (Zhang, 3 Jan 2025)
• "Combinatorial p-th Calabi flows on surfaces" (Lin et al., 2018)
• "Deformation of discrete conformal structures on surfaces" (Xu, 2023)
• "2-Dimensional Combinatorial Calabi Flow in Hyperbolic Background Geometry" (Ge et al., 2013) Combinatorial Calabi flow is a discrete curvature flow on triangulated or polyhedral surfaces in which discrete conformal parameters are evolved by the negative gradient of an $L^2$-type curvature energy. In its original surface form, it was introduced for triangulated surfaces with circle packing metrics as an analogue of smooth Calabi flow (Ge, 2012). Subsequent work extended the construction to hyperbolic background geometry (Ge et al., 2013), to vertex scaling of Euclidean and hyperbolic polyhedral metrics with surgery by edge flips (Zhu et al., 2018), to fractional and $p$-th order variants (Wu et al., 2021), and to settings with boundary or ideal circle patterns where the prescribed data are boundary lengths or total geodesic curvatures rather than angle deficits (Luo et al., 2021). Across these formulations, the central objective is uniformization or prescribed-curvature realization inside a discrete conformal class.

1. Foundational definition and geometric data

The basic setting is a triangulated surface $T=(V,E,F)$, where a discrete metric is encoded either by circle packing radii, by polyhedral edge lengths, or by a more general discrete conformal structure. In the standard triangulated-surface formulation, the discrete curvature at a vertex $i$ is the angle deficit

$K_i = 2\pi - \sum_{\{i,j,k\}\in F} \theta_i^{jk},$

with $\theta_i^{jk}$ the angle at $i$ in the corresponding geometric triangle (Ge, 2012). This definition persists, with geometric reinterpretation, in circle packing, vertex-scaling, ideal-pattern, and polyhedral settings.

In Euclidean circle packing geometry, a circle packing metric is a positive radius assignment $r:V\to(0,+\infty)$, with edge lengths

$$l_{ij} = \sqrt{r_i^2 + r_j^2 + 2 r_i r_j \cos(\Phi_{ij})},$$

for edge weights $\Phi:E\to[0,\pi/2]$ (Ge, 2012). In hyperbolic background geometry, the analogous length law is

$p$0

and the discrete curvature is again the angle deficit at vertices (Ge et al., 2013).

A second major realization is vertex scaling of polyhedral metrics. For a Euclidean polyhedral metric with edge lengths $p$1 and conformal factor $p$2, Zhu–Xu use the update rule

$p$3

while in the hyperbolic case vertex scaling acts on “sinh half-lengths” (Zhu et al., 2018). In this framework, the triangles’ angles depend smoothly on $p$4 over the admissible region where triangle inequalities hold.

Glickenstein’s axiomatic discrete conformal structures unify these constructions. In the Euclidean case, the edge length formula

$p$5

contains Thurston’s circle packings, Bowers–Stephenson inversive distance packings, and Luo’s vertex scaling as special cases (Xu, 2023). This unifying viewpoint clarifies that combinatorial Calabi flow is not tied to a single geometric model, but to a curvature map on a discrete conformal parameter space.

2. Energy, Jacobians, and discrete Laplacians

The governing functional is the combinatorial Calabi energy. In one common convention,

$p$6

while some papers omit the factor $p$7; both conventions occur in the literature (Ge, 2012). Here $p$8 is a prescribed target curvature, and the unprescribed case corresponds to constant curvature or zero curvature, depending on the background geometry.

Let

$p$9

be the Jacobian of the curvature map. On a fixed triangulation, the combinatorial Calabi flow is

$T=(V,E,F)$0

so it is the negative gradient flow of the Calabi energy in the $T=(V,E,F)$1-coordinates (Ge, 2012). The induced curvature evolution is a discrete biharmonic-type equation: $T=(V,E,F)$2 When $T=(V,E,F)$3, this is equivalently $T=(V,E,F)$4 (Zhu et al., 2018).

The analytic content of the flow is concentrated in the structure of $T=(V,E,F)$5. For Euclidean circle packings, Ge showed that $T=(V,E,F)$6 is symmetric, semipositive definite, of rank $T=(V,E,F)$7, with kernel $T=(V,E,F)$8; equivalently, the discrete dual-Laplacian $T=(V,E,F)$9 annihilates constants (Ge, 2012). In hyperbolic circle packing geometry, $i$0 is symmetric positive definite (Ge et al., 2013). For vertex scaling on a Delaunay triangulation, Zhu–Xu identify $i$1 with the cotan Laplacian in the Euclidean case and prove positive semidefiniteness, while in the hyperbolic case $i$2 is symmetric positive definite on the admissible set (Zhu et al., 2018).

The $i$3-th generalization replaces the linear Laplacian by a discrete $i$4-Laplacian

$i$5

and yields $i$6-th Calabi flows for any $i$7 (Lin et al., 2018). Fractional versions are defined spectrally from powers of $i$8, producing nonlocal flows

$i$9

which recover Ricci flow at $K_i = 2\pi - \sum_{\{i,j,k\}\in F} \theta_i^{jk},$0 and the standard Calabi flow at $K_i = 2\pi - \sum_{\{i,j,k\}\in F} \theta_i^{jk},$1 (Wu et al., 2021).

3. Geometric realizations and target quantities

Although the classical target of combinatorial Calabi flow is vertex angle-deficit curvature, later work broadened the meaning of the “curvature” being prescribed. This is a fundamental point: combinatorial Calabi flow is a variational mechanism, not a single fixed curvature model.

In hyperbolic background circle packing geometry, the natural target is zero curvature. Ge–Xu proved that the flow converges to a ZCCP-metric, meaning a zero curvature circle packing metric, when such a metric exists (Ge et al., 2013). In Euclidean background, the target is typically constant curvature $K_i = 2\pi - \sum_{\{i,j,k\}\in F} \theta_i^{jk},$2, reflecting the discrete Gauss–Bonnet constraint (Ge, 2012).

For surfaces with boundary, the relevant discrete datum is no longer an angle deficit at an interior vertex. On ideally triangulated surfaces with boundary, the generalized combinatorial curvature is the length of each boundary component, and the combinatorial Calabi flow is designed to prescribe those boundary lengths inside a hyperbolic metric with totally geodesic boundary (Luo et al., 2021). In the $K_i = 2\pi - \sum_{\{i,j,k\}\in F} \theta_i^{jk},$3 generalized circle packing metric of Guo–Luo, the same principle is expressed through boundary lengths $K_i = 2\pi - \sum_{\{i,j,k\}\in F} \theta_i^{jk},$4, and the associated Calabi flow exists for all time and converges exponentially fast for any positive target boundary lengths (Xu et al., 2022).

Ideal circle pattern theory provides another major realization. In Euclidean and hyperbolic ideal circle patterns, Li–Wang formulate a Calabi flow for the curvature vector $K_i = 2\pi - \sum_{\{i,j,k\}\in F} \theta_i^{jk},$5 and prove global existence with exponential convergence to the unique metric realizing any attainable curvature vector (Li et al., 3 Jan 2025). A closely related 2025 treatment proves all-time existence and exponential convergence to a flat cone metric in the Euclidean case and a hyperbolic metric in the hyperbolic case for any given initial ideal circle pattern (Zhang, 3 Jan 2025).

In spherical background geometry, the controlled variable changes again. For ideal circle patterns on a closed surface with a closed 2-cell embedding, the relevant quantity is the total geodesic curvature

$K_i = 2\pi - \sum_{\{i,j,k\}\in F} \theta_i^{jk},$6

not the angle deficit. Lei–Zhou define a spherical combinatorial Calabi flow in the coordinates $K_i = 2\pi - \sum_{\{i,j,k\}\in F} \theta_i^{jk},$7 and prove that convergence is equivalent to the existence of an ideal circle pattern with the prescribed total geodesic curvatures (Lei et al., 2023). The same shift from angle deficit to total geodesic curvature appears in generalized hyperbolic circle packings with circles, horocycles, and hypercycles, where $K_i = 2\pi - \sum_{\{i,j,k\}\in F} \theta_i^{jk},$8 is the total geodesic curvature carried by a generalized circle (Ba et al., 2023).

4. Surgery, Delaunay conditions, and singularity control

One of the main technical obstacles in combinatorial Calabi flow is the development of singularities as the triangulation ceases to be geometrically adapted to the evolving metric. In vertex-scaling formulations, the decisive remedy is surgery by edge flipping.

For an interior edge $K_i = 2\pi - \sum_{\{i,j,k\}\in F} \theta_i^{jk},$9 shared by triangles $\theta_i^{jk}$0 and $\theta_i^{jk}$1, Zhu–Xu define the opposite-angle sum

$\theta_i^{jk}$2

When $\theta_i^{jk}$3, the edge is locally non-Delaunay and is flipped from $\theta_i^{jk}$4 to $\theta_i^{jk}$5 (Zhu et al., 2018). The flip is an isometry at that instant: it changes the combinatorics but not the underlying metric, and it preserves the discrete conformal class. Maintaining a Delaunay triangulation keeps cotan weights nonnegative and restores the positive semidefinite or positive definite structure needed in the convergence analysis.

The same principle appears in inversive-distance circle packing. Xu–Zheng introduce combinatorial Calabi, fractional Calabi, and $\theta_i^{jk}$6-th Calabi flows with surgery under the weighted Delaunay condition

$\theta_i^{jk}$7

where the $\theta_i^{jk}$8-angles are defined using the face-circle geometry of a decorated metric (Xu et al., 2023). When the weighted Delaunay condition fails, an edge flip is performed; the metric is unchanged, but the triangulation is updated to restore the sign conditions on the discrete Laplacian.

This surgery mechanism is not merely a numerical convenience. Zhu–Xu prove that only finitely many flips occur along the entire flow and that, after finitely many surgeries, the flow stays Delaunay and converges exponentially fast (Zhu et al., 2018). By contrast, in the broader Glickenstein framework, Xu shows that the combinatorial Calabi flow cannot be extended by the same angle-extension technique used for Ricci flow; surgery by weighted Delaunay flips is presented as the natural alternative, and global convergence in that generality is formulated as a conjectural direction (Xu, 2023).

5. Existence, rigidity, and convergence theory

The foundational existence-and-convergence theorem for surfaces appears in Ge’s original circle-packing paper: for fixed $\theta_i^{jk}$9, the solution of the combinatorial Calabi flow exists for all time, and it converges if and only if Thurston’s circle packing exists; when convergence occurs, it is exponentially fast (Ge, 2012). The proof relies on the combinatorial Calabi energy, the combinatorial Ricci potential, and the discrete dual-Laplacian.

In hyperbolic background geometry, Ge–Xu sharpen this picture. They prove that the two-dimensional hyperbolic combinatorial Calabi flow converges to a ZCCP-metric if the initial energy is small enough, and, assuming the curvature has a uniform upper bound less than $i$0, the flow exists for all time and converges to a ZCCP-metric if and only if such a metric exists (Ge et al., 2013). The geometric meaning of the condition $i$1 is explicit: if $i$2, then the incident angles tend to $i$3, forcing $i$4.

The vertex-scaling theorem of Zhu–Xu is stronger in a different direction. For any initial Euclidean or hyperbolic polyhedral metric on a closed surface, the combinatorial Calabi flow with surgery exists for all time and converges exponentially fast after a finite number of surgeries; the limit is the unique constant-curvature or prescribed-curvature metric in the same discrete conformal class, and the convergence is independent of the combinatorial structure of the initial triangulation (Zhu et al., 2018). This triangulation independence is one of the most distinctive structural results in the subject.

Fractional extensions preserve the same convergence picture in several important settings. Wu–Xu prove long-time existence and global exponential convergence of the fractional combinatorial Calabi flow for Thurston’s Euclidean and hyperbolic circle packings, and, with surgery, for vertex scaling on polyhedral surfaces (Wu et al., 2021). The finite-$i$5 theory developed by Lin–Zhang shows that for any $i$6 the combinatorial $i$7-th Calabi flow exists for all time and converges if and only if a constant curvature metric exists in Euclidean background or a zero curvature metric exists in hyperbolic background (Lin et al., 2018).

6. Algorithmic role and contemporary extensions

Algorithmically, combinatorial Calabi flow is used to compute uniformizing or prescribed-curvature metrics by iterating the same cycle: initialize discrete conformal factors, compute geometric quantities such as angles and curvatures, assemble the Jacobian or Laplacian, advance the flow, and stop when the curvature residual is sufficiently small. This workflow is explicit in the vertex-scaling, boundary, and inversive-distance papers, where the linear algebra is sparse and the geometric updates are local (Zhu et al., 2018).

The boundary setting illustrates the computational role especially clearly. On ideally triangulated surfaces with boundary, Luo–Xu prove that both the combinatorial Calabi flow and its fractional generalization exist for all time and converge exponentially fast, thereby providing effective algorithms to construct hyperbolic surfaces with totally geodesic boundaries of prescribed lengths (Luo et al., 2021). In ideal circle pattern geometry, Li–Wang similarly derive an algorithm from the all-time existence and exponential convergence of the Calabi flow for attainable curvatures in both Euclidean and hyperbolic backgrounds (Li et al., 3 Jan 2025).

Recent work has pushed the theory beyond the classical finite triangulated-surface setting. For finite and infinite ideal circle patterns, the combinatorial $i$8-th Calabi flow with $i$9 converges in the finite case if and only if a constant curvature metric exists, while in the infinite case long-time existence is proved for $r:V\to(0,+\infty)$0 (Yang et al., 8 Jun 2025). A further transposition replaces surface angle-deficit geometry by Lin–Lu–Yau curvature on finite graphs of girth at least $r:V\to(0,+\infty)$1; in that graph-theoretic analogue, the modified, fractional, and $r:V\to(0,+\infty)$2-th Calabi flows exist globally, and convergence is equivalent to realizability of the prescribed curvature (Li et al., 3 Apr 2026).

Taken together, these developments show that “combinatorial Calabi flow” now denotes a family of discrete variational flows unified by three structural features: a discrete conformal parameter space, a curvature map with symmetric Jacobian, and a curvature-squared Lyapunov functional. What changes from one theory to another is the geometry encoded by the discrete conformal class, the exact form of the Laplacian, and the meaning of the target curvature.