Combinatorial Yamabe Flow on Hyperbolic Surfaces

• Combinatorial Yamabe flow is a discrete curvature evolution method on triangulated surfaces that adjusts boundary conformal factors to achieve prescribed geodesic boundary lengths.
• The approach utilizes variational principles and strictly convex energy functionals to ensure global rigidity, long-time existence, and exponential convergence.
• Its framework provides practical numerical algorithms for discrete uniformization, with applications extending to various hyperbolic and combinatorial geometric settings.

Searching arXiv for relevant papers on combinatorial Yamabe flow and hyperbolic bordered surfaces. Combinatorial Yamabe flow is a discrete curvature flow on triangulated surfaces that deforms a combinatorial conformal factor so as to control geometric boundary data. In the setting of hyperbolic surfaces with geodesic boundary, the flow is formulated on an ideal triangulation by assigning one real parameter to each boundary component and evolving these parameters according to the discrepancy between the current and target boundary lengths. For hyperbolic bordered surfaces, Li–Xu–Zhou proved that the prescribed-boundary-length flow exists for all time and converges exponentially fast to the unique conformal factor producing the desired boundary lengths (Li et al., 2022). This development both refines and corrects earlier rigidity results due to Guo, whose 2010 work established the variational structure and long-time behavior of the unforced flow toward cusped metrics (Guo, 2010). Later work generalized the boundary-prescribing dynamics to a parameter family of combinatorial flows with analogous global convergence properties (Li et al., 15 Apr 2025).

1. Geometric setting and discrete conformal data

Let $\Sigma$ be a compact surface with $n$ boundary components, and let $T$ be an ideal triangulation of $\Sigma$. In the formulation used for hyperbolic bordered surfaces, $\Sigma$ is obtained from a triangulation of a closed surface by removing small open balls around the vertices; the boundary components of $\Sigma$ correspond to the removed vertex neighborhoods, ideal edges join boundary components, and ideal faces are triangles whose vertices lie on three boundary components (Li et al., 2022).

A discrete hyperbolic metric on $(\Sigma,T)$ is given by positive lengths on the ideal edges. Each ideal face is realized as a right-angled hyperbolic hexagon, determined by the three alternating edge lengths. Gluing these hexagons along the corresponding pair edges produces a complete hyperbolic surface with geodesic boundary (Li et al., 2022). This hexagonal model is also the starting point in Guo’s earlier treatment of hyperbolic surfaces with boundary (Guo, 2010).

The discrete conformal degree of freedom is a vector attached to boundary components rather than to ordinary vertices. In Li–Xu–Zhou, the conformal factor is $w=(w_1,\dots,w_n)\in\mathbb R^n$, and the conformally changed edge lengths are defined by

$\cosh l_{ij}(w)=e^{w_i+w_j}\cosh l^0_{ij},$

equivalently

$$l_{ij}(w)=\operatorname{arcosh}\!\bigl(e^{w_i+w_j}\cosh l^0_{ij}\bigr).$$

The admissible domain $n$0 is the convex set of all $n$1 satisfying

$n$2

for every ideal edge $n$3 (Li et al., 2022). In Guo’s notation the same structure is expressed with a conformal factor $n$4 and edge lengths $n$5 satisfying

$n$6

with the resulting boundary lengths defined face by face through the associated right-angled hexagons (Guo, 2010).

This framework makes the conformal deformation finite-dimensional and explicitly computable. A plausible implication is that it places the prescribed-boundary-length problem in a setting analogous to discrete uniformization, but with geodesic boundary lengths replacing interior curvature as the primary target datum.

2. Boundary-length map and rigidity

For each boundary component, the geodesic boundary length is obtained by summing the relevant $n$7-arc lengths contributed by adjacent hexagons. Li–Xu–Zhou denote the resulting map by

$n$8

They prove that $n$9 is a smooth diffeomorphism from $T$0 onto $T$1 (Li et al., 2022). This is the core rigidity statement for the hyperbolic bordered-surface problem: prescribed positive boundary lengths determine a unique discrete conformal factor.

The proof is variational. Writing $T$2, one obtains a Jacobian matrix that is symmetric, strictly diagonally dominant, and negative-definite on $T$3 (Li et al., 2022). Symmetry implies that the $T$4-form

$T$5

is closed, hence exact on the convex domain $T$6. Its primitive

$T$7

satisfies $T$8, and the negativity of $T$9 gives convexity of $\Sigma$0 (Li et al., 2022).

Properness of the boundary-length map is established by analyzing the behavior as $\Sigma$1 approaches the boundary of $\Sigma$2: some $\Sigma$3 tends to $\Sigma$4 or $\Sigma$5, which prevents loss of compactness in the interior variables (Li et al., 2022). These ingredients yield global injectivity and surjectivity of $\Sigma$6.

Guo’s 2010 paper proved an analogous rigidity result in the earlier formulation: the boundary-length map

$\Sigma$7

is a global homeomorphism, and the proof also uses a variational principle based on a strictly concave facewise potential summed to a global energy $\Sigma$8 (Guo, 2010). The later work sharpens this picture by identifying the smooth diffeomorphism structure on the admissible domain and by using corrected Jacobian estimates (Li et al., 2022).

A common misconception is to view the boundary lengths as only local or partially constrained data. In this discrete hyperbolic setting, the rigidity theorem states the opposite: within a fixed ideal triangulation and fixed initial edge-length data, the full vector of positive boundary lengths globally parametrizes the conformal factors (Li et al., 2022).

3. Flow equations and geometric interpretation

Given a prescribed target vector $\Sigma$9, the unique factor $\Sigma$0 satisfying $\Sigma$1 exists by the rigidity theorem. Li–Xu–Zhou define the combinatorial Yamabe flow by the ODE

$\Sigma$2

for arbitrary initial data $\Sigma$3 (Li et al., 2022). This is the prescribed-boundary-length version of combinatorial Yamabe flow on hyperbolic bordered surfaces.

The earlier unforced flow studied by Guo is

$\Sigma$4

which drives all boundary lengths to zero and hence deforms the geodesic-boundary surface toward a complete finite-area cusped surface (Guo, 2010). In that sense, Guo’s flow solves a discrete degeneration-to-cusps problem, whereas the later flow solves a discrete prescription problem for arbitrary positive boundary lengths (Li et al., 2022).

The generalized formulation introduced by Li–Wang replaces the unit mobility by a positive coefficient depending on the parameter $\Sigma$5:

$\Sigma$6

where

$\Sigma$7

They note that $\Sigma$8 recovers Li–Xu–Zhou’s Yamabe flow, while $\Sigma$9 and $\Sigma$0 recovers Guo’s flow (Li et al., 15 Apr 2025). This identifies the earlier flows as special cases in a one-parameter family.

The geometric meaning of the prescribed-length flow is direct: it evolves the conformal factor on boundary components so that the induced hyperbolic metric acquires boundary lengths exactly equal to the target vector. This suggests an interpretation as a discrete analogue of a boundary-value uniformization procedure, though the exact continuous counterpart is not stated in the cited works.

4. Variational structure and Lyapunov theory

The prescribed-boundary-length flow is governed by a strictly convex energy. Li–Xu–Zhou define

$\Sigma$1

with

$\Sigma$2

Since $\Sigma$3 is strictly convex and $\Sigma$4 is linear, $\Sigma$5 is strictly convex and has a unique critical point at $\Sigma$6 (Li et al., 2022).

They also introduce the error-energy

$\Sigma$7

which measures the residual in boundary lengths (Li et al., 2022). The negative-definite Jacobian $\Sigma$8 yields a differential inequality for $\Sigma$9 along the flow, and this is the key to exponential decay.

In Guo’s formulation the variational structure is dual in sign: the facewise potentials sum to a strictly concave global energy

$(\Sigma,T)$0

with

$(\Sigma,T)$1

Accordingly, Guo writes the combinatorial Yamabe flow as the positive-gradient flow of a concave functional, or equivalently as the negative-gradient flow of the convex functional $(\Sigma,T)$2 (Guo, 2010). The sign conventions differ, but both formulations are gradient-type dynamics driven by the boundary-length map.

Li–Wang use a Lyapunov functional adapted to the generalized mobility. They define a potential $(\Sigma,T)$3 from the closed $(\Sigma,T)$4-form $(\Sigma,T)$5, then set

$(\Sigma,T)$6

and further define

$(\Sigma,T)$7

They show that $(\Sigma,T)$8 is non-increasing along the generalized flow and that its sublevel sets are compactly contained in $(\Sigma,T)$9 (Li et al., 15 Apr 2025). This reproduces the same coercive variational mechanism in a more flexible dynamical family.

The essential analytic point across these papers is that the relevant Jacobian is symmetric and definite with the correct sign. That structure simultaneously gives rigidity of the boundary-length map, convexity or concavity of the potential, and monotonicity of the associated flow [(Li et al., 2022); (Guo, 2010)].

5. Long-time existence and exponential convergence

Li–Xu–Zhou prove that for any initial $w=(w_1,\dots,w_n)\in\mathbb R^n$0, the prescribed-boundary-length flow exists for all $w=(w_1,\dots,w_n)\in\mathbb R^n$1, remains in $w=(w_1,\dots,w_n)\in\mathbb R^n$2, and converges exponentially fast to the unique solution $w=(w_1,\dots,w_n)\in\mathbb R^n$3 of $w=(w_1,\dots,w_n)\in\mathbb R^n$4 (Li et al., 2022). The argument has two parts.

First, global existence follows from the Lyapunov functional $w=(w_1,\dots,w_n)\in\mathbb R^n$5. Along the negative-gradient dynamics, the trajectory remains in a compact sublevel set of $w=(w_1,\dots,w_n)\in\mathbb R^n$6, and therefore cannot approach the boundary of $w=(w_1,\dots,w_n)\in\mathbb R^n$7 or develop finite-time blow-up (Li et al., 2022). The details in the source summarize this as global existence and uniform boundedness.

Second, exponential convergence follows from the residual energy $w=(w_1,\dots,w_n)\in\mathbb R^n$8. Using symmetry and definiteness of $w=(w_1,\dots,w_n)\in\mathbb R^n$9,

$\cosh l_{ij}(w)=e^{w_i+w_j}\cosh l^0_{ij},$0

for a uniform lower bound $\cosh l_{ij}(w)=e^{w_i+w_j}\cosh l^0_{ij},$1 on the smallest eigenvalue of $\cosh l_{ij}(w)=e^{w_i+w_j}\cosh l^0_{ij},$2 along the compact trajectory (Li et al., 2022). Grönwall’s inequality then yields exponential decay of $\cosh l_{ij}(w)=e^{w_i+w_j}\cosh l^0_{ij},$3, and the diffeomorphism property of $\cosh l_{ij}(w)=e^{w_i+w_j}\cosh l^0_{ij},$4 transfers this to exponential convergence of $\cosh l_{ij}(w)=e^{w_i+w_j}\cosh l^0_{ij},$5 itself (Li et al., 2022).

Guo’s 2010 theorem establishes global existence for the unforced flow and proves that the boundary lengths satisfy $\cosh l_{ij}(w)=e^{w_i+w_j}\cosh l^0_{ij},$6 as $\cosh l_{ij}(w)=e^{w_i+w_j}\cosh l^0_{ij},$7, while the edge lengths tend to $\cosh l_{ij}(w)=e^{w_i+w_j}\cosh l^0_{ij},$8 so that each original right-angled hexagon degenerates to an ideal hyperbolic triangle. The limiting surface is therefore a complete finite-area hyperbolic surface with $\cosh l_{ij}(w)=e^{w_i+w_j}\cosh l^0_{ij},$9 cusps (Guo, 2010). The monotonicity

$$l_{ij}(w)=\operatorname{arcosh}\!\bigl(e^{w_i+w_j}\cosh l^0_{ij}\bigr).$$0

captures the boundary-length decay in that setting (Guo, 2010).

Li–Wang extend the global convergence theory to the generalized combinatorial Yamabe flow for $$l_{ij}(w)=\operatorname{arcosh}\!\bigl(e^{w_i+w_j}\cosh l^0_{ij}\bigr).$$1. Their Theorem A states that the flow exists uniquely for all time, never leaves $$l_{ij}(w)=\operatorname{arcosh}\!\bigl(e^{w_i+w_j}\cosh l^0_{ij}\bigr).$$2, and converges exponentially fast to the unique $$l_{ij}(w)=\operatorname{arcosh}\!\bigl(e^{w_i+w_j}\cosh l^0_{ij}\bigr).$$3 solving $$l_{ij}(w)=\operatorname{arcosh}\!\bigl(e^{w_i+w_j}\cosh l^0_{ij}\bigr).$$4 (Li et al., 15 Apr 2025). This shows that exponential convergence is not peculiar to the unit-mobility flow, but persists under a family of positive reweightings of the residual.

6. Algorithms and computational role

The existence and uniqueness theorems imply an algorithmic solution of the discrete boundary prescription problem. Li–Xu–Zhou explicitly state that the convergence result provides an algorithm for finding the conformal factor producing prescribed positive boundary lengths (Li et al., 2022).

Because $$l_{ij}(w)=\operatorname{arcosh}\!\bigl(e^{w_i+w_j}\cosh l^0_{ij}\bigr).$$5 is strictly convex and satisfies $$l_{ij}(w)=\operatorname{arcosh}\!\bigl(e^{w_i+w_j}\cosh l^0_{ij}\bigr).$$6, two numerical strategies are singled out in the paper summary:

Method	Update
Gradient descent	$$l_{ij}(w)=\operatorname{arcosh}\!\bigl(e^{w_i+w_j}\cosh l^0_{ij}\bigr).$$7
Newton’s method	$$l_{ij}(w)=\operatorname{arcosh}\!\bigl(e^{w_i+w_j}\cosh l^0_{ij}\bigr).$$8

Here $$l_{ij}(w)=\operatorname{arcosh}\!\bigl(e^{w_i+w_j}\cosh l^0_{ij}\bigr).$$9, and the Hessian is the symmetric negative-definite Jacobian $n$00 (Li et al., 2022). On compact sublevel sets, Newton’s method converges quadratically once sufficiently close to $n$01, whereas gradient descent converges at least linearly; a typical stopping criterion is $n$02 (Li et al., 2022).

Li–Wang write an explicit iterative scheme corresponding to the generalized flow: compute $n$03, set

$n$04

then update

$n$05

They remark that $n$06 may be chosen by a Courant-type condition or an Armijo-backtracking rule to preserve admissibility, and that one may also implement an implicit or semi-implicit Euler step (Li et al., 15 Apr 2025). They further suggest that the parameter $n$07 may offer an “acceleration” in transient regimes, although the paper is described as purely theoretical and does not provide detailed numerical experiments (Li et al., 15 Apr 2025).

The computational significance is therefore not incidental. In these works, combinatorial Yamabe flow is simultaneously a geometric ODE, a variational gradient system, and a constructive solver for the prescribed-boundary-length problem.

7. Relation to broader combinatorial Yamabe-flow theory

Combinatorial Yamabe flow appears in several geometrically distinct settings, and the hyperbolic bordered-surface theory occupies one branch of a broader program. Guo’s surface-with-boundary flow extends Luo’s combinatorial Yamabe flow for piecewise-flat surfaces to the hyperbolic category with boundary (Guo, 2010). Li–Xu–Zhou then adapt this framework from degeneration to cusps to exact prescription of positive boundary lengths (Li et al., 2022).

In higher dimension, a different but related notion arises from sphere or ball packings on triangulated $n$08-manifolds. Ge–Hua study the $n$09-dimensional combinatorial Yamabe flow in hyperbolic background geometry,

$n$10

together with an extended flow using extended solid angles. They prove existence and non-existence results indexed by tetra-degree, as well as exponential convergence to a zero-curvature packing when such a real packing exists (Ge et al., 2018). Ge–Jiang–Shen analyze the Euclidean $n$11-dimensional setting via ball packings, extended angles, and the Cooper–Rivin–Glickenstein functional, obtaining small-energy convergence and exponential convergence in the regular case (Ge et al., 2018).

On closed triangulated surfaces and polyhedral surfaces, parameterized versions of combinatorial curvature and curvature flows generalize both Chow–Luo’s Ricci flow and Luo’s Yamabe flow. Ge–Xu introduce $n$12-flows in dimensions two and three and show, in particular settings, equivalences between convergence and the existence of constant $n$13-curvature metrics (Ge et al., 2015). Xu–Zheng develop parameterized combinatorial $n$14-Ricci flows for discrete conformal structures on polyhedral surfaces, prove local and global rigidity, and establish long-time existence and exponential convergence for extended flows with prescribed curvature under suitable sign conditions (Xu et al., 2021).

Most recently, Bohao Ji studies infinite combinatorial Yamabe flows on noncompact Euclidean triangulated surfaces, proving short-time existence and uniqueness under uniform nondegeneracy, bounded degree, and uniform Delaunay assumptions, as well as long-time existence for an extended flow on arbitrary infinite triangulations (Ji, 16 Jul 2025). Although this Euclidean infinite-surface theory is structurally different from the finite-dimensional hyperbolic bordered-surface case, the shared pattern is notable: extension of geometric data across degeneracies, variational or maximum-principle control, and convergence to a distinguished metric when a solvable target problem is available.

Taken together, these works show that “combinatorial Yamabe flow” is not a single equation but a family of discrete conformal flows adapted to different background geometries, dimensions, and target data. In the particular case of hyperbolic bordered surfaces, its defining feature is the strict global rigidity of the boundary-length map and the resulting ability to prescribe geodesic boundary lengths by a provably convergent finite-dimensional flow [(Li et al., 2022); (Guo, 2010); (Li et al., 15 Apr 2025)].