Combinatorial pth Calabi flows for total geodesic curvatures in spherical background geometry

Bin Liu; Lishan Li; Yi Qi

doi:10.1515/anona-2025-0067

Article Open Access

Combinatorial pth Calabi flows for total geodesic curvatures in spherical background geometry

Bin Liu , Lishan Li and Yi Qi

Published/Copyright: February 20, 2025

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Advances in Nonlinear Analysis Volume 14 Issue 1

Abstract

The concept of combinatorial pth Calabi flow is proposed for finding circle packing metrics with prescribed curvatures. It has been studied under the Euclidean and hyperbolic background geometries. This study investigates the combinatorial pth Calabi flow for total geodesic curvatures under the spherical background geometry. It is established that the combinatorial pth Calabi flow, subject to given total geodesic curvatures under the spherical background geometry, exists for all time t ∈ [ 0 , + ∞ ) and converges to an ideal circle pattern metric. This finding introduces a new algorithm for constructing ideal circle pattern metrics with prescribed total geodesic curvatures.

Keywords: spherical circle pattern; combinatorial pth Calabi flow

MSC 2010: 52C26; 53E99

1 Introduction

It is important to explore a metric with prescribed curvatures on a given manifold under various background geometries. The classical Nirenberg problem aims to determine the achievable Gaussian curvatures for a conformal metric on S 2 [7]. Geometric flow is a powerful tool for solving problems involving prescribed curvatures. The Ricci flow introduced by Hamilton [15] is effective in generating Riemann metric from Ricci curvatures. Perelman successfully used this tool to resolve the Poincaré conjecture [23,24]. To find a Kähler metric with constant curvature, Calabi [5,6] studied a variational problem for minimizing the Calabi energy.

In discrete geometry, geometric flow can also be employed to investigate circle pattern metrics on polyhedral surfaces. Let S be a surface and μ denote a metric on it. A circle packing (circle pattern) consists of a collection of oriented disks D on ( S , μ ) , and there is a contact graph G D = { V D , E D , F D } attached to it. Initially, each vertex is associated with a disk, and an edge exists between two vertices if and only if the disks associated with them intersect at their boundaries. In application, Thurston [26] utilized circle patterns in the construction of hyperbolic 3-manifolds. It is natural to parameterize the analog of Teichmüller space of all circle patterns metrics on ( S , G , θ ) with radius. Regarding discrete Gaussian curvatures, the rigidity of certain types of circle patterns under Euclidean and hyperbolic background geometries is encapsulated in the Koebe-Andreev-Thurston theorem [1,2,17,26]. Then, many people generalized the theorem [3,4,19,25].

To identify circle packing metric with prescribed discrete Gaussian curvatures, Chow and Luo [8] introduced the combinatorial Ricci flow as follows:

(1) d r d t = − K ⋅ s ( r ) ,

where

r = ( r v 1 , r v 2 , … , r v n ) T = ( r 1 , r 2 , … , r n ) T , K = diag ( K v 1 , K v 2 , … , K v n ) = diag ( K 1 , K 2 , … , K n ) , s ( r ) = ( r 1 , r 2 , … , r n ) T , in Euclidean geometry , ( sinh r 1 , sinh r 2 , … , sinh r n ) T , in hyperbolic geometry ( sin r 1 , sin r 2 , … , sin r n ) T , in spherical geometry ,

where { r v i } v i ∈ V = { r i } i = 1 n are the radius of the circles in D and { K v i } v i ∈ V = { K i } i = 1 n are the discrete Gaussian curvatures at the vertices on surface S . The discrete Gaussian curvature is defined as K i = 2 π − α i , where α i is the cone angle of the singularity at the vertex v i .

The combinatorial Ricci flow evolves as the gradient flow of discrete entropy energy [14]. The solution to the normalized combinatorial Ricci flow converges to Thurston’s circle packing exponentially fast [8].

Subsequently, the combinatorial Calabi flow, as an analog of smooth Calabi flow, exhibits a solution that exists for all time. Ge demonstrated that it converges exponentially fast to a constant curvature metric in both Euclidean and hyperbolic background geometries as well [11]. Lin and Zhang [20,21] introduced a type of higher-order curvature flow known as combinatorial pth Calabi flow to generalize the combinatorial Calabi flow. In the context of Euclidean background geometry, here is how it is defined:

(2) d r d t = Δ p K ⋅ r ,

where Δ p K = diag ( Δ p K 1 , … , Δ p K n ) and

Δ p y i = ∑ j ∼ i B i j ∣ y j − y i ∣ p − 2 ( y j − y i ) , ∀ y ∈ R n ,

with

B i j = ∂ ( θ i j k + θ i j l ) ∂ r j r j ,

where j ∼ i denotes that the vertex v j is adjacent to v i . θ i j k is the inner angle at the vertex v i in a triangle Δ v i v j v k ∈ F .

In hyperbolic background geometry, the combinatorial pth Calabi flow is defined as

(3) d r d t = ( Δ p K − A ⋅ K ) sinh r ,

where A = diag ( A 1 , … , A n ) and

A i = sinh r i ∂ ∂ r i ∑ { i j k } ∈ F Area ( Δ v i v j v k ) , 1 ≤ i ≤ n .

Those combinatorial curvature flows mentioned earlier are investigated on polyhedral surface characterized by discrete Gaussian curvatures. To address the rigidity of circle pattern in spherical background geometry, Nie [22] parameterized the analog of Teichmüller space of all the ideal circle pattern metrics on ( S , G , θ ) using total geodesic curvatures. Each vertex is linked to an intersection point of disk boundaries, and each edge is corresponding to two intersection points with adjacent disks. Consequently, each face is associated with a conical disk D f and C f denotes the boundary circle. The weight function θ : E → 0 , π 2 ∣ E ∣ of G represents the intersection angle between adjacent disks. Singularities may occur at the vertices and centers of these disks.

Then, combinatorial Ricci flow and combinatorial Calabi flow with total geodesic curvatures are introduced for circle patterns in hyperbolic and spherical background geometry [12,13,18]. Hu [16] redefined the combinatorial pth Calabi flow for total geodesic curvatures in hyperbolic background geometry.

Inspired by Nie and Hu’s work, we construct combinatorial pth Calabi flow based on total geodesic curvatures in spherical background geometry, which is depicted in Definition 2.4:

(4) d k f d t = k f ( Δ p − A f ) ( T f − T ˆ f ) , ∀ f ∈ F ,

where Δ p denotes the discrete Laplace operator and A f is a function on geodesic curvature k f . T f is the total geodesic curvature of C f , which varies with k f . T ˆ f ∈ T that means the prescribed total geodesic curvature. Here,

(5) T = { ( T 1 , T 2 , … , T ∣ F ∣ ) ∈ R + ∣ F ∣ ∣ ∑ f ∈ F ′ T f < ∑ e ∈ E ( F ′ ) 2 θ e , ∀ F ′ ⊂ F and F ′ ≠ ∅ } ,

where E ( F ′ ) = { e ∈ E ∣ e is an edge incident with some face f ∈ F ′ } .

We establish the existence of the combinatorial pth Calabi flow indefinitely, demonstrating its convergence contingent upon the fulfillment of a specific combinatorial criterion by the prescribed total geodesic curvatures.

Theorem 1.1

Let { T f } f ∈ F = ( T 1 , … , T ∣ F ∣ ) T ∈ R + ∣ F ∣ denote a vector defined on V F . For the combinatorial pth Calabi flow (4), the equivalence between the following statements holds.

{ T f } f ∈ F ∈ T , where T f is the total geodesic curvature defined on the circle corresponding to the face f;
The solution of the combinatorial pth Calabi flow exists for all time t ∈ [ 0 , + ∞ ) and converges to an ideal circle pattern metric with prescribed total geodesic curvatures { T f } f ∈ F .

Assuming θ e ∈ 0 , π 2 , and the range of radii { r f } f ∈ F is r f ∈ ( 0 , π 2 ) , we establish that the geodesic curvature k f lies in ( 0 , + ∞ ) . Consequently, the bigons within this framework are mutually exclusive. Theorem 1.1 outlines a methodology for constructing circle pattern metrics on surfaces with prescribed total geodesic curvatures, thereby substantiating the principal assertion in [22].

Organization of this article. This article is structured as follows. Section 2 provides an introduction to the foundational concepts of ideal circle patterns in spherical background geometry and the combinatorial pth Calabi flow. Section 3 outlines our primary approach, focusing on the variational principle used in our proof. Section 4 is dedicated to proving the main theorem.

2 Preliminaries

2.1 Spherical ideal circle pattern

Informally, a circle pattern refers to a finite or potentially infinite arrangement of open circular disks, situated within a spherical, Euclidean, or hyperbolic geometric context on a compact surface S . This study focuses specifically on a variant of circle patterns featuring spherical conical disks.

The spherical conical disk D α ( r ) (refer to Figure 1) is derived by locally joining both sides of a sector modeled on the spherical space S 2 , where the radius r varies within ( 0 , π 2 ) , and the apex angle α ranges over ( 0 , + ∞ ) .

$Figure 1 Spherical conical disk D α ( r ) {D}_{\alpha }\left(r) .$

Figure 1

Spherical conical disk D α ( r ) .

Definition 2.1

(Spherical ideal circle pattern) Consider a closed surface S and a finite set of points { x 1 , … , x n } . Let D = { D 1 , D 2 , … , D m } represent a finite collection of spherical conical disks. A locally homomorphism p : ∪ i = 1 m D i → S constitutes a spherical circle pattern under the following conditions.

p ( ∪ i = 1 m D i ) = S \ { x 1 , … , x n } and the homomorphism p is at most two to one;
The images of every spherical conical disk p ( D i ) do not contain each other.

In this context, we consider the intersections of the spherical circles as the vertices. If the two vertices coincide with the same bigon, an edge is subsequently drawn between them. There is a simple weighted graph G D corresponding to the spherical circle pattern D (Figure 2).

Figure 2

Local picture of a circle pattern with its weighted graph.

G D ≔ { V D , E D } ,

where V D denotes the set of vertices and E D is the set of edges. Each e ∈ E D is weighted by the interior angle θ e of the corresponding bigon Ω e . The graph is simple, means that it excludes both parallel edges and self-loops.

Based on the weighted graph, there exists a closed 2-cell embedding η : G D → S such that

η ( V D ) = { x 1 , … , x n } ;
η ( E D ) = { e ∣ e is the geodesic edge connecting two vertices of some bigon } .

Moreover, η ( G D ) determines a polygonal cellular decomposition on the surface S . A face P is a connected component of S \ η ( V D ∪ E D ) . To simplify, we denote the weighted graph G D by { V D , E D , F D } , where F D = { η − 1 ( P ) ∣ P is a face of the polygonal cellular decomposition η ( G D ) on S } .

The spherical circle pattern induces a unique spherical circle pattern metric σ on the surface S via the pull-back mapping p . Given the edge lengths of each polygon in Σ D , the metric σ is uniquely determined.

Definition 2.2

(Polyhedral surfaces) Consider a compact surface S with its embedding weighted graph G ≔ { V , E , F } . Here, V , E , and F denote the sets of vertices, edges, and faces, in that order. The function d : E → R + defines a metric on S ensuring that each face in G is isometric to a Euclidean (or hyperbolic, or spherical) convex polygon. Therefore, the triple ( S , G , d ) denotes a polyhedral surface that is Euclidean, hyperbolic, or spherical in nature.

The circle pattern D on S defines a circle pattern metric ( σ , D ) on S . Let v f denote the center of the spherical conical disk corresponding to face f ∈ F , with V F = { v f ∣ f ∈ F D } representing the set of these centers. The cone angle α f of Σ at singularity v f ranges from ( 0 , + ∞ ) , while the cone angle α v at vertex v ∈ V D is determined by the weight θ e according to

(6) α v = ∑ e ∈ E ( v ) ( π − θ e ) ,

where E ( v ) ≔ { e ∈ E ∣ one of the boundary point of edge e is vertex v } .

We establish auxiliary geodesic lines from the center v f ∈ V F (for f ∈ F D ) and vertex v ∈ V D , where v is the vertex of the polygon f . Thus, we could view the surface S equipped with ( σ , D ) as composed of finite spherical quadrilaterals, as depicted in Figure 3. The singularities of the circle pattern metric ( σ , D ) comprise V D ∪ V F .

Figure 3

Spherical quadrilateral.

From the perspective of three-dimensional hyperbolic geometry, a spherical ideal circle pattern is equivalent to a convex hyperbolic ideal polyhedron. Since the Poincaré ball B 3 represents the hyperbolic 3-space prominently, the boundary of hyperbolic 3-space corresponds to the 2-sphere S 2 . Let us consider a convex hyperbolic ideal polyhedron where its vertices lie on the boundary and its faces manifest as geodesic hyperbolic surfaces (Figure 4). Extending each hyperbolic face so that it intersects vertically with B 3 , we obtain a spherical circle on S 2 . Therefore, the arrangement of ideal circles on the spherical background geometry corresponds to a convex hyperbolic polyhedron in B 3 , where the dihedral angles match the intersection angles of the ideal circle pattern.

Figure 4

Hyperbolic ideal polyhedron and its corresponding spherical ideal circle pattern.

2.2 Parameterization of the analog of Teichmüller space

Geodesic curvature and circumference of the boundary C f = ∂ D f , denoted by k f and l f respectively, are expressed as

(7) k f = cot r f , l f = α f sin r f .

Define C f , e as the arc of C f facing edge e . We denote the arc length l f , e and central angle α f , e of C f truncated by edge e (refer to Figure 5). Integrating the geodesic curvature along C f , e yields the total geodesic curvature T f , e . Given our understanding of spherical geometry, it follows that

(8) l f , e = α f , e sin r f , T f , e = k f l f , e = α f , e cos r f .

Figure 5

Face with its corresponding conical circle.

For the ideal circle pattern, the circumference l f and the total geodesic curvature T f are given by

(9) l f = ∑ e ∈ E ( f ) l f , e , T f = ∑ e ∈ E ( f ) T f , e .

Given a closed surface S with a graph G = { V , E , F } embedded on it, where θ : E → 0 , π 2 ∣ E ∣ represents the edge weights of G , Nie established the analog of the Teichmüller space encompassing all spherical ideal circle pattern metrics in his work [22].

(10) T ( S , G , θ ) ≔ ( σ , D ) D is a circle pattern metric on S such that G D is isotopy to G and there weights match, σ is the spherical cone metric induced by D ∕ Homeo 0 ( S ) .

In other words, two circle pattern metrics ( σ 1 , D 1 ) and ( σ 2 , D 2 ) on the same surface S and weighted graph G are equivalent if and only if there exists an isometric mapping h : ( σ 1 , D 1 ) → ( σ 2 , D 2 ) that is homotopic to the identity. As per the proposition that follows, the system of geodesic curvatures { ln k f } f ∈ F ∈ ( 0 , + ∞ ) ∣ F ∣ uniquely determines ( σ , D ) as a point of T ( S , G , θ ) , suggesting a parameterization T ( S , G , θ ) ≅ ( 0 , + ∞ ) ∣ F ∣ .

Proposition 2.3

The map T ( S , G , θ ) → R + ∣ F ∣ given by ( σ , D ) ↦ { ln k f } f ∈ F is a homeomorphism.

Proof

In the context of spherical ideal circle patterns, there is a homeomorphism between radius space { r f } f ∈ F and geodesic curvatures space { ln k f } f ∈ F for k f = ln ( cot r f ) .

The map is well defined for every ( σ , D ) since there is a nature radius vector. To demonstrate bijectivity, we confirm that for every radius vector, there exists a unique circle pattern metric ( σ , D ) ∈ T ( S , G , θ ) .

By connecting the centers of each polyhedral face with its vertices, a triangulation on S is obtained. The cotangent 4-part formula in classical spherical geometry, deduced from the cosine law of Δ v v f v g (refer to Figure 6), is given by

(11) cot α f , e 2 = 1 sin θ e ( cot r g sin r f + cos r f cos θ e ) .

Thus, the cone angles are entirely determined by the radius based on ( S , G , θ ) . Each triangle is isomorphic to a spherical triangle, and from the concept of polyhedral surfaces, there corresponds a spherical ideal circle pattern metric ( σ , D ) . Therefore, the mapping is surjective and continuous.

Given a topological surface S and its weighted graph G , the circle pattern metrics determined by the same geodesic curvature vector s = ( ln k 1 , ln k 2 , … , ln k ∣ F ∣ ) are isomorphic, and any isometry between them is homotopic to the identity. Thus, this constitutes a homeomorphism.□

Figure 6

Two intersecting conical circles.

For hyperbolic ideal circle pattern, the analog of Teichmüller space could be parameterized by system of discrete Gaussian curvatures (cone angles) while fail in spherical and Euclidean background geometry. In their work [3], Bobenko and Springborn established the existence of a unique circle pattern up to Möbius transformation based on ( S , G , θ ) contingent upon cone angles satisfying a condition akin to (5) across Euclidean, hyperbolic, and spherical backgrounds. Since the Möbius transformation is an isometry in the hyperbolic background geometry, the system of cone angles determines the ideal circle pattern metric up to isotopy.

The analog of Teichmüller space can also be parameterized by system of total geodesic curvatures through a variable substitution for the 1-form in the original variational method as is shown in equation (18).

2.3 Combinatorial pth Calabi flow

The combinatorial pth Calabi flow constitutes a class of combinatorial geometric flows. It primarily investigates the temporal evolution of the circle packing metric on the topological surface S , associated with the same underlying graph G . In this study, we introduce the combinatorial pth Calabi flow with respect to total geodesic curvatures, aiming to elucidate the dynamic changes in the ideal circle pattern metric within the moduli space T ( S , G , θ ) over time.

Definition 2.4

(Combinatorial pth Calabi flow)

(12) d k f d t = k f ( Δ p − A f ) ( T f − T ˆ f ) , ∀ f ∈ F ,

where y : V F → R is a function on V F . For p > 1 , we construct the discrete pth Laplace operator Δ p on y , i.e.,

(13) Δ p y f = ∑ g ∼ f B f g ∣ y g − y f ∣ p − 2 ( y g − y f ) , ∀ f ∈ F ,

where g ∼ f denotes that the face g is adjacent to the face f . T ˆ = ( T ˆ 1 , … , T ˆ ∣ F ∣ ) T is a given vector in R + ∣ F ∣ which represents the prescribed total geodesic curvatures:

(14) B f g = − ∂ ∑ e ∈ E ( f ) T f , e ∂ k g k g = − ∂ T f ∂ k g k g ,

(15) A f = − k f ∂ ∂ k f ∑ e ∈ E ( f ) Area ( Ω e ) .

Remark 2.5

The combinatorial pth Calabi flow is the generalization of combinatorial Calabi flow, and it comes to the combinatorial Calabi flow when p = 2 .

3 Variational principle

de Verdière [9,10] introduced the variational principle to address the circle pattern problem. Nie [22] extended this principle to focus on the total geodesic curvatures of ideal circle pattern within a spherical background geometry. This approach transforms the task of discovering circle patterns with constant total geodesic curvatures into extremum problem for a specific functional. A critical component involves minimizing a suitable functional to achieve this transformation.

Lemma 3.1

[22] Let C f and C g be two circles intersecting each other at angle θ e ∈ 0 , π 2 . Let Ω e be the bigon corresponding to the edge e, which is formed by C f , e and C g , e . Then, we have

(16) Area ( Ω e ) = 2 θ e − T f , e − T g , e ,

where T f , e and T g , e denote the total geodesic curvatures of C f , e and C g , e .

Proof

This lemma can be directly deduced by the Gauss-Bonnet theorem.□

Lemma 3.2

[12] The symbols Ω e and C f , e , C g , e , k f , k g , T f , e , and T g , e are defined as above. Let s f = ln k f and s g = ln k g . Then, we have

(17) ∂ T f , e ∂ s g = ∂ T g , e ∂ s f < 0 , ∂ T f , e ∂ s f , ∂ T g , e ∂ s g > 0 , ∂ Area ( Ω e ) ∂ s f , ∂ Area ( Ω e ) ∂ s g < 0 .

Let ℳ θ be the space of all the spherical bigons ( s f , s g ) whose angle is θ . By ω Ω e , we denote the 1-form defined on the space ℳ θ :

ω Ω e ≔ − α f , e 1 sin r f d r f − α g , e 1 sin r g d r g .

Through a variable substitution, it is equivalent to the form presented as follows:

(18) ω Ω e = l f , e d k f + l g , e d k g = T f , e d s f + T g , e d s g .

Next, we can demonstrate the closeness of the 1-form ω Ω e subsequently:

d ( ω Ω e ) = d ( T f , e d s f + T g , e d s g ) = ∂ T g , e ∂ s f − ∂ T f , e ∂ s g d s f ∧ d s g .

By Lemma 3.2, ∂ T g , e ∂ s f − ∂ T f , e ∂ s g = 0 . Hence, the 1-form ω Ω e is closed. For the sum of that 1-form of all the bigon,

ω ≔ ∑ e ∈ E ω Ω e = ∑ f ∈ F ∑ e ∈ E ( f ) T f , e d s f = ∑ f ∈ F T f d s f

on ( S , G ) is also closed.

For F = { f i } i = 1 ∣ F ∣ , by

s = ( s f 1 , … , s f ∣ F ∣ ) T = ( s 1 , … , s ∣ F ∣ ) T , T = ( T f 1 , … , T f ∣ F ∣ ) T = ( T 1 , … , T ∣ F ∣ ) T .

Let us denote the vector containing the logarithms of geodesic curvatures and total geodesic curvatures across all faces, and let s 0 represent some initial value. Based on this, we define a potential function in the following manner:

(19) Φ ( s ) = ∫ s 0 s ω − ∫ s 0 s ∑ i = 1 ∣ F ∣ T ˆ i d s i = ∫ s 0 s ∑ i = 1 ∣ F ∣ ( T i − T ˆ i ) d s i .

It is easy to see that ∇ Φ = ( T 1 , … , T ∣ F ∣ ) T . The Hess matrix of Φ ( s ) equals to a Jacobi matrix:

(20) Hess Φ = M = ∂ T 1 ∂ s 1 ⋯ ∂ T 1 ∂ s ∣ F ∣ ⋮ ⋱ ⋮ ∂ T ∣ F ∣ ∂ s 1 ⋯ ∂ T ∣ F ∣ ∂ s ∣ F ∣ .

Lemma 3.2 indicates that the Hessian matrix is symmetric and diagonally dominant. For each column, by subtracting the rest of the elements from which on the main diagonal, we obtain that

∂ T i ∂ s i − ∑ j ≠ i ∂ T j ∂ s i = ∂ T i ∂ s i + ∑ j ≠ i ∂ T j ∂ s i = ∑ f j ∼ f i ∂ T f i , e i j ∂ s i + ∂ T f j , e i j ∂ s i > 0 , 1 ≤ i ≤ ∣ F ∣ ,

where e i j denotes the common edge of the adjacent faces f j and f i .

Consequently, if M is positive definite, then Φ ( s ) becomes strictly convex. Furthermore, a unique critical point s ˆ exists for the potential function Φ ( s ) .

Lemma 3.3

The gradient map of Φ

(21) ∇ Φ : R + ∣ F ∣ → T s ↦ T − T ˆ

is a homeomorphism. Here, T is the set of parameters with total geodesic curvatures, which is defined as equation (5).

Proof

It is established that for a C 2 -smooth function ρ : R n → R , strict convexity with a positively definite Hessian matrix ensures that its gradient ∇ ρ : R n → R n acts as a smooth embedding. Consequently, the gradient map of Φ smoothly embeds R + ∣ F ∣ into T ⊂ R ∣ F ∣ .

Furthermore, we illustrate that the range of the gradient map precisely corresponds to T . Since we are dealing with a polygonal region, our focus narrows down to showcasing limit behaviors.

By equation (11),

α f , e = 2 arccot 1 sin θ e ( cot r g sin r f + cos r f cos θ e ) = 2 arccot 1 sin θ e k g 1 1 + k f 2 + cos θ e k f 1 + k f 2 < π .

Therefore, T f , e = α f , e cos r f = α f , e k f 1 + k f 2 → 0 as k f → 0 . It is indisputable that the lower bound of T f is zero. Moreover, for any subset F ′ ⊂ F , ∑ f ∈ F ′ T f → 0 as T f → 0 , ∀ f ∈ F ′ .

Our attention now pivots toward establishing the upper bound. By the Gauss-Bonnet formula,

Area ( Ω e ) = 2 θ e − ( T f , e + T g , e ) > 0 .

Thus, T f , e + T g , e < 2 θ e and T f , e + T g , e → 2 θ e as Area ( Ω e ) → 0 . For any subset F ′ ⊂ F , we observe that ∑ f ∈ F ′ T f < ∑ e ∈ E ( F ′ ) 2 θ e . Furthermore, by the spherical four-part formula, we know that

α f , e = 2 arccot 1 sin θ e k g 1 1 + k f 2 + cos θ e k f 1 + k f 2 → 2 θ e , k f → + ∞ ,

where k g is fixed in ( 0 , + ∞ ) .

Consequently, T f , e = α f , e k f 1 + k f 2 → 2 θ e as k f → + ∞ with k g fixed. Therefore, ∑ f ∈ F ′ T f → ∑ e ∈ E ( F ′ ) 2 θ e as k f → + ∞ , ∀ f ∈ F ′ .

This confirms the surjective nature of the mapping. Ultimately, the gradient map of Φ is established as a homeomorphism from R + ∣ F ∣ to the set of parameters with total geodesic curvatures T .□

4 Proof of Theorem 1.1

Assuming the solution to the combinatorial pth Calabi flow converges to a spherical circle packing, according to Lemma 3.3, the total geodesic curvature { T f } f ∈ F ∈ T . Therefore, the 2-to-1 aspect of Theorem 1.1 is self-evident. Our focus then shifts to proving long-term existence and convergence under prescribed total geodesic curvature.

4.1 Long-time existence

We use the change of variables s f = ln k f , ∀ f ∈ F , then the combinatorial pth Calabi flow has the following equivalent form, i.e.,

(22) d s f d t = ( Δ p − A f ) ( T f − T ˆ f ) , ∀ f ∈ F .

Since all ( Δ p − A f ) ( T f − T ˆ f ) are continuous functions on R ∣ F ∣ , the combinatorial pth Calabi flow exists in some interval [ 0 , ε ] for any initial value by Peano’s existence theorem in classical ordinary differential equation theory. Consequently, the existence of the solution is established for all time.

Proposition 4.1

For any initial value s ( 0 ) ∈ R ∣ F ∣ , the solution of combinatorial pth Calabi flow exists for all time t ∈ [ 0 , + ∞ ) .

Proof

∣ B f g ∣ = ∑ e ∈ E ( f ) ∂ T f , e ∂ k g k g ≤ ∑ e ∈ E ( f ) ∂ T f , e s g = ∑ e ∈ E ( f ) 2 cos r f cos r g sin α f , e 2 sin α g , e 2 sin θ e ≤ ∑ e ∈ E ( f ) 1 sin θ e .

Now, we consider the function A f

A f = − k f ∑ e ∈ E ( f ) ∂ Area ( Ω e ) ∂ k f = k f ∑ g ∼ f ∂ ( T f + T g ) ∂ k f = ∑ g ∼ f sin r f 2 cos r f ( α f , e − sin α f , e ) ≤ ∑ e ∈ E ( f ) α f , e ≤ n π ,

where n is the number of edges of polygon corresponding to f .

Fixed a face f ∈ F for the polygonal cellular decomposition of surface S , by Lemma 3.2, we have 0 < K f ≤ n π .

We now establish the existence of a constant μ , which relies solely on the polygonal cellular decomposition, such that

∣ B f g ∣ , ∣ A f ∣ ≤ μ , ∀ f ∈ F , ∀ g ∼ f .

By the definition of Δ p , it follows that

∣ ( Δ p − A f ) ( T f − T ˆ f ) ∣ ≤ μ ∣ F ∣ ( π ∣ E ∣ + ‖ T ˆ ‖ ∞ ) p − 1 + μ ( π ∣ E ∣ + ‖ T ˆ ‖ ∞ ) , ∀ f ∈ F .

Hence, all ∣ ( Δ p − A f ) ( T f − T ˆ f ) ∣ are uniformly bounded by a constant. Then, by the malleability theorem, the solution exists for all time t ∈ [ 0 , + ∞ ] .□

4.2 Convergence

Let T ˆ = ( T ˆ 1 , … , T ˆ ∣ F ∣ ) T denote the prescribed total geodesic curvatures. Assuming s ( t ) = ( s 1 ( t ) , … , s ∣ F ∣ ( t ) ) T represents a solution to the combinatorial pth Calabi flow with d s d t = ( s 1 ′ ( t ) , … , s ∣ F ∣ ′ ( t ) ) T . We define T ( s ( t ) ) = ( T 1 ( s ( t ) ) , … , T ∣ F ∣ ( s ( t ) ) ) T . Consequently, we derive that

(23) Φ ( s ( t ) ) = ∫ 0 t ( T ( s ( t ) ) − T ˆ ) T d s d t d t .

Proposition 4.2

The function Φ ( s ( t ) ) is convergent under the combinatorial pth Calabi flow.

Proof

Let g : V F → R be a function on V F . For M f g = M g f , if we exchange f and g , it is easy to know

∑ f ∈ F y f Δ p y f = − 1 2 ∑ f ∈ F ∑ g ∼ f M f g ∣ y g − y f ∣ p ≤ 0 .

As a consequence, for the total geodesic curvature

T ( s ( t ) ) = ( T 1 ( s ( t ) ) , … , T ∣ F ∣ ( s ( t ) ) ) T

and

T ˆ = ( T ˆ 1 , … , T ˆ ∣ F ∣ ) T ∈ T ,

we have

( T ( s ( t ) ) − T ˆ ) T Δ p ( T ( s ( t ) ) − T ˆ ) ≤ 0 .

Moreover, for the matrix A ( s ( t ) ) = diag ( A 1 ( s ( t ) ) , … , A ∣ F ∣ ( s ( t ) ) ) , we have

( T ( s ( t ) ) − T ˆ ) T A ( s ( t ) ) ( T ( s ( t ) ) − T ˆ ) = ∑ i = 1 ∣ F ∣ A i ( s ( t ) ) ( T i ( s ( t ) ) − T ˆ i ) 2 ≥ 0 .

For Φ ( s ( t ) ) , we have

Φ ′ ( s ( t ) ) = ∇ Φ ( s ( t ) ) T ⋅ d s d t = ( T ( s ( t ) ) − T ˆ ) T ⋅ d s d t = ( T ( s ( t ) ) − T ˆ ) T Δ p ( T ( s ( t ) ) − T ˆ ) − ( T ( s ( t ) ) − T ˆ ) T A ( s ( t ) ) ( T ( s ( t ) ) − T ˆ ) ≤ 0 .

Therefore, Φ ( s ( t ) ) is decreasing on [ 0 , + ∞ ) , and since Φ ( s ( t ) ) ≥ 0 , it follows that Φ ( s ( t ) ) converges, i.e., lim t → + ∞ Φ ( s ( t ) ) exists.□

Proposition 4.3

Suppose { s ( t ) ∣ t ∈ [ 0 , + ∞ ) } is a long time solution to the combinatorial pth Calabi flow and is compactly supported in R ∣ F ∣ , then Φ ( s ( t ) ) converges to 0 , i.e.,

(24) lim t → + ∞ Φ ( s ( t ) ) = 0 .

Proof

Using the mean value theorem for the function Φ ( s ( t ) ) , there are ξ n ∈ [ n , n + 1 ] such that

Φ ′ ( s ( ξ n ) ) = Φ ( s ( n + 1 ) ) − Φ ( s ( n ) ) .

By the assumption, the solution to the combinatorial pth Calabi flow { s ( t ) } t ≥ 0 is compactly supported in R + ∣ F ∣ . We infer that { s ( ξ n ) ∣ ξ n ∈ [ n , n + 1 ] , n ∈ N } possesses a convergent subsequence. To simplify, denote this convergent subsequence by { s ( ξ n ) } n ≥ 0 . Hence, we ascertain the existence of a point s * ∈ R + ∣ F ∣ such that

lim n → + ∞ s ( ξ n ) = s * .

According to Proposition 4.2, lim t → + ∞ Φ ( s ( t ) ) exists.

Thus,

lim n → + ∞ Φ ′ ( s ( ξ n ) ) = 0 . ( T ( s * ) − T ˆ ) T Δ p ( T ( s * ) − T ˆ ) = ( T ( s * ) − T ˆ ) T A ( s * ) ( T ( s * ) − T ˆ ) = 0 ,

which implies that T * − T ˆ = 0 .

Hence, we have

∇ Φ ( s * ) = T ( s * ) − T ˆ = 0 ,

i.e., s * is a critical point of the function Φ ( s ) .

Since we have that s ˆ is the unique critical point of the potential function Φ ( s ) , which implies that s * = s ˆ ,

Φ ( s ˆ ) = lim n → + ∞ Φ ( s ( ξ n ) ) = 0 ,

which implies that

□ lim t → + ∞ Φ ( s ( t ) ) = 0 .

We are now in a position to complete the proof of Theorem 1.1. The convergence is established through a proof by contradiction.

Suppose the solution s ( t ) of the combinatorial pth Calabi flow does not converge. Then, for any T > 0 , there exists δ > 0 and t 0 > T such that the difference ∣ s ( t 0 ) − s ˆ ∣ exceeds δ .

Since Φ ( s ( t ) ) is a convex function, there exists λ > 0 such that

∣ Φ ( s ( t 0 ) ) − Φ ( s ˆ ) ∣ = ∣ Φ ( s ( t 0 ) ) ∣ > λ .

Upon encountering conditions that contradict Proposition 4.3, the function s ( t ) subsequently converges toward s ˆ , i.e.,

(25) lim t → + ∞ s ( t ) = s ˆ .

For any initial value, the solution of the combinatorial pth Calabi flow exists for all time as stated in Proposition 4.1. As time progresses indefinitely, the solution converges. Hence, the proof of Theorem 1.1 is complete.

By Lemma 3.3, the gradient map of Φ is a homeomorphism from R ∣ F ∣ to T .

Thus,

(26) lim t → ∞ T ( s ( t ) ) = T ˆ .

That is to say, along the combinatorial pth Calabi flow, the spherical circle pattern metric is convergent to a unique metric with prescribed total geodesic curvatures.

Acknowledgment

The authors are very grateful to the referee for a detailed reading of this manuscript and recommendations.

Funding information: This work was supported by NSFC (Grant Number 12271017).
Author contributions: All authors contributed equally to the writing of this article. All authors read and approved the final manuscript.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

[1] E. M. Andreev, On Convex polyhedra in Lobačevskǐ spaces, Math. USSR-Sb. 10 (1970), no. 3, 413–440, DOI: https://doi.org/10.1070/SM1970v010n03ABEH001677.10.1070/SM1970v010n03ABEH001677Search in Google Scholar

[2] E. M. Andreev, On Convex polyhedra of finite volume in Lobačevskǐ space, Math. USSR-Sb. 12 (1970), no. 2, 255–259, DOI: https://doi.org/10.1070/SM1970v012n02ABEH000920.10.1070/SM1970v012n02ABEH000920Search in Google Scholar

[3] A. I. Bobenko and B. A. Springborn, Variational principles for circle patterns and Koebe’s theorem, Trans. Amer. Math. Soc. 356 (2004), no. 2, 659–689, https://doi.org/10.48550/arXiv.math/0203250. Search in Google Scholar

[4] P. L. Bowers and K. Stephenson, Uniformizing dessins and Belyĭ maps via circle packing, Mem. Amer. Math. Soc. 170 (2004), no. 805, xii+97 pp. DOI: https://doi.org/10.1090/memo/0805.10.1090/memo/0805Search in Google Scholar

[5] E. Calabi, Extremal Kähler metrics, Annals of Mathematics Studies, vol 102, Princeton University Press, Princeton, NJ, 1982, pp. 259–290. DOI: https://doi.org/10.1515/9781400881918-016.10.1515/9781400881918-016Search in Google Scholar

[6] E. Calabi, Extremal Kähler metrics. II, Differential geometry and complex analysis, Springer-Verlag, Berlin, 1985, pp. 95–114. DOI: https://doi.org/10.1007/978-3-642-69828-6_8.10.1007/978-3-642-69828-6_8Search in Google Scholar

[7] S. Y. A. Chang and P. C. Yang, Prescribing Gaussian curvature on S2, Acta Math. 159 (1987), no. 3–4, 215–259, 10.1007/BF02392560. Search in Google Scholar

[8] B. Chow and F. Luo, Combinatorial Ricci flows on surfaces, J. Differential Geom. 63 (2003), no. 1, 97–129, https://doi.org/10.4310/jdg/1080835659. Search in Google Scholar

[9] Y. C. de Verdière, Un principe variationnel pour les empilements de cercles, Invent. Math. 104 (1991), no. 3, 655–669, https://doi.org/10.1007/BF01245096. Search in Google Scholar

[10] J. Dai, D. Gu, and F. Luo, Variational principles for discrete surfaces, Advanced Lectures in Mathematics, vol. 4, International Press, Somerville, MA; Higher Education Press, Beijing, 2008, iv+146 pp. ISBN: 978-1-57146-172-8. Search in Google Scholar

[11] H. Ge, Combinatorial Calabi flows on surfaces, Trans. Amer. Math. Soc. 370 (2018), no. 2, 1377–1391, DOI: https://doi.org/10.1090/tran/7196.10.1090/tran/7196Search in Google Scholar

[12] H. Ge, B. Hua, and P. Zhou, A combinatorial curvature flow in spherical background geometry, J. Funct. Anal. 286 (2024), no. 7, Paper No. 110335, 12 pp. https://doi.org/10.1016/j.jfa.2024.110335. Search in Google Scholar

[13] H. Ge, B. Hua, and Z. Zhou, Combinatorial Ricci flows for ideal circle patterns, Adv. Math. 383 (2021), Paper No. 107698, 26 pp. https://doi.org/10.1016/j.aim.2021.107698. Search in Google Scholar

[14] D. X. Gu, F. Luo, and S. T. Yau, Fundamentals of computational conformal geometry, Math. Comput. Sci. 4 (2010), no. 4, 389–429. https://doi.org/10.1007/s11786-011-0065-6. Search in Google Scholar

[15] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. 10.4310/jdg/1214436922Search in Google Scholar

[16] G. Hu, Z. Lei, Y. Qi, and P. Zhou, Combinatorial p-th Calabi flows for total geodesic curvatures in hyperbolic background geometry, J. Geom. Anal. 35 (2025), 18, https://doi.org/10.1007/s12220-024-01846-9. Search in Google Scholar

[17] P. Koebe, Kontaktprobleme der konformen Abbildung, Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl. 88 (1936), 141–164. Search in Google Scholar

[18] Z. Lei and P. Zhou, Combinatorial Calabi flows for ideal circle patterns in spherical background geometry, J. Differential Eqns. 427 (2025), 676–688. DOI: https://doi.org/10.1016/j.jde.2025.02.002.10.1016/j.jde.2025.02.002Search in Google Scholar

[19] G. Leibon, Characterizing the Delaunay decompositions of compact hyperbolic surfaces, Geom. Topol. 6 (2002), 361–391, 10.2140/gt.2002.6.361. Search in Google Scholar

[20] A. Lin and X. Zhang, Combinatorial pth Calabi flows on surfaces, Adv. Math. 346 (2019), 1067–1090, DOI: https://doi.org/10.1016/j.aim.2019.02.011. 10.1016/j.aim.2019.02.011Search in Google Scholar

[21] A. Lin and X. Zhang, Combinatorial pth Ricci flows on surfaces, Nonlinear Anal. 211 (2021), Paper No. 112417, 12 pp. DOI: https://doi.org/10.1016/j.na.2021.112417.10.1016/j.na.2021.112417Search in Google Scholar

[22] X. Nie, On circle patterns and spherical conical metrics, Proc. Amer. Math. Soc. 152 (2024), no. 2, 843–853, DOI: https://doi.org/10.1090/proc/16557.10.1090/proc/16557Search in Google Scholar

[23] G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv: math/0307245, DOI: https://doi.org/10.48550/arXiv.math/0307245.Search in Google Scholar

[24] G. Perelman, Ricci flow with surgery on three-manifolds, arXiv: math/0303109DOI: https://doi.org/10.48550/arXiv.math/0303109.Search in Google Scholar

[25] I. Rivin, Euclidean structures on simplicial surfaces and hyperbolic volume, Ann. Math. (2) 139 (1994), no. 3, 553–580, https://doi.org/10.2307/2118572. Search in Google Scholar

[26] W. P. Thurston, The Geometry and Topology of 3-manifolds, Lecture Notes, Princeton University, 1978.Search in Google Scholar

Received: 2024-07-27

Revised: 2024-11-26

Accepted: 2025-01-22

Published Online: 2025-02-20

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/anona-2025-0067

Keywords for this article

spherical circle pattern; combinatorial pth Calabi flow

Creative Commons

BY 4.0