Combinatorial Ricci Flow

• Combinatorial Ricci flow is a discrete curvature flow on triangulated spaces where conformal parameters evolve via curvature defects.
• It employs convex and proper energy functionals, symmetry properties, and convergence theorems to ensure effective discrete uniformizations.
• The theory extends across Euclidean, hyperbolic, and spherical geometries, and even into three-dimensional triangulations for hyperbolization.

Combinatorial Ricci flow denotes a family of discrete curvature flows on triangulated or cell-decomposed spaces in which discrete conformal parameters evolve by curvature defects. On surfaces, the basic parameters are typically circle-packing radii or logarithmic conformal factors, and the curvature is usually an angle defect of the form $K_i=2\pi-\sum\theta_i^{jk}$; for surfaces with boundary or in spherical background geometry, the prescribed quantity may instead be a boundary length or a total geodesic curvature; in three dimensions, the curvature becomes the cone-angle deficit along edges of ideal or hyper-ideal tetrahedral decompositions. Across these settings, the theory is organized by convex or proper energy functionals, symmetry and definiteness properties of curvature Jacobians, extensions through degeneracies, and convergence theorems that make the flow an effective algorithm for constructing discrete uniformizations and hyperbolic structures (Takatsu, 2018, Xu et al., 2022, Ge et al., 2023, Xu, 2020).

1. Discrete curvature models on triangulated surfaces

On a weighted triangulation $(T,\Theta)$ of a connected, oriented, closed surface, a circle-packing metric is a vector of radii $r=(r_v)_{v\in V}\in \mathbb R_{>0}^V$. The edge length attached to an edge $e=uv$ is

$\ell(e;r)= \begin{cases} \sqrt{r_u^2+r_v^2+2\,r_u r_v\cos\Theta(e)},&\chi(S)=0,\[4pt] \operatorname{arccosh}\!\bigl(\cosh r_u\cosh r_v+\sinh r_u\sinh r_v\cos\Theta(e)\bigr),&\chi(S)<0, \end{cases}$

and each face becomes a geodesic triangle with inner angles $\phi_{f,v}(r)$. The combinatorial curvature at $v$ is then

$K_v(r)=2\pi-\sum_{f\ni v}\phi_{f,v}(r).$

A circle-pattern metric is characterized by $K_v(r)=0$ for all vertices (Takatsu, 2018).

A broader discrete conformal framework assigns metrics through vertex functions $f:V\to\mathbb R$ together with coefficients $(T,\Theta)$0 and $(T,\Theta)$1. In this setting the classical curvature remains $(T,\Theta)$2, while the parameterized combinatorial $(T,\Theta)$3-curvature is

$(T,\Theta)$4

with $(T,\Theta)$5 in Euclidean background and the corresponding hyperbolic $(T,\Theta)$6-coordinates determined by the chosen discrete conformal structure. The case $(T,\Theta)$7 recovers the classical curvature (Xu et al., 2021).

For ideally triangulated surfaces with boundary, the basic object is a type $(T,\Theta)$8 generalized circle-packing metric. If $(T,\Theta)$9 has components labeled by $r=(r_v)_{v\in V}\in \mathbb R_{>0}^V$0, with positive radii $r=(r_v)_{v\in V}\in \mathbb R_{>0}^V$1 and edge weights $r=(r_v)_{v\in V}\in \mathbb R_{>0}^V$2, then the edge lengths satisfy

$r=(r_v)_{v\in V}\in \mathbb R_{>0}^V$3

Gluing the corresponding right-angled hyperbolic hexagons produces a hyperbolic structure on $r=(r_v)_{v\in V}\in \mathbb R_{>0}^V$4 with totally geodesic boundary. In this case the combinatorial curvature at boundary component $r=(r_v)_{v\in V}\in \mathbb R_{>0}^V$5 is

$r=(r_v)_{v\in V}\in \mathbb R_{>0}^V$6

where $r=(r_v)_{v\in V}\in \mathbb R_{>0}^V$7 is the boundary-arc length at $r=(r_v)_{v\in V}\in \mathbb R_{>0}^V$8 in the relevant hexagon (Xu et al., 2022).

2. Canonical flow equations and normalization conventions

The literature uses several equivalent or analogous formulations, depending on the discrete metric variable, the background geometry, and whether a target curvature is prescribed.

Setting	Variable	Flow law
Chow–Luo surface flow	$r=(r_v)_{v\in V}\in \mathbb R_{>0}^V$9	$e=uv$0
Closed surface, log variable	$e=uv$1 or $e=uv$2	$e=uv$3
Surface with boundary, prescribed $e=uv$4	$e=uv$5	$e=uv$6
Prescribed $e=uv$7-curvature	$e=uv$8	$e=uv$9
Spherical total geodesic curvature	$\ell(e;r)= \begin{cases} \sqrt{r_u^2+r_v^2+2\,r_u r_v\cos\Theta(e)},&\chi(S)=0,\[4pt] \operatorname{arccosh}\!\bigl(\cosh r_u\cosh r_v+\sinh r_u\sinh r_v\cos\Theta(e)\bigr),&\chi(S)<0, \end{cases}$0	$\ell(e;r)= \begin{cases} \sqrt{r_u^2+r_v^2+2\,r_u r_v\cos\Theta(e)},&\chi(S)=0,\[4pt] \operatorname{arccosh}\!\bigl(\cosh r_u\cosh r_v+\sinh r_u\sinh r_v\cos\Theta(e)\bigr),&\chi(S)<0, \end{cases}$1
Three-dimensional edge flow	$\ell(e;r)= \begin{cases} \sqrt{r_u^2+r_v^2+2\,r_u r_v\cos\Theta(e)},&\chi(S)=0,\[4pt] \operatorname{arccosh}\!\bigl(\cosh r_u\cosh r_v+\sinh r_u\sinh r_v\cos\Theta(e)\bigr),&\chi(S)<0, \end{cases}$2	$\ell(e;r)= \begin{cases} \sqrt{r_u^2+r_v^2+2\,r_u r_v\cos\Theta(e)},&\chi(S)=0,\[4pt] \operatorname{arccosh}\!\bigl(\cosh r_u\cosh r_v+\sinh r_u\sinh r_v\cos\Theta(e)\bigr),&\chi(S)<0, \end{cases}$3

These formulas exhibit a common pattern: the state variable is driven by a discrete curvature defect. The multiplicative factors $\ell(e;r)= \begin{cases} \sqrt{r_u^2+r_v^2+2\,r_u r_v\cos\Theta(e)},&\chi(S)=0,\[4pt] \operatorname{arccosh}\!\bigl(\cosh r_u\cosh r_v+\sinh r_u\sinh r_v\cos\Theta(e)\bigr),&\chi(S)<0, \end{cases}$4, $\ell(e;r)= \begin{cases} \sqrt{r_u^2+r_v^2+2\,r_u r_v\cos\Theta(e)},&\chi(S)=0,\[4pt] \operatorname{arccosh}\!\bigl(\cosh r_u\cosh r_v+\sinh r_u\sinh r_v\cos\Theta(e)\bigr),&\chi(S)<0, \end{cases}$5, $\ell(e;r)= \begin{cases} \sqrt{r_u^2+r_v^2+2\,r_u r_v\cos\Theta(e)},&\chi(S)=0,\[4pt] \operatorname{arccosh}\!\bigl(\cosh r_u\cosh r_v+\sinh r_u\sinh r_v\cos\Theta(e)\bigr),&\chi(S)<0, \end{cases}$6, or $\ell(e;r)= \begin{cases} \sqrt{r_u^2+r_v^2+2\,r_u r_v\cos\Theta(e)},&\chi(S)=0,\[4pt] \operatorname{arccosh}\!\bigl(\cosh r_u\cosh r_v+\sinh r_u\sinh r_v\cos\Theta(e)\bigr),&\chi(S)<0, \end{cases}$7 reflect the chosen coordinates rather than a contradiction in the underlying idea. In particular, prescribed-curvature versions replace $\ell(e;r)= \begin{cases} \sqrt{r_u^2+r_v^2+2\,r_u r_v\cos\Theta(e)},&\chi(S)=0,\[4pt] \operatorname{arccosh}\!\bigl(\cosh r_u\cosh r_v+\sinh r_u\sinh r_v\cos\Theta(e)\bigr),&\chi(S)<0, \end{cases}$8 by $\ell(e;r)= \begin{cases} \sqrt{r_u^2+r_v^2+2\,r_u r_v\cos\Theta(e)},&\chi(S)=0,\[4pt] \operatorname{arccosh}\!\bigl(\cosh r_u\cosh r_v+\sinh r_u\sinh r_v\cos\Theta(e)\bigr),&\chi(S)<0, \end{cases}$9, $\phi_{f,v}(r)$0, $\phi_{f,v}(r)$1, or their higher-dimensional analogues (Saucan, 2011, Takatsu, 2018, Xu et al., 2022, Xu et al., 2021, Ge et al., 2023, Xu, 2020, Feng et al., 2020).

3. Variational structure, Jacobians, and discrete parabolicity

A central feature of combinatorial Ricci flow is its variational formulation. For generalized circle packings on surfaces with boundary, the discrete conformal coordinates

$\phi_{f,v}(r)$2

make the curvature map $\phi_{f,v}(r)$3 smooth, and

$\phi_{f,v}(r)$4

defines a symmetric negative-definite $\phi_{f,v}(r)$5 matrix. The associated Ricci energy

$\phi_{f,v}(r)$6

is strictly convex, and the linearization has strictly negative spectrum at the target metric (Xu et al., 2022).

In spherical background geometry with prescribed total geodesic curvatures, the local 1-form

$\phi_{f,v}(r)$7

is closed on each edge, hence admits a local convex potential $\phi_{f,v}(r)$8. These edge potentials assemble into a global strictly convex functional

$\phi_{f,v}(r)$9

and the flow in $v$0-variables is

$v$1

Properness of this energy is characterized exactly by the coherence inequalities

$v$2

for every nonempty subset $v$3 (Ge et al., 2023).

For parameterized combinatorial curvature flows, the Ricci energy

$v$4

satisfies

$v$5

so the modified $v$6-Ricci flow is the negative-gradient flow of $v$7. A key technical device is the continuous extension of triangle angles across degenerate configurations, which extends $v$8 to a $v$9 function on $K_v(r)=2\pi-\sum_{f\ni v}\phi_{f,v}(r).$0 (Xu et al., 2021).

In three dimensions the analogous energy is the co-volume functional. On an ideally triangulated compact $K_v(r)=2\pi-\sum_{f\ni v}\phi_{f,v}(r).$1-manifold with boundary one defines

$K_v(r)=2\pi-\sum_{f\ni v}\phi_{f,v}(r).$2

while in the cusped case

$K_v(r)=2\pi-\sum_{f\ni v}\phi_{f,v}(r).$3

Both identities come from Schläfli’s formula, and both realize the flow as a negative-gradient evolution of a convex functional (Xu, 2020, Xu, 2020).

A plausible implication is that the various versions of combinatorial Ricci flow are best understood not as a single discretization, but as a variational paradigm: choose a discrete metric space, identify a curvature map with symmetric derivative structure, and run the negative-gradient flow of the corresponding energy.

4. Long-time existence and convergence on finite surfaces

For compact polyhedral surfaces without boundary and bounded combinatorial or Wald curvature, one can prove forward and backward short-time existence and uniqueness of the discrete or metric Ricci flow by approximating the polyhedral surface with smoothings, applying classical Ricci-flow results on the smooth approximants, and then passing to the limit. In the normalized flow on a compact closed surface, convergence to constant curvature occurs exponentially fast in the negative- and positive-curvature cases and at rate $K_v(r)=2\pi-\sum_{f\ni v}\phi_{f,v}(r).$4 in the flat case. The same source also states that on genus $K_v(r)=2\pi-\sum_{f\ni v}\phi_{f,v}(r).$5 polyhedral surfaces the unnormalized combinatorial flow never develops combinatorial singularities, whereas on the torus or sphere with Euclidean background triangle inequalities may fail late in the flow and local retriangulations by edge-flips may be used to continue it (Saucan, 2011).

For hyperbolic background geometry, the modified curvature

$K_v(r)=2\pi-\sum_{f\ni v}\phi_{f,v}(r).$6

yields the flow $K_v(r)=2\pi-\sum_{f\ni v}\phi_{f,v}(r).$7. The convergence theorem is an if and only if statement: the flow converges exactly when a circle-packing metric with zero curvature exists. A stronger sufficient condition is also available: if the initial modified curvatures are all non-positive, then the flow exists for all time, the curvatures remain non-positive, the trajectory stays in a compact subset of $K_v(r)=2\pi-\sum_{f\ni v}\phi_{f,v}(r).$8, and the solution converges exponentially fast to the unique zero-curvature metric. The paper also states the same equivalence for arbitrary smooth positive area elements $K_v(r)=2\pi-\sum_{f\ni v}\phi_{f,v}(r).$9 (Ge et al., 2015).

For ideally triangulated surfaces with boundary and prescribed boundary lengths $K_v(r)=0$0, the combinatorial Ricci flow

$K_v(r)=0$1

has a particularly strong global theory. For any initial $K_v(r)=0$2, the solution exists for all $K_v(r)=0$3, remains in a compact subset of $K_v(r)=0$4, and converges exponentially fast to the unique $K_v(r)=0$5 with $K_v(r)=0$6. The same paper proves the analogous result for the combinatorial Calabi flow. It also records an explicit Euler-type algorithm in the $K_v(r)=0$7-variables, with complexity $K_v(r)=0$8 per step and $K_v(r)=0$9 steps to reach accuracy $f:V\to\mathbb R$0, up to CFL-type restrictions on the step size (Xu et al., 2022).

For the parameterized $f:V\to\mathbb R$1-Ricci flow, singularities are handled by extending the angle functions by constants across degenerate triangles. Under the structure conditions

$f:V\to\mathbb R$2

and assuming the prescribed curvature is realized by some discrete conformal structure, the extended modified flow has a unique global solution. If additionally $f:V\to\mathbb R$3 in Euclidean background or $f:V\to\mathbb R$4 in hyperbolic background, then the solution converges exponentially fast to the unique target metric. In particular, the $f:V\to\mathbb R$5 case recovers long-time convergence of Chow–Luo’s flow and Luo’s Yamabe flow under the usual sign conditions (Xu et al., 2021).

5. Degeneration, prescribed data, and spherical background geometry

A recurring issue is whether failure of the strict existence criterion destroys the asymptotic structure of the flow. For closed surfaces with $f:V\to\mathbb R$6, Takatsu analyzed the marginal case

$f:V\to\mathbb R$7

In this regime the combinatorial Ricci flow still satisfies $f:V\to\mathbb R$8 for every vertex, but some radii collapse: in the Euclidean case the full radius vector does not converge in $f:V\to\mathbb R$9, while in the hyperbolic case the flow does converge in $(T,\Theta)$00 and the limiting metric vanishes exactly on the collapsing set. The limiting metric on the complement is the unique circle-pattern metric on the reduced complex (Takatsu, 2018).

In hyperbolic background geometry, degenerated generalized circle packings can be treated directly by prescribing the total geodesic curvature data

$(T,\Theta)$01

For the interior target region $(T,\Theta)$02, the map $(T,\Theta)$03, $(T,\Theta)$04 is a homeomorphism, so every $(T,\Theta)$05 determines a unique nondegenerate metric and the Ricci flow converges exponentially fast to it. The map extends continuously to the boundary, and for $(T,\Theta)$06 on the partial boundary $(T,\Theta)$07 there is a unique degenerated metric solving $(T,\Theta)$08. The corresponding flow exists for all time on the relevant boundary stratum and converges to that unique degenerated solution (Hu et al., 2024).

The spherical theory replaces angle deficit by total geodesic curvature. For a circle of radius $(T,\Theta)$09 one has $(T,\Theta)$10, total length $(T,\Theta)$11, and total geodesic curvature

$(T,\Theta)$12

The spherical prescribed-curvature flow converges for every initial data if and only if the target data satisfy the coherence inequalities

$(T,\Theta)$13

for every nonempty subset $(T,\Theta)$14; under these inequalities the flow converges exponentially fast to the unique circle pattern realizing $(T,\Theta)$15. The same source gives an explicit or semi-implicit Euler implementation with $(T,\Theta)$16 work per step and overall complexity $(T,\Theta)$17 (Ge et al., 2023).

A further spherical refinement treats degenerated circle pattern metrics and spherical conical metrics with prescribed total geodesic curvatures. The existence theorem identifies the admissible sets $(T,\Theta)$18, $(T,\Theta)$19, and $(T,\Theta)$20 for nondegenerate, degenerate, and general spherical conical metrics, respectively. The same work proves convergence of the prescribed combinatorial Ricci flows in three forms: the nondegenerate flow for $(T,\Theta)$21, the restricted flow on an index subset $(T,\Theta)$22, and a mixed flow on the full space in which some geodesic curvatures remain zero. It also gives two distinct constructions of degenerated solutions, one on the reduced coordinate subspace and one on the full space with vanishing components preserved (Hu et al., 2024).

6. Infinite triangulations and noncompact analogues

Combinatorial Ricci flow extends beyond finite complexes to infinite, locally finite triangulations. For infinite disk triangulations in Euclidean and hyperbolic background geometries, the flow

$(T,\Theta)$23

is defined on the entire infinite vertex set. Long-time existence is obtained by exhausting the triangulation by finite simply connected subcomplexes, solving Dirichlet-boundary flows on the exhaustion pieces, and then using a maximum principle together with an Arzelà–Ascoli diagonal argument. Uniqueness holds under uniformly bounded curvature. In hyperbolic background, if the initial metric satisfies $(T,\Theta)$24 for all vertices, then the flow converges to a zero-curvature packing. In the Euclidean case, the regular infinite hexagonal triangulation admits a small-data convergence theorem in which the Dirichlet energy

$(T,\Theta)$25

tends to zero and $(T,\Theta)$26 (Ge et al., 8 Apr 2025).

The same noncompact philosophy appears in the theory of infinite ideal circle patterns. For a locally finite infinite cell decomposition $(T,\Theta)$27 of $(T,\Theta)$28 or $(T,\Theta)$29, one defines vertex curvature by angle deficits and runs

$(T,\Theta)$30

Under the ideal-pattern condition

$(T,\Theta)$31

on each face, the flow exists for all $(T,\Theta)$32. Uniqueness holds under either uniformly bounded curvature or uniformly bounded degrees. In the hyperbolic case, if the initial metric has $(T,\Theta)$33 for all vertices, then the flow converges to a good ideal circle pattern with $(T,\Theta)$34. The same work uses Thurston’s correspondence between ideal circle patterns on $(T,\Theta)$35 and convex ideal polyhedra in the Poincaré ball to characterize infinite ideal polyhedra in $(T,\Theta)$36 and thereby solve Rivin’s infinite version under the stated combinatorial conditions (Ge et al., 5 Jun 2025).

In spherical background geometry on infinite cellular decompositions, the basic variables are radii $(T,\Theta)$37 or logarithmic coordinates $(T,\Theta)$38, and the target quantity is the total geodesic curvature

$(T,\Theta)$39

Under the conditions

$(T,\Theta)$40

for every finite subset $(T,\Theta)$41, the flow exists uniquely for all time. If in addition the initial data satisfy

$(T,\Theta)$42

then $(T,\Theta)$43 converges and $(T,\Theta)$44 for every vertex. The proof uses finite exhaustions, uniform $(T,\Theta)$45–$(T,\Theta)$46 estimates, a maximum principle on infinite graphs, and a Lyapunov functional built from convex edge potentials (Li et al., 9 May 2025).

7. Three-dimensional extensions and hyperbolization

In dimension three, combinatorial Ricci flow is formulated on ideal or hyper-ideal triangulations of pseudo-manifolds, compact manifolds with boundary, or cusped manifolds. The discrete metric variable is an edge-length vector, and the curvature at an edge is the cone-angle deficit

$(T,\Theta)$47

or $(T,\Theta)$48 for certain boundary edges in the partially truncated setting. Zero curvature means that the tetrahedra glue to a complete hyperbolic metric with totally geodesic boundary or cusps, depending on the model (Xu, 2020, Feng et al., 2020, Ke et al., 10 Feb 2025).

For ideally triangulated compact $(T,\Theta)$49-manifolds with boundary, the basic flow is Luo’s additive equation

$(T,\Theta)$50

which Xu shows can be extended through tetrahedral degenerations by Luo–Yang’s continuous extension of dihedral angles to all of $(T,\Theta)$51. The extended flow exists uniquely for all $(T,\Theta)$52, and if there exists an admissible zero-curvature hyper-ideal metric $(T,\Theta)$53, then every initial datum converges exponentially fast to $(T,\Theta)$54. This verifies Luo’s conjecture in that setting (Xu, 2020).

A multiplicative variant on pseudo-$(T,\Theta)$55-manifolds evolves

$(T,\Theta)$56

or, in the extended form, $(T,\Theta)$57. Under the hypothesis that every edge has valence at least $(T,\Theta)$58, there exists a unique zero-curvature hyper-ideal metric and the extended flow converges exponentially fast from arbitrary positive initial data. The paper also gives the estimate

$(T,\Theta)$59

A later refinement lowers the valence threshold to $(T,\Theta)$60 and proves that for any compact $(T,\Theta)$61-manifold with boundary admitting an ideal triangulation with all edge valences at least $(T,\Theta)$62, there exists a unique complete hyperbolic metric with totally geodesic boundary realizing that triangulation, together with explicit upper and lower bounds contained in $(T,\Theta)$63 (Feng et al., 2020, Zhao, 21 Jan 2026).

For cusped $(T,\Theta)$64-manifolds, one works with decorated ideal tetrahedra and signed edge lengths. The co-volume functional on the affine cusp slice has gradient $(T,\Theta)$65, so the extended flow

$(T,\Theta)$66

is again a negative-gradient flow of a convex energy. The main theorem states that an ideal triangulated cusped $(T,\Theta)$67-manifold admits a complete hyperbolic metric if and only if the extended combinatorial Ricci flow converges to a zero-curvature point, and when convergence occurs the rate is exponential. This provides a curvature-flow formulation dual to the Casson–Rivin volume-maximization program (Xu, 2020, Feng et al., 2020).

The relation to geometric triangulations is made explicit in the partially truncated and decorated setting. There the extended combinatorial Ricci flow exists uniquely for all initial data in $(T,\Theta)$68, and it converges if and only if the triangulation is geometric. Equivalently, there exists a complete hyperbolic structure on the manifold with the given ideal triangulation realized as a geometric ideal triangulation if and only if the flow converges to a zero-curvature metric. The same framework is presented as a systematic approach to Thurston’s triangulation conjecture (Ke et al., 10 Feb 2025).

Taken together, these results show that combinatorial Ricci flow has evolved from the original Chow–Luo surface equation into a broad discrete uniformization and hyperbolization program. The common structure is curvature-defect evolution, convex energy descent, and rigidity of the target metric; the major technical differences arise from background geometry, boundary behavior, degenerations, and the passage from finite to infinite or from two-dimensional to three-dimensional complexes.