A combinatorial Yamabe problem on two and three dimensional manifolds
AA combinatorial Yamabe problem on two andthree dimensional manifolds
Huabin Ge, Xu XuSeptember 24, 2018
Abstract
In this paper, we define a new discrete curvature on two and three dimensionaltriangulated manifolds, which is a modification of the well-known discrete curvatureon these manifolds. The new definition is more natural and respects the scalingexactly the same way as Gauss curvature does. Moreover, the new discrete curvaturecan be used to approximate the Gauss curvature on surfaces. Then we study thecorresponding constant curvature problem, which is called the combinatorial Yamabeproblem, by the corresponding combinatorial versions of Ricci flow and Calabi flowfor surfaces and Yamabe flow for 3-dimensional manifolds. The basic tools are thediscrete maximal principle and variational principle.
Mathematics Subject Classification (2010).
Contents a r X i v : . [ m a t h . DG ] J a n Combinatorial Calabi flow on surfaces 315 Combinatorial α -curvature and combinatorial α -flows 346 3-dimensional combinatorial Yamabe problem 39 One of the central topics in differential geometry is the existence of canonical metrics on agiven smooth manifold, especially metrics with constant curvatures. One can ask similarquestions on triangulated manifolds. Different from the standard processes on smoothmanifolds, where all calculations are done in local coordinates, Regge [44] suggested anentirely different way, where the calculations are done in a geometric simplex. Sincethe geometric information in a single simplex is included in the lengths of edges, thegeometric approach to a triangulated manifold may be started with a piecewise flat metric,which assigns each edge a Euclidean distance such that each simplex can be realized asa Euclidean simplex. Piecewise flat metric brings singularities on codimension two sub-simplices, which are explained as simplicial curvature tensors in [1, 9, 38, 39]. Besidesdefining edge lengths directly, one can also define discrete metrics at vertices and derivethe edge lengths indirectly. In fact, Thurston [48] introduced the notion of circle packingmetric on a triangulated surface, while Cooper and Rivin introduced the notion of spherepacking metric on a triangulated 3-manifold. We shall study the circle (sphere) packingmetrics and the corresponding discrete curvatures in this paper, and we want to knowwhen there exist canonical discrete metrics and how to find them.Suppose M is a closed surface with a triangulation T = { V, E, F } , where V, E, F represent the sets of vertices, edges and faces respectively. Let Φ : E → [0 , π ] be a functionassigning each edge { ij } a weight Φ ij ∈ [0 , π ]. The triple ( M, T , Φ) will be referred to asa weighted triangulation of M in the following. All the vertices are ordered one by one,marked by v , · · · , v N , where N = V (cid:93) is the number of vertices. We use i ∼ j to denotethat the vertices i and j are adjacent if there is an edge { ij } ∈ E with i , j as end points.Throughout this paper, all functions f : V → R will be regarded as column vectors in R N and f i is the value of f at i . And we use C ( V ) to denote the set of functions defined2n V . Each map r : V → (0 , + ∞ ) is called a circle packing metric. Given ( M, T , Φ), weattach each edge { ij } a length l ij = (cid:113) r i + r j + 2 r i r j cos Φ ij . (1.1)Thurston proved [48] that the lengths { l ij , l jk , l ik } satisfy the triangle inequalities, whichensures that the face { ijk } could be realized as an Euclidean triangle (cid:52) v i v j v k with lengths { l ij , l jk , l ik } . In this sense, the triangulated surface ( M, T , Φ) could be taken as gluingmany Euclidean triangles coherently. However, this gluing procedure produces singulari-ties at the vertices, which are described by the discrete curvatures. Suppose θ jki is the innerangle of the triangle (cid:52) v i v j v k at the vertex i , the classical well-known discrete Gaussiancurvature at i is defined as K i = 2 π − (cid:88) (cid:52) v i v j v k ∈ F θ jki , (1.2)where the sum is taken over all the triangles with v i as one of its vertices. Discrete Gaussiancurvature K i satisfies the following discrete version of Gauss-Bonnet formula [14]: (cid:88) i ∈ V K i = 2 πχ ( M ) . (1.3)Denote K av = πχ ( M ) N , it is natural to ask the following combinatorial Yamabe problem: Question 1.
Given a weighted triangulated surface ( M, T , Φ), does there exist a circlepacking metric r that determines a constant K -curvature? How to find it if it exists?Different from the smooth case, the constant K -curvature metric, i.e. a metric r with K i = K av for all i ∈ V , does not always exist. Thurstion [48] found that, besidesthe topological structure, the combinatorial structure plays an essential role for theexistence of constant K -curvature metric. In fact, for any proper subset I ⊂ V , let F I bethe subcomplex whose vertices are in I and let Lk ( I ) be the set of pairs ( e, v ) of an edge e and a vertex v satisfying the following three conditioins: (1) The end points of e are notin I ; (2) v is in I ; (3) e and v form a triangle. Thurston [48] proved that Theorem 1.1. (Thurston)
Given a weighted triangulated surface ( M, T , Φ), the ex-istence of constant K -curvature metric is equivalent to the following combinatorial-topological conditions2 πχ ( M ) | I || V | > − (cid:88) ( e,v ) ∈ Lk ( I ) ( π − Φ( e )) + 2 πχ ( F I ) , ∀ I : φ (cid:36) I (cid:36) V. (1.4)Moreover, the constant K -curvature metric is unique, if exists, up to the scaling of r .3n fact, the following result is proved for the classical discrete curvature K [48, 37, 14]. Theorem 1.2. (Thurston-Marden&Rodin-Chow&Luo)
Given a weighted triangu-lated surface ( M, T , Φ), denote Y (cid:44) K GB ∩ ( ∩ φ (cid:54) = I (cid:36) V Y I ) , (1.5)where K GB (cid:44) (cid:110) x ∈ R N (cid:12)(cid:12)(cid:12) N (cid:88) i =1 x i = 2 πχ ( M ) (cid:111) (1.6)and for any nonempty proper subset I ⊂ V , Y I (cid:44) (cid:110) x ∈ R N (cid:12)(cid:12)(cid:12) (cid:88) i ∈ I x i > − (cid:88) ( e,v ) ∈ Lk ( I ) ( π − Φ( e )) + 2 πχ ( F I ) (cid:111) . (1.7)The space of all admissible K -curvatures (cid:8) K = K ( r ) (cid:12)(cid:12) r ∈ R N> (cid:9) is exactly Y .Chow and Luo [14] first established an intrinsic connection between Thurston’s circlepacking metric and the surface Ricci flow. They introduced a combinatorial Ricci flow dr i dt = − K i r i with the normalization dr i dt = ( K av − K i ) r i . (1.8)Then they reproved Theorem 1.1 by the curvature flow methods. In fact, they proved: Theorem 1.3. (Chow&Luo)
Given a weighted triangulated surface ( M, T , Φ), the so-lution to the normalized combinatorial Ricci flow (1.8) converges (exponentially fast to aconstant K -curvature metric) if and only if there exists a constant K -curvature metric.Inspired by [14], the first author introduced a combinatorial Calabi flow dr i dt = ∆ K i r i (1.9)in [20] and proved similar results: Theorem 1.4. (Ge)
Given a weighted triangulated surface ( M, T , Φ), the solution to thecombinatorial Calabi flow (1.9) converges (exponentially fast to a constant K -curvaturemetric) if and only if there exists a constant K -curvature metric.Theorem 1.1, Theorem 1.2, Theorem 1.3 and Theorem 1.4 answer Question 1 perfectly.The first two theorems give a necessary and sufficient combinatorial-topological conditionfor the existence of a constant K -curvature metric, while the last two theorems give twodifferent efficient ways to find a constant K -curvature metric, i.e. for any initial value r (0), the solutions to the combinatorial Ricci flow and Calabi flow converge automaticallyto a constant K -curvature metric exponentially fast.4 .2 Main results It seems that the classical discrete Gauss curvature K is a suitable analogue for smoothGauss curvature. However, there are some intrinsic disadvantages for the classical discreteGauss curvature. For one thing, the smooth Gauss curvature changes under the scaling ofRiemannian metrics while the discrete Gauss curvature dose not. For another, the discreteGauss curvature K can not be used directly to approximate the smooth Gauss curvaturewhen the triangulation is finer and finer. Motivated by the two disadvantages, we modifythe classical discrete Gauss curvature K i to R i = K i r i (1.10)at the vertex i , which is called as R -curvature in the following. In some sense, this newdefinition is more natural and respects the scaling exactly the same way as the Gauss cur-vature does. Furthermore, this new definition can be used directly to approximate smoothGauss curvature (see Section 2). It is natural to consider the following combinatorialYamabe problem for R -curvature. Question 2.
Given a weighted triangulated surface ( M, T , Φ), does there exist a circlepacking metric r that determines a constant R -curvature? How to find it if it exists?We introduce the corresponding combinatorial versions of the Ricci flow and the Calabiflow for surfaces and the Yamabe flow for 3-dimensional manifolds using the new definitionof discrete curvature. We give an answer to Question 2 by discrete curvature flow methodsand variational methods. We just state the main results in this subsection.We introduce a new combinatorial Ricci flow dg i dt = ( R av − R i ) g i , (1.11)where R av = 2 πχ ( M ) / (cid:107) r (cid:107) and g i = r i is a discrete version of Riemann metric ten-sor. This flow has the same form as combinatorial Yamabe flow in two dimension. Byintroducing a new discrete Laplace operator∆ f i = 1 r i (cid:88) j ∼ i ( − ∂K i ∂u j )( f j − f i ) , where f is a function defined on all vertices and u j = ln g j is a coordinate transformation,we find that the flow (1.11) exhibits similar properties to the smooth Ricci flow on surfaces.For example, the curvature R i evolves according to a heat-type equation dR i dt = ∆ R i + R i ( R i − R av ) , dg i dt = ∆ R i · g i , (1.12)which is similar to the smooth Calabi flow on surfaces. By considering a discrete quadraticenergy functional (cid:101) C ( r ) = (cid:80) Ni =1 ϕ i , where ϕ i = K i − R av r i , we can define a modified Calabiflow ˙ u = − ∇ u (cid:101) C. (1.13)Combining Corollary 3.19, Theorem 4.2, Theorem 4.6, Theorem 3.24 and Theorem3.26, we have the following main theorem. Theorem 1.5.
Given a weighted triangulated surface ( M, T , Φ) with χ ( M ) ≤
0. Thenthe constant R -curvature metric is unique (if exists) up to scaling, and the followingstatements are mutually equivalent:(1) There exists a constant R -curvature circle packing metric;(2) The solution to the combinatorial Ricci flow (1.11) converges;(3) The solution to the combinatorial Calabi flow (1.12) converges;(4) The solution to the modified Calabi flow (1.13) converges;(5) There exists a circle packing metric r ∗ such that, for any nonempty proper subset I of V , 2 πχ ( M ) (cid:80) i ∈ I r ∗ i (cid:107) r ∗ (cid:107) > − (cid:88) ( e,v ) ∈ Lk ( I ) ( π − Φ( e )) + 2 πχ ( F I ); (1.14)(6) The combinatorial-topological conditions Y ∩ R N< (cid:54) = φ when χ ( M ) <
0, and Y ∩{ } (cid:54) = φ when χ ( M ) = 0 are valid, where Y is defined in (1.5). Remark 1.
When χ ( M ) >
0, the conclusions in Theorem 1.5 are not true. Example 2shows that the combinatorial Ricci flow for the sphere with the tetrahedron triangulationmay not converge, while Example 3 shows that there are more than one constant curvaturemetric on the sphere with the tetrahedron triangulation.Theorem 1.5 and Remark 1 provide a suitable answer to the Question 2.As pointed out in Remark 1, it’s surprising that the combinatorial flows are quite dif-ferent from the smooth flows. The combinatorial structure of the triangulation bring some6xtra trouble that needs special considerations. However, for the surfaces with χ ( M ) > r i by r − i or, more generally, r αi with α ≤
0. In fact, for any α ∈ R , we candefine the α -curvature as R α,i = K i r αi (1.15)at each vertex i . The idea behind this definition is to consider r α as a metric (of α order). From the viewpoint of Riemannian geometry, a piecewise flat metric is a singularRiemannian metric on M or M , which produces conical singularities at all vertices. Forany α ∈ R , a metric g with conical singularity at a point can be expressed as g ( z ) = e f ( z ) | z | α − dzd ¯ z locally. Choosing f ( z ) = − ln α , then g ( z ) = | dz α | . Comparing r α with | dz α | , the α -metric r α may be taken as a discrete analogue of conical metric to someextent. By introducing the corresponding α -flows, we get the following result for constant α -curvature problem, which is a combination of Theorem 5.4, Theorem 5.6 and Corollary5.8. Theorem 1.6.
Given a weighted triangulated surface ( M, T , Φ) with αχ ( M ) ≤
0. Thenthe constant α -curvature metric is unique (if exists) up to scaling, and the followingstatements are mutually equivalent:(1) There exists a constant α -curvature circle packing metric;(2) The solution to the α -Ricci flow (5.4) converges;(3) The solution to the α -Calabi flow (5.7) converges;(4) The solution to the modified α -Calabi flow (5.8) converges;(5) There exists a circle packing metric r ∗ such that, for any nonempty proper subset I of V , 2 πχ ( M ) (cid:80) i ∈ I r ∗ αi (cid:107) r ∗ (cid:107) αα > − (cid:88) ( e,v ) ∈ Lk ( I ) ( π − Φ( e )) + 2 πχ ( F I ); (1.16)(6) The following combinatorial-topological conditions(i) Y ∩ R N< (cid:54) = φ , when α > χ ( M ) < Y ∩ R N> (cid:54) = φ , when α < χ ( M ) > πχ ( M ) | V | (1 , · · · , T ∈ Y , when αχ ( M ) = 0.are valid, where Y is defined in (1.5). 7 emark 2. Theorem 1.6 contains Theorem 1.1, 1.3, 1.4, 1.5 as special cases. Take α = 2,then Theorem 1.6 is reduced to Theorem 1.5; Take α = 0, then the equivalence between(1) and (5) is in fact Theorem 1.1, the equivalence between (1), (2) and (4) is in factTheorem 1.3 and Theorem 1.4.Theorem 1.6 can not be generalized to the case of αχ ( M ) > α -curvature metric, or more ideally, admits a solution to discrete Ricciflow that converges to a constant α -curvature metric. By the discrete maximum principle,we have the following existence result for non-negative constant α -curvature metric. Theorem 1.7.
Suppose ( M, T , Φ) is a weighted triangulated surface. If there is a metric r satisfying R α,i ≥ i ∈ V , and − (cid:88) ( e,v ) ∈ Lk ( I ) ( π − Φ( e )) + 2 πχ ( F I ) < , ∀ I : ∅ (cid:36) I (cid:36) V, then there exists a non-negative constant α -curvature metric r ∗ .For a compact manifold M with a triangulation T = { V, E, F, T } , where T is the set ofall tetrahedrons. A ball packing metric r : V → (0 , + ∞ ) evaluates each { i, j } ∈ E a length l ij = r i + r j . Denote α ijkl as the solid angle at a vertex i in the tetrahedron { i, j, k, l } ∈ T .Cooper and Rivin [18] defined a discrete scalar curvature (we call it “CR-curvature” forsimple in the following ) at each vertex i as K i = 4 π − (cid:88) { i,j,k,l }∈ T α ijkl . To study the constant CR-curvature problem, Glickenstein [24, 25] first introduced acombinatorial Yamabe flow dr i dt = − K i r i . (1.17)Inspired by Glickenstein’s work, the first author of this paper and Jiang [21] modify Glick-enstein’s flow (1.17) to dr i dt = ( λ − K i ) r i , (1.18)where λ = (cid:80) Ni =1 K i r i / (cid:107) r (cid:107) l , and get better convergence results. Similar to the two dimen-sional case, we give a new definition of the combinatorial scalar curvature R i = K i r i . Similar to the smooth case, we find that the sphere packing metrics with constant com-binatorial scalar curvature are isolated and are exactly the critical points of a normalized8instein-Hilbert-Regge functional (cid:80) R i r i / ( (cid:80) r i ) / . Hence we propose to study a com-binatorial Yamabe problem with respect to the new combinatorial scalar curvature R i . Question 3.
Given a 3-dimensional manifold M with triangulation T , find a sphere pack-ing metric with constant combinatorial scalar curvature in the combinatorial conformalclass M T . Here M T is the space of admissible sphere packing metrics determined by T .Set R av = (cid:80) R i r i / (cid:80) r i , we introduce the following combinatorial Yamabe flow dg i dt = ( R av − R i ) g i (1.19)to study the combinatorial Yamabe problem. Nonpositive constant curvature metrics arelocal attractors of the flow (1.19), which implies the following result. Theorem 1.8.
Suppose r ∗ is a sphere packing metric on ( M, T ) with nonpositive con-stant combinatorial scalar curvature. If || r (0) − r ∗ || is small enough, the solution of thenormalized combinatorial Yamabe flow (6.14) exists for all time and converges to r ∗ .The paper is organized as follows. In Section 2, We introduce the new definition ofdiscrete Gauss curvature for triangulated surfaces with circle packing metrics. In Section 3,We introduce the combinatorial Ricci flow and use it to give a solution to the 2-dimensionalYamabe problem and the prescribing curvature problem. In Section 4, we introduce thecombinatorial Calabi flow and study its properties. In Section 5, we introduce the notionof α -curvature and the corresponding α -flows and then study the corresponding constantcurvature problem and prescribing curvature problem using the α -flows. In Section 6, weintroduce the new definition of combinatorial scalar curvature on 3-dimensional manifoldsand give a proof of Theorem 1.8. In Section 7, We list some unsolved problems closelyrelated to the paper. The well-known discrete Gauss curvature K i has been widely studied in discrete geometry,we refer to [14, 27, 28] for recent progress. However, there are two disadvantages of theclassical definition of K i . For one thing, classical discrete Gauss curvature does not performso perfectly in that it is scaling invariant, i.e. if ˜ r i = λr i for some positive constant λ , then˜ K i = K i , which is different from the transformation of scalar curvature R λg = λ − R g inthe smooth case. For another, classical discrete Gauss curvature can not be used directlyto approximate smooth Gauss curvature when the triangulation is finer and finer. As thetriangulation of a fixed surface is finer and finer, we can get a sequence of polyhedronswhich approximate the surface. However, the classical discrete Gauss curvature at each9igure 1: a local triangulation of spherevertex i tends to zero, for all triangles surrounding the vertex i will finally run into thetangent plane and the triangulation of the fixed surface locally becomes a triangulation ofthe tangent plane at vertex i . To approximate the smooth Gauss curvature, dividing K i by an “area elment” A i is necessary. Since we are considering the circle packing metrics,we may choose the area of the disk packed at i , i.e. A i = πr i , as the “area elment”attached to the vertex i . Omitting the coefficient π , we introduce the following definitionof discrete curvature on triangulated surfaces. Definition 2.1.
Given a weighted triangulated surface ( M, T , Φ) with circle packing met-ric r : V → (0 , + ∞ ), the discrete Gauss curvature at the vertex i is defined to be R i = K i r i , where K i is the classical discrete Gauss curvature defined as the angle deficit at i by (1.2).The curvature R i still locally measures the difference of the triangulated surface fromthe Euclidean plane at the vertex i . In the rest of this subsection, we will give someevidences that the Definition 2.1 is a good candidate to avoid the two disadvantages wementioned in the above paragraph. Firstly, let us have a look at the following example. Example 1.
Consider the standard sphere S imbedded in R . The smooth Gauss curva-ture of S is +1 everywhere. Consider a local triangulation of S at the north pole, whichis denoted as a vertex i . O is the origin of R . h is a point lying in Oi , see Figure 1. Theintersection between the horizonal plane passing through h and S is a circle. Divide thiscircle into n equal portions. Denote any two conjoint points by j and k , thus i , j , k formsa triangle. Denote x = ∠ iOj . Taking r i = r j = r k = ij/ x , Φ ij = Φ ik = 0 and suitablychoosing Φ jk , we can always get a circle packing metric locally with l jk = 2 sin πn sin x .Moreover, we can get an infinitesimal triangulation at vertex i by letting n → + ∞ and x →
0. By a trivial calculation we can get θ jki = ∠ kij = 2 arcsin( sin πn cos x ), hence theclassical discrete Gauss curvature is K i = 2 π − n arcsin( sin πn cos x ). It’s obviously thatlim n →∞ , x → K i = 0 . R i approaches the smooth Gauss curvature in thefollowing way R i π = K i πr i = 2 π − n arcsin( sin πn cos x ) π sin x → +1 . (cid:4) Secondly, let’s analyse from the view point of the scaling law of Riemann tensor andGauss curvature. Recall that, for a C curve γ : [ a, b ] → M in a Riemannian manifold( M, g ), the length of the curve is defined to be L ( γ, g ) = (cid:82) ba (cid:112) g ( ˙ γ ( t ) , ˙ γ ( t )) dt , where ˙ γ ( t )is the tangent vector of γ ( t ) in M . So we have L ( γ, ˜ g ) = λ / L ( γ, g ) if ˜ g = λg for somepositive constant λ . Note that, for a weighted triangulated surface ( M, T , Φ) with circlepacking metric r : V → (0 , + ∞ ), the length l ij of the edge { ij } is given by (1.1). So thequantity corresponding to the Riemannian metric g on the weighted triangulated surface( M, T , Φ) with circle packing metric r should be quadratic in r , among which r i is thesimplest. Furthermore, according to the definition R i = K i r i , we have R i ( λr i ) = λ − R i ( r i ),which has the same form as the transformation for the smooth scalar curvature.Finally, the Definition 2.1 is geometrically reasonable. Let us recall the original def-inition of Gauss curvature. We just give a sketch of the definition here and the readerscould refer to [47] for details. Suppose M is a surface embedded in R and ν is the unitnormal vector of M in R . ν defines the well-known Gauss map ν : M → S ⊂ R . Thenthe Gauss curvature at p ∈ M is defined as K ( p ) = lim A → p Aera ν ( A ) Aera A , where the limit is taken as the region A around p becomes smaller and smaller. Forthe weighted triangulated surface ( M, T , Φ) with circle packing metric r , assume it isembedded in R and take the normal vector as a set-valued function, then the definitionof discrete Gauss curvature R i = K i r i is an approximation of the original Gauss curvatureup to a uniform constant π , as the numerator K i is the measure of the set-valued function ν and the denominator r i is the area of the disk with radius r i up to a uniform constant π . Note that the classical discrete Gauss curvature satisfies the Gauss-Bonnet identity(1.3). If we define a discrete measure µ on the vertices by µ i = r i , then we have (cid:90) M f dµ = N (cid:88) i =1 f i r i (2.1)for any f ∈ C ( V ). Using this discrete measure, we have the following discrete version ofGauss-Bonnet formula (cid:90) M Rdµ = (cid:88) i R i dµ i = 2 πχ ( M ) . (2.2)11n this sense, the average curvature is R av = (cid:82) M Rdµ (cid:82) M dµ = 2 πχ ( M ) || r || , (2.3)where || r || = (cid:80) Ni =1 r i is the total measure of M with respect to µ .We give the following table for the corresponding quantities for smooth surfaces andtriangulated surfaces. Smooth surfaces Weighted triangulated surfacesMetric g = g ij dx i dx j g i = r i Length L ( γ, g ) = (cid:82) ba (cid:112) g ( ˙ γ ( t ) , ˙ γ ( t )) dt l ij = (cid:113) r i + r j + 2 r i r j cos Φ ij Measure dµ = (cid:112) det g ij dx dµ i = r i Curvature Gauss curvature
K R i = K i r i Total curvature (cid:82) M Kdµ = 2 πχ ( M ) (cid:80) i R i dµ i = 2 πχ ( M )Table 1: Corresponding between smooth surfaces and triangulated surfacesFor the curvature R i defined by (2.1), it is natural to consider the correspondingcombinatorial Yamabe problem, i.e. does there exist any circle packing metric r withconstant curvature R i ? Inspired by [14, 20, 24, 35], we study the combinatorial Yamabeproblem by the combinatorial curvature flows introduced in the following. Ricci flow was introduced by Hamilton [29] to study the low dimension topology. For aclosed Riemannian manifold ( M n , g ij ), the Ricci flow is defined as ∂∂t g ij = − R ij (3.1)with normalization ∂∂t g ij = 2 n rg ij − R ij , (3.2)where r is the average of the scalar curvature. Ricci flow is a powerful tool and has lotsof applications. Specially, it has been used to prove the Poincar´e conjecture [40, 41, 42].For a closed surface ( M , g ij ), the Ricci flow equation (3.1) is reduced to ∂∂t g ij = − Rg ij (3.3)12ith the normalization ∂∂t g ij = ( r − R ) g ij , (3.4)where R is the scalar curvature. It is proved [30, 13] that, for any closed surface withany initial Riemannian metric, the solution of the normalized Ricci flow (3.4) exists for alltime and converges to a constant curvature metric conformal to the initial metric as timegoes to infinity. It is further pointed out by Chen, Lu and Tian [12] that the Ricci flow canbe used to give a new proof of the famous uniformization theorem of Riemann surfaces.The discrete version of surface Ricci flow was first introduce by Chow and Luo in [14], inwhich they gave another proof of Thurston’s existence of circle packing theorem. In thissubsection, we will introduce a different combinatorial Ricci flow on surfaces to study theconstant curvature problem of R i . Definition 3.1.
For a weighted triangulated surface ( M, T , Φ) with circle packing metric r , the combinatorial Ricci flow is defined as dg i dt = − R i g i , (3.5)where g i = r i .Following Hamilton’s approach, we introduce the following normalization of the flow(3.5) dg i dt = ( R av − R i ) g i , (3.6)where R av is the average curvature of R i defined by (2.3). It is easy to check that thetotal measure µ ( M ) = || r || of M with respect to µ is invariant along the normalized flow(3.6), from which we know that the average curvature R av = πχ ( M ) || r || is invariant along(3.6). As our goal is to study the existence of constant combinatorial curvature metric,we will focus on the properties of the flow (3.6) in the following. We assume r (0) ∈ S N − and then r ( t ) ∈ S N − along the flow in the following.The flows (3.5) and (3.6) differ only by a change of scale in space and a change ofparametrization in time. Let t, r, R denote the variables for the unnormalized flow (3.5),and ˜ t, ˜ r, ˜ R for the normalized flow (3.6). Suppose r ( t ), t ∈ [0 , T ), is a solution of (3.5).Set ˜ r (˜ t ) = ϕ ( t ) / r ( t ), where ϕ ( t ) = || r || − and ˜ t = (cid:82) t ϕ ( τ ) dτ . Then we have || ˜ r || = 1 , ˜ R i = ϕ ( t ) − R i , ˜ R av = ϕ ( t ) − R av = 2 πχ ( M ) . This gives d ˜ g i d ˜ t = d ˜ r i dt dtd ˜ t = ( ϕ (cid:48) r i + ϕ dr i dt ) ϕ − = ( ϕ (cid:48) ϕ − ˜ r i − ϕR i r i ) ϕ − = ( ˜ R av − ˜ R i )˜ g i , ϕ (cid:48) = ddt || r || − = −|| r || − ddt || r || = 2 πχ ( M ) ϕ . Conversely, if ˜ r (˜ t ) , ˜ t ∈ [0 , ˜ T ), is a solution of (3.6), set r ( t ) = ϕ (˜ t ) / ˜ r (˜ t ), where ϕ (˜ t ) = e − ˜ R av ˜ t , t = (cid:90) ˜ t ϕ ( τ ) dτ. Then it is easy to check that dg i dt = − R i g i .For simplicity, we will set u i = ln g i in the following. Then the flows (3.5) and (3.6)could be written as ˙ u = − R (3.7)and ˙ u = R av − R (3.8)respectively, where u = ( u , · · · , u N ) T , = (1 , · · · , T and R = ( R , · · · , R N ) T . Since µ is an analogy of the area element, we can define an inner product (cid:104)· , ·(cid:105) on ( M, T , Φ)with circle packing metric r by (cid:104) f, h (cid:105) = N (cid:88) i =1 f i h i r i = h T Σ f (3.9)for any real functions f, h ∈ C ( V ), whereΣ (cid:44) diag { r , · · · , r N } . A combinatorial operator S : C ( V ) → C ( V ) is said to be self-adjoint if (cid:104) Sf, h (cid:105) = (cid:104) f, Sh (cid:105) for any f, h ∈ C ( V ).The classical discrete Laplace operator [17] “∆” is often written in the following form∆ f i = (cid:88) i ∼ j ω ij ( f j − f i ) , where the weight ω ij can be arbitrarily selected for different purpose. Bennett Chow andFeng Luo [14] first gave a special weight which comes from the dual structure of circle14atterns, while Glickenstein [24] gave a similar weight in three dimension. The curvature K i is determined by the circle packing metric r . Consider the curvature as a function of u , where u i = ln g i = 2 ln r i are coordinate transformations, then Bennett Chow and FengLuo’s discrete Laplacian ∆ CL can be interpreted [20] as the Jacobian of curvature map,i.e. ∆ CL = − L , where L = ∂ ( K , · · · , K N ) ∂ ( u , · · · , u N ) . (3.10)It is noticeable that symbol L above is different from that in [20] by a factor 2, whichcomes from the coordinate transformations.Using the discrete measure µ i = r i , we can give a new definition of discrete Laplaceoperator, which is slightly different from that in [14, 20]. Definition 3.2.
For a weighted triangulated surface ( M, T , Φ) with circle packing metric r : V → (0 , + ∞ ), the discrete Laplace operator ∆ : C ( V ) → C ( V ) is defined as∆ f i = 1 r i (cid:88) j ∼ i ( − ∂K i ∂u j )( f j − f i ) (3.11)for f ∈ C ( V ).The following property of L will be used frequently in the rest of the paper. Lemma 3.3. ([14], Proposition 3.9.) L is positive semi-definite with rank N -1 and kernel { t | t ∈ R } . Moreover, ∂K i ∂u i > ∂K i ∂u j < i ∼ j and ∂K i ∂u j = 0 for others.Using this lemma, we can write discrete Laplace operator as a matrix, that is,∆ = − Σ − L with ∆ f = − Σ − Lf.
Note that the term − ∂K i ∂u j > i ∼ j in the definition of the operator ∆, thus it could betaken as a weight on the edges. We sometimes denote this weight as w ij in the followingif there is no confusion. This implies that ∆ is a standard Laplacian operator defined onthe weighted triangulated surface ( M, T , Φ) with circle packing metric r and measure µ .And it is easy to get (cid:104) ∆ f, g (cid:105) = g T Σ( − Σ − Lf ) = − g T Lf = (cid:104) f, ∆ g (cid:105) . Hence the Laplacian operator ∆ is self-adjoint with respect to (cid:104)· , ·(cid:105) . Moreover, if we define |∇ f | i = 12 r i (cid:88) j ∼ i ( − L ij )( f j − f i ) , (cid:104)|∇ f | , (cid:105) = N (cid:88) i =1 |∇ f | i r i = (cid:88) j ∼ i ( − L ij )( f j − f i ) = f T Lf = −(cid:104) f, ∆ f (cid:105) . (3.12)Using the discrete measure µ introduced in (2.1), (3.12) can be written in an integral form (cid:90) M |∇ f | dµ = − (cid:90) M f ∆ f dµ. When the circle packing metric evolves along the flow (3.6), so is the curvature R i .The evolution of R i is very simple and it has almost the same form as the evolution ofscalar curvature along the Ricci flow on surfaces derived by Hamilton in [30]. Lemma 3.4.
Along the normalized combinatorial Ricci flow (3.6), the curvature R i sat-isfies the evolution equation dR i dt = ∆ R i + R i ( R i − R av ) . (3.13) Proof. As u i = ln r i , we have ∂∂u j = r j ∂∂r j . Then we have dR i dt = (cid:88) j ∂R i ∂u j du j dt = (cid:88) j ( 1 r i ∂K i ∂u j − K i r i r i r j δ ij )( R av − R j )= 1 r i (cid:88) j ∂K i ∂u j ( R av − R j ) − R i ( R av − R i )= − r i (cid:88) j ∂K i ∂u j R j + R i ( R i − R av )= 1 r i (cid:88) j ∼ i ( − ∂K i ∂u j )( R j − R i ) + R i ( R i − R av ) . In the last two steps, Lemma 3.3 is used. (cid:4)
Thus the evolution equation (3.13) of R i is a reaction-diffusion equation. In fact,suppose Ω is a subset of V , set ∇ ij f = f j − f i for f ∈ C ( V ) and i ∼ j , and denote ∂ Ω = {{ ij } ∈ E | i ∈ Ω , j ∈ Ω c } , then it is easy to check that ddt (cid:90) Ω Rdµ = (cid:88) i ∈ Ω ,j ∈ Ω c ,i ∼ j ( − ∂K i ∂u j )( R j − R i ) = (cid:90) ∂ Ω ∇ Rdw.
Then we have similar explanation as the smooth Ricci flow for equation (3.6).16 .3 Maximum principle
As the evolution equation (3.13) is a heat-type equation, we have the following discretemaximum principle for such equations. Such maximum principle is almost standard, forcompleteness, we write it down here.
Theorem 3.5. (Maximum Principle) Let f : V × [0 , T ) → R be a C function such that ∂f i ∂t ≥ ∆ f i + Φ i ( f i ) , ∀ ( i, t ) ∈ V × [0 , T )where Φ i : R → R is a local Lipschitz function. Suppose there exists C ∈ R such that f i (0) ≥ C for all i ∈ V . Let ϕ be the solution to the associated ODE (cid:40) dϕdt = Φ i ( ϕ ) ϕ (0) = C , then f i ( t ) ≥ ϕ ( t )for all ( i, t ) ∈ V × [0 , T ) such that ϕ ( t ) exists.Similarly, suppose f : V × [0 , T ) → R be a C function such that ∂f i ∂t ≤ ∆ f i + Φ i ( f i ) , ∀ ( i, t ) ∈ V × [0 , T ) . Suppose there exists C ∈ R such that f i (0) ≤ C for all i ∈ V . Let ψ be the solution tothe associated ODE (cid:40) dψdt = Φ i ( ψ ) ψ (0) = C , then f i ( t ) ≤ ψ ( t )for all ( i, t ) ∈ V × [0 , T ) such that ψ ( t ) exists. Remark 3.
In fact, Theorem 3.5 is valid for general Laplacian operators defined as∆ f i = (cid:88) j ∼ i a ij ( t )( f j − f i ) , where a ij ≥
0, but not required to satisfy the symmetry condition a ij = a ji .The proof of Theorem 3.5 is almost the same as that in [15], we give a proof of it herejust for completeness. 17 roposition 3.6. Let f : V × [0 , T ) → R N be a C solution to the heat equation ∂f∂t = ∆ f, i.e. ∂f i ∂t = ∆ f i = (cid:88) j ∼ i a ij ( t )( f j − f i ) . If there are constants C ≤ C ∈ R such that C ≤ f i (0) ≤ C for all i ∈ V , then C ≤ f i ( t ) ≤ C , ∀ ( i, t ) ∈ V × [0 , T ) . This proposition follows from the following more general theorem.
Theorem 3.7.
Suppose f : V × [0 , T ) → R N be a C function with f i (0) ≥ α for someconstant α ∈ R , and f satisfies ∂f i ∂t ≥ ∆ f i at any ( i, t ) ∈ V × [0 , T ) such that f i ( t ) < α . Then f i ( t ) ≥ α for all ( i, t ) ∈ V × [0 , T ). Proof.
Note that if H : V × [0 , T ) → R is a C function and ( i, t ) is the vertex andtime where H i ( t ) attains its minimum among all vertices and earlier times, namely H i ( t ) = min ( j,t ) ∈ V × [0 ,t ] H j ( t ) , then ∂H i ∂t ( t ) ≤ , ∆ H i ( t ) ≥ . Consider a function H defined by H i ( t ) = ( f i ( t ) − α ) + εt + ε, where ε is any positiveconstant. Then H i (0) ≥ ε >
0, and ∂H i ∂t = ∂f i ∂t + ε ≥ ∆ f i + ε = ∆ H i + ε for ( i, t ) ∈ V × [0 , T ) satisfies f i ( t ) ≤ α . Thus we just need to prove that H i ( t ) > i, t ) ∈ V × [0 , T ).Suppose that ( i, t ) is the first point in V × [0 , T ) such that H i ( t ) = 0. Then f i ( t ) = H i ( t ) + α − εt − ε = α − εt − ε < α. Then we have 0 ≥ ∂H i ∂t ( t ) ≥ ∆ H i ( t ) + ε ≥ ε > , which is a contradiction. And the theorem follows from ε → (cid:4) Now suppose β : V × [0 , T ) → R be a function.18 roposition 3.8. Let f : V × [0 , T ) → R be a C function such that ∂f i ∂t ≥ ∆ f i + β i f i . And for each τ ∈ [0 , T ), ∃ C τ < ∞ such that β i ( t ) < C τ for ( i, t ) ∈ V × [0 , τ ]. If f i ≥ i ∈ V , then f i ( t ) ≥ i, t ) ∈ V × [0 , T ). Proof.
Given τ ∈ (0 , T ), define J i ( t ) = e − C τ t f i ( t ) , then ∂J i ∂t ≥ ∆ J i + ( β i − C τ ) J i . Since β i ( t ) ≤ C τ for all ( i, t ) ∈ V × [0 , τ ], then we have ∂J i ∂t ≥ ∆ J i for ( i, t ) ∈ V × [0 , τ ] such that J i ( t ) <
0. Then by the Theorem 3.7, we have J i ( t ) ≥ f i ( t ) ≥ i, t ) ∈ V × [0 , T ). (cid:4) Now we give the proof of Theorem 3.5.
Proof of Theorem 3.5:
We take the lower bound to prove, and proof for the upperbound is similar. By assumption, we have ∂∂t ( f i − ϕ ) ≥ ∆( f i − ϕ ) + Φ i ( f i ) − Φ i ( ϕ ) . The assumptions on the initial data imply that( f i − ϕ )(0) ≥ , ∀ i ∈ V. Let τ ∈ (0 , T ), then ∃ C τ < ∞ such that | f i ( t ) | < C τ for all ( i, t ) ∈ V × [0 , τ ] and | ϕ ( t ) | < C τ for t ∈ [0 , τ ]. Since F is locally Lipschitz, then we have | Φ i ( f i ) − Φ i ( ϕ ) | ≤ L τ | f i − ϕ | , ∀ i ∈ V for some positive constant L τ >
0. Then ∂∂t ( f i − ϕ ) ≥ ∆( f i − ϕ ) − L τ Sgn ( f i − ϕ ) · ( f i − ϕ ) , ∀ ( i, t ) ∈ V × [0 , τ ] . Applying Proposition 3.8, we get f i ( t ) − ϕ ( t ) ≥ , ∀ ( i, t ) ∈ V × [0 , τ ] . And the theorem follows from τ ∈ [0 , T ) is arbitrary. (cid:4) Applying Theorem 3.5 to (3.13) with vanishing initial value, we can easily get thefollowing corollary. 19 orollary 3.9. If R i (0) ≥ R i (0) ≤
0) for all i ∈ V , then R i ( t ) ≥ R i ( t ) ≤
0) for all i ∈ V as long as the flow exists, i.e. the positive and negative curvatures are preservedalong the normalized flow (3.6).Set R max ( t ) = max i ∈ V R i ( t ) and R min ( t ) = min i ∈ V R i ( t ). Applying Theorem 3.5 to(3.13) with general initial data, we get the following lower bounds for R i ( t ). For simplicity,We just state it in the form parallel to that given in [15]. Lemma 3.10. (Lower Bound) Let r i ( t ) be a solution to the normalized combinatorialRicci flow (3.13) on a closed triangulated surface ( M, T,
Φ).(1) If χ ( M ) <
0, then R i − R av ≥ R av − (1 − R av R min (0) ) e R av t − R av ≥ ( R min (0) − R av ) e R av t . (2) If χ ( M ) = 0, then R i ≥ R min (0)1 − R min (0) t > − t . (3) If χ ( M ) > R min (0) <
0, then R ≥ R av − (1 − R av R min (0) ) e R av t ≥ R min (0) e − R av t . Notice that in each case, the right hand side of the lower bound estimate tends to 0 as t → ∞ . However, for the upper bound, the situation is not so good. The solution of theODE (cid:40) dsdt = s ( s − R av ) s (0) = s corresponding to (3.13) is s ( t ) = , s = 0 s − s t , s (cid:54) = 0 , χ ( M ) = 0 R av − (1 − Ravs ) e Ravt , s (cid:54) = 0 , χ ( M ) (cid:54) = 0 . If s > max { R av , } , there is T < ∞ given by T = (cid:40) − R av ln (cid:16) − R av s (cid:17) > , χ ( M ) (cid:54) = 0 s , χ ( M ) = 020uch that lim t → T − s ( t ) = + ∞ . The implies that, in the case of R max (0) > max { R av , } , Theorem 3.5 will not give usgood upper bounds for the curvature along the combinatorial Ricci flow (3.13). However,in the case of R max (0) <
0, we have good upper bounds.
Lemma 3.11. If R i (0) < i ∈ V , then we have R i ( t ) − R av ≤ R av (cid:18) − R av R max (0) (cid:19) e R av t . These results are almost parallel to that of the smooth Ricci flow on surfaces.
Combining Lemma (3.10) with (3.11) gives
Theorem 3.12.
Suppose ( M, T , Φ) is a weighted triangulated surface with initial circlepacking metric r (0) satisfying R i (0) < i ∈ V , then there exists a negative con-stant curvature metric r ∗ on ( M, T , Φ). Furthermore, the solution r ( t ) to the normalizedcombinatorial Ricci flow (3.6) exists for all time and converges exponentially fast to r ∗ as t → + ∞ .Theorem 3.12, derived by combinatorial maximum principle, implies more than itseems. It claims that, if there is a metric with all curvatures negative, then there alwaysexists a negative constant curvature metric, and vice versa. Using this fact, we will givea combinatorial and topological condition which is equivalent to the existence of negativeconstant curvature metric in subsection 3.6. For the convergence of combinatorial Ricciflow, the negative initial curvature condition in Theorem 3.12 is a little restrictive. It isnatural to consider Hamilton’s approach to generalize this result. However, we found thatthere is some technical difficulties to go further with Hamilton’s approach in this case. Bystudying critical points of (3.6), which is an ODE system, we can get the following localconvergence result. Theorem 3.13.
Suppose r ∗ is a constant R -curvature metric on a weighted triangulatedsurface ( M, T , Φ). If the first positive eigenvalue λ of − ∆ at r ∗ satisfies λ ( − ∆) > R ∗ av = 2 πχ ( M ) || r ∗ || (3.14)and || r (0) − r ∗ || is small enough, then the solution to the normalized combinatorial Ricciflow (3.6) exists for t ∈ [0 , + ∞ ) and converges exponentially fast to r ∗ .21 roof. We can rewrite the normalized combinatorial Ricci flow (3.6) as dr i dt = 12 ( R av − R i ) r i . Set Γ i ( r ) = ( R av − R i ) r i , then the Jacobian matrix of Γ( r ) is given by( D r Γ( r )) ij = ∂∂r j (cid:18)
12 ( R av − R i ) r i (cid:19) = 12 ( R av − R i ) δ ij + ( R i δ ij − R av r i r j || r || ) − r i r j ∂K i ∂u j . Denote Λ = Σ − L Σ − , then Λ ∼ Σ − ΛΣ = − ∆, hence λ ( − ∆) = λ (Λ). At theconstant curvature metric point r ∗ , we have D r Γ | r ∗ = R av (cid:18) I − rr T || r || (cid:19) − Σ − L Σ − = R av (cid:18) I − rr T || r || (cid:19) − Λ . Select an orthonormal matrix P such that P T Λ P = diag { , λ (Λ) , · · · , λ N − (Λ) } . Suppose P = ( e , e , · · · , e N − ), where e i is the ( i + 1)-column of P . Then Λ e = 0 andΛ e i = λ i e i , ≤ i ≤ N −
1, which implies e = r/ (cid:107) r (cid:107) and r ⊥ e i , ≤ i ≤ N −
1. Hence (cid:0) I N − rr T (cid:107) r (cid:107) (cid:1) e = 0 and (cid:0) I N − rr T (cid:107) r (cid:107) (cid:1) e i = e i , 1 ≤ i ≤ N −
1, which implies P T (cid:0) I N − rr T (cid:107) r (cid:107) (cid:1) P = diag { , , · · · , } . Therefore, D r Γ (cid:12)(cid:12) r ∗ = P · diag { , R av − λ (Λ) , · · · , R av − λ N − (Λ) } · P T . If λ (Λ) > R ∗ av = πχ ( M ) || r ∗ || , then D r Γ (cid:12)(cid:12) r ∗ is negative semi-definite with kernel { cr | c ∈ R } and rank ( D r Γ (cid:12)(cid:12) r ∗ ) = N −
1. Note that, along the flow (3.6), || r || is invariant. Thus the kernelis transversal to the flow. This implies that D r Γ | r ∗ is negative definite on S N − and r ∗ is a local attractor of the normalized combinatorial Ricci flow (3.6). Then the conclusionfollows from the Lyapunov Stability Theorem([43], Chapter 5). (cid:4) If χ ( M ) ≤
0, we always have λ ( − ∆) > ≥ R ∗ av = πχ ( M ) || r ∗ || , thus we get Corollary 3.14.
Suppose there is a nonpositive constant curvature metric r ∗ on aweighted triangulated surface ( M, T , Φ) with χ ( M ) ≤
0. If || r (0) − r ∗ || is small enough,then the solution to the normalized combinatorial Ricci flow (3.6) exists for t ∈ [0 , + ∞ )and converges exponentially fast to r ∗ .Before proving the global results, we first check that the convergence of the flow (3.6)ensures the existence of the constant curvature metric. More generally, we have the fol-lowing result. 22 emma 3.15. Suppose that the solution to the flow (3.6) lies in a compact region in R N> , then there exists a constant curvature metric r ∗ . Moreover, there exists a sequenceof metrics which converge to r ∗ along this flow. Proof.
Consider the Ricci potential F ( u ) = (cid:90) uu N (cid:88) i =1 ( K i − R av r i ) du i , where u i = ln r i and u ∈ R N is an arbitrary point. If we set ϕ i = K i − πχ ( M ) || r || r i , bydirect calculations, we have ∂ϕ i ∂u j = ∂ϕ j ∂u i . As the domain of u is simply connected, this implies that the integration is path indepen-dent and then the Ricci potential is well defined. Suppose t ∈ [0 , T ) and T is the maximalexisting time of u ( t ). Then { u ( t ) } ⊂⊂ R N implies T = + ∞ , otherwise u ( t ) will run outof the compact region. { u ( t ) } ⊂⊂ R N also implies that F ( u ( t )) is bounded. Moreover, ddt F ( u ( t )) = ( ∇ u F ) T · ˙ u = − (cid:88) (cid:18) K i r i − πχ ( M ) || r || r i (cid:19) = − (cid:88) i ( R i − R av ) r i ≤ . Hence lim t → + ∞ F ( u ( t )) exists. Then by the mean value theorem, there exists a sequence ξ n ∈ ( n, n + 1) such that F ( u ( n + 1)) − F ( u ( n )) = ( F ( u ( t ))) (cid:48) | t = ξ n = − (cid:88) i ( R i − R av ) r i ( ξ n ) → n → + ∞ . Since { r ( t ) } ⊂⊂ R N> , we can further select a subsequence t k = ξ n k suchthat r ( t k ) → r ∗ as k → + ∞ . Combining (cid:80) i r i ( R i − R av ) ( ξ n k ) →
0, we know that r ∗ determines a constant R -curvature. (cid:4) Corollary 3.16.
Suppose that the solution to flow (3.6) exists for all time and convergesto r (+ ∞ ), then r (+ ∞ ) is a constant curvature metric. Remark 4.
For the Ricci potential functional F used in the proof of Lemma 3.15, theprimitive form of this type of functional was first constructed by Colin de Verdi`ere in [19]and then further studied by Chow and Luo in [14]. Our definition is a modification oftheir constructions.For the Ricci potential F introduced in the proof of Lemma 3.15, we further have thefollowing property. 23 emma 3.17. Given a weighted triangulated surface ( M, T , Φ). Assume there is a con-stant curvature circle packing metric r ∗ on ( M, T , Φ). Define the Ricci potential as F ( u ) = (cid:90) uu ∗ N (cid:88) i =1 ( K i − R av r i ) du i . (3.15)Denote U a (cid:44) { u ∈ R N | (cid:88) i u i = a } , a ∈ R . If λ ( − ∆) > R av for all r ∈ R N> , then(1) Hess u F is positive semi-definite with rank N − { t | t ∈ R } .(2) Restricted to U a , F | U a is strictly convex and proper. F | U a has a unique zero pointwhich is also the unique minimum point. Moreover, lim u ∈ U a , u →∞ F ( u ) = + ∞ . Proof.
By direct calculations, we have
Hess u F = L − R av Σ (cid:18) I − rr T || r || (cid:19) Σ = Σ (cid:18) Λ − R av (cid:18) I − rr T || r || (cid:19)(cid:19) Σ . (3.16)By the same analysis as that of Theorem 3.13, we know that, if λ ( − ∆) > R av , Hess u F is positive semi-definite with rank ( Hess u F ) = N − Ker ( Hess u F ) = { t | t ∈ R } .For the second part of the proof, we follow that of Colin de Verdi`ere [19] and Chow andLuo[14]. It’s easy to see F ( u ) = F ( u + t ) for any t ∈ R and u ∈ R N . As F is invariantalong the direction , we just need to prove it on U = { u ∈ R N | (cid:80) u i = 0 } . A rigorousproof is formulated in Theorem B.2 of [20]. In addition, u ∗ + a − T u ∗ N is the unique zeropoint and minimum point of F | U a . (cid:4) Theorem 3.18.
Suppose ( M, T , Φ) is a weighted triangulated surface and λ ( − ∆) >R av for all r ∈ R N> . Then the solution to the flow (3.6) converges if and only if thereexists a constant curvature metric r ∗ . Furthermore, if the solution converges, it convergesexponentially fast to the metric of constant curvature. Proof.
The necessary part can be seen from Corollary 3.16. For the sufficient part,without loss of generality, we may assume r ∗ ∈ S N − . Consider the flow (3.6) with initialmetric r (0) ∈ S N − ∩ R N> . Then r ( t ) ∈ S N − ∩ R N> and u ( t ) ∈ V ∗ (cid:44) { u ∈ R N | (cid:80) i e u i = 1 } for all t . Let Π be the orthogonal projection from V ∗ to the plane U = { u ∈ R N | (cid:80) i u i =0 } . Then F ( u ) = F (Π( u )) and Π( u ) → ∞ as u → ∞ . In fact, for a sequence { u ( n ) } ,Π( u ( n ) ) is unbounded if and only if | Π( u ( n ) ) i − Π( u ( n ) ) j | = | u ( n ) i − u ( n ) j | is unbounded.Then Lemma 3.17 implies lim u →∞ , u ∈ V ∗ F ( u ) = + ∞ , (3.17)24hich means that F ( u ) is still proper on V ∗ . Since F ( u ( t )) (cid:48) ≤ u ( t ) lies in a compact re-gion in V ∗ . By Lemma 3.15, the solution exists for t ∈ [0 , + ∞ ) and there exists a sequenceof metrics r ( t k ) which converge to r ∗ as t k ↑ + ∞ . Hence F ( u (+ ∞ )) = lim k → + ∞ F ( u ( t k )) = F ( u ∗ ) = 0. While Lemma 3.17 says that u ∗ is the unique zero and minimum point of F ,thus we get u ( t ) converges to u ∗ as t → + ∞ . The exponentially convergent rate comesfrom Theorem 3.13. (cid:4) Corollary 3.19.
Suppose ( M, T , Φ) is a weighted triangulated surface with χ ( M ) ≤ R -curvature metric r ∗ . Furthermore, if the solution converges, it converges exponentiallyfast to the metric of constant curvature.We expect Corollary 3.19 to be still valid for surfaces with χ ( M ) >
0. However, thingsare not so satisfactory for general triangulations. In fact, things may become very differentand complicated. Assume there exists a constant curvature metric r ∗ , we may comparethe Ricci potential (3.15) with G ( u ) = (cid:90) uu ∗ N (cid:88) i =1 ( K i − K ∗ i ) du i , where K ∗ i = πχ ( M ) (cid:107) r ∗ (cid:107) r ∗ i , then F ( u ) − G ( u ) = (cid:90) uu ∗ N (cid:88) i =1 ( K ∗ i − R av r i ) du i = 2 πχ ( M ) (cid:32) (cid:80) Ni =1 r ∗ i ( u i − u ∗ i ) (cid:107) r ∗ (cid:107) − (cid:90) uu ∗ e u du + · · · + e u N du N e u + · · · + e u N (cid:33) = 2 πχ ( M ) (cid:80) Ni =1 r ∗ i ( u i − u ∗ i ) (cid:107) r ∗ (cid:107) − πχ ( M ) ln( e u + · · · + e u N ) (cid:12)(cid:12)(cid:12)(cid:12) uu ∗ . Restricted to the hypersurface V ∗ (cid:44) { u ∈ R N | (cid:80) i e u i = (cid:80) i e u ∗ i } , the last term becomeszero, hence F ( u ) = G ( u ) + 2 πχ ( M ) (cid:80) Ni =1 r ∗ i ( u i − u ∗ i ) (cid:107) r ∗ (cid:107) , ∀ u ∈ V ∗ . By similar arguments in the proof of Theorem 3.18, we further havelim u →∞ , u ∈ V ∗ G ( u ) = + ∞ . One may expect that the growth behavior of F is similar to that of G . Unfortunately,there is an obstruction which makes things very complicated. It’s easy to seelim u →∞ , u ∈ V ∗ (cid:80) Ni =1 r ∗ i ( u i − u ∗ i ) (cid:107) r ∗ (cid:107) = −∞ . F is uncertain unless one can compare the growth rates of thesetwo terms concretely. It seems that we can not expect that G ( u ) succeeds the last termdue to the following example. Example 2.
Given a topological sphere, triangulate it into four faces of a single tetra-hedron. Fix weights Φ ≡
0. Denote r ∗ = , then r ∗ is a constant curvature metric with R i ≡ π . By direct calculation, we get Hess u F | u ∗ = ( √ − π )(4 I − T ), which is negativesemi-definite with three negative eigenvalues. Up to scaling, Hess u F is negative definiteat r ∗ . Thus the fixed point r ∗ = of the ODE system r (cid:48) i ( t ) = ( R av − R i ) r i (i.e. the flow(3.6)) is a source. For this particular weighted triangulation, the flow (3.6) can never con-verge to the constant curvature metric r ∗ for any initial value r (0), as t → + ∞ . However,when t → −∞ , the solution of (3.6) converges to r ∗ if r (0) ∈ S N − (2) is close enough to r ∗ . From the example we know that it is impossible to get an analogous version of Corollary3.19 when χ ( M ) >
0. However, we can get the following long time existence of (3.6) bymaximum principle.
Theorem 3.20.
Given a weighted triangulated surface ( M, T , Φ) with χ ( M ) > − (cid:88) ( e,v ) ∈ Lk ( I ) ( π − Φ( e )) + 2 πχ ( F I ) < , ∀ I : ∅ (cid:36) I (cid:36) V. (3.18)Suppose r is a circle packing metric with R i ≥ i ∈ V . Then the solution of (3.6)with initial metric r exists for all time and lies in a compact region in R N> . Furthermore,there exists t n ↑ + ∞ , such that r ( t n ) converges to a constant curvature metric r ∗ . Proof.
By Lemma 3.15, we just need to prove that { r ( t ) } ⊂⊂ R N> . We prove it bycontradiction. Suppose { r ( t ) } ∩ ∂ ( R N> ∩ S N − ) (cid:54) = ∅ , then there exists a sequence t n such that r ( t n ) → r ∗ ∈ ∂ ( R N> ∩ S N − ) , where r ∗ = (0 , · · · , , r ∗| I | +1 , · · · , r ∗ N ) for some nonempty proper subset I of V . By Propo-sition 4.1 of [14], we havelim n → + ∞ (cid:88) i ∈ I K i ( r ( t n )) = − (cid:88) ( e,v ) ∈ Lk ( I ) ( π − Φ( e )) + 2 πχ ( F I ) < . (3.19)However, by Corollary 3.9, R i ≥ K i ≥ i ∈ V , which implies lim n → + ∞ (cid:80) i ∈ I K i ( r ( t n )) ≥
0. This contradicts (3.19). (cid:4)
Remark 5.
It is easy to see that the triangulation of sphere in Example 2 does not satisfythe condition (3.18). 26 .5 The prescribing curvature problem
Using modified combinatorial Ricci flow, we can consider the prescribing curvature problemon a weighted triangulated surface ( M, T , Φ).
Definition 3.21.
Suppose ( M, T , Φ) is a weighted triangulated surface with circle packingmetric r , R ∈ C ( V ) is a function defined on M . The modified combinatorial Ricci flowwith respect to R is defined to be dg i dt = ( R i − R i ) g i , (3.20)where g i = r i as before.Note that the total measure µ ( M ) = || r || may vary along the modified combinatorialRicci flow (3.20), which is different from that of (3.6). R is called admissible if there is a circle packing metric r with curvature R . For agiven function R ∈ C ( V ), we can introduce the following modified Ricci potential F ( u ) = (cid:90) uu N (cid:88) i =1 (cid:0) K i − R i r i (cid:1) du i . (3.21)It is easy to check that the modified Ricci potential F is well-defined. Furthermore, bydirect calculations, we have Hess u F = L − Σ R . . . R N Σ . The following lemma is useful.
Lemma 3.22. ([14]) Suppose Ω ⊂ R N is convex, the function h : Ω → R is strictlyconvex, then the map ∇ h : Ω → R N is injective. Theorem 3.23.
Suppose ( M, T , Φ) is a weighted triangulated surface and R ∈ C ( V ) isa function defined on M .(1) If the solution to the modified flow (3.20) converges, then R is admissible.(2) If R i ≤ i , but not identically zero, and R is admissible by a metric r . Then r is the unique metric in R N> such that its curvature is R . Moreover, the solutionto the modified flow (3.20) converges exponentially fast to r .27 roof. The first part is obviously, and R is admissible by metric r (+ ∞ ). For thesecond part, notice that Hess u F = L − Σ diag { R , · · · , R N } Σ , it is easy to check that, if R i ≤ i = 1 , · · · , N and not identically zero, Hess u F ispositive definite. By Lemma 3.22, ∇ u F = ( K − R r , · · · , K N − R N r N ) T is an injectivemap from u ∈ R N to R N . Hence r is the unique zero point of ∇ u F . This fact implies that r is the unique metric in R N> such that it’s curvature is R . By Lemma B.1 in [20], we knowthat F is proper and lim u →∞ F ( u ) = + ∞ . Furthermore, ddt F ( u ( t )) = − (cid:80) i r − i ( K i − R i r i ) ≤ (cid:4) Remark 6.
Theorem 3.23 could be taken as a discrete version of a result obtained byKazdan and Warner in [33].
Remark 7.
The second part of Theorem 3.23 implies that χ ( M ) <
0. If R i = 0 for all i , the corresponding prescribing curvature problem is already solved in Corollary 3.19. Inthis case, the metric r is not unique. However, it is unique up to scaling. This is slightlydifferent from Theorem 3.23. We first condider the uniqueness of constant R -curvature metric. Theorem 3.24.
Suppose ( M, T , Φ) is a weighted triangulated surface. c ∗ ∈ R is aconstant.(1) If c ∗ <
0, there exists at most one circle packing metric with curvature R = c ∗ .(2) If c ∗ = 0, there exists at most one metric with curvature R = c ∗ up to scaling. Proof.
The first part is just a corollary of the second part of Theorem 3.23. Thesecond part is already proved by Thurston [48]. (cid:4) If χ ( M ) ≤
0, Theorem 3.24 implies that the constant curvature metric (if exists) isunique or unique up to scaling. If χ ( M ) >
0, there may be several different constantcurvature metrics on S N − and we have the following example. Example 3.
Consider the weighted triangulation of the sphere in Example 2 with tan-gential circle packing metric. Let r = 1, r i = x >
0, 2 ≤ i ≤ x = cos θ − cos θ ,where θ is the inner angle of (cid:52) v v v at the vertex v .28y direct calculations, K = 6 θ − π , K i = π − θ , 2 ≤ i ≤
4. Constant R -curvatureimplies that π − θx = 6 θ − π . Using MATLAB, we can solve the equation and get two solutions x = 1 and x ≈ . r ∗ = and r ∗ ≈ (1 , . , . , . R -curvaturemetric. For the classical discrete Gauss curvature K i , Theorem 1.1 states that the existenceof constant K -curvature metric is equivalent to2 πχ ( M ) | I || V | > − (cid:88) ( e,v ) ∈ Lk ( I ) ( π − Φ( e )) + 2 πχ ( F I ) (3.22)for any nonempty proper subset I of V . Theorem 1.2 further states that the space of alladmissible classical discrete Gauss curvature K is Y (cid:44) K GB ∩ (cid:16) ∩ φ (cid:54) = I (cid:36) V Y I (cid:17) , where K GB and Y I are defined by (1.6) and (1.7) respectively. If r ∗ determines a constant R -curvature, then the K -curvature K ∗ = ( K ∗ , · · · , K ∗ N ) T is admissible, where K ∗ i = πχ ( M ) || r ∗ || r ∗ i . Hence K ∗ ∈ Y I , i.e.2 πχ ( M ) (cid:80) i ∈ I r ∗ i || r ∗ || > − (cid:88) ( e,v ) ∈ Lk ( I ) ( π − Φ( e )) + 2 πχ ( F I ) (3.23)for any nonempty proper subset I of V . (3.23) is necessary for the existence of constant R -curvature metric, which involves the combinatorial, topological and metric structure ofthe triangulated surface. In the case of χ ( M ) ≤
0, the combinatorial-topological-metriccondition (3.23) is also sufficient. In fact, we have the following result.Figure 2: tetrahedron triangulation of sphere29 heorem 3.25.
Suppose ( M, T , Φ) is a weighted triangulated surface with χ ( M ) ≤ R -curvature metric if and only if there exists a circle packingmetric r ∗ such that, for any nonempty proper subset I of vertices V , (3.23) is valid. Proof.
In the case of χ ( M ) = 0, a zero R -curvature metric is exactly a zero K -curvature metric. The conclusion is contained in Theorem 1.1.In the case of χ ( M ) <
0, We just need to prove the sufficient part. Set K ∗ i =2 πχ ( M ) (cid:80) i ∈ I r ∗ i (cid:107) r ∗ (cid:107) . By Theorem 1.2, (3.23) implies that K ∗ = ( K ∗ , · · · , K ∗ N ) T is an ad-missible K -curvature. Suppose K ( r ) = K ∗ . Then r is a circle packing metric with R i ( r ) < i ∈ V . By Theorem 3.12, the combinatorial Ricci flow (3.6) with r asthe initial metric converges to a negative constant R -curvature metric. Therefore, thereexists a constant R -curvature metric on ( M, T , Φ). (cid:4)
The key point of the proof of Theorem 5.7 is to find a circle packing metric with all R -curvature negative and then use the combinatorial Ricci flow (3.6) to evolve it to aconstant R -curvature metric. Therefore, we can generalize Theorem 5.7 to the followingform. Theorem 3.26.
Suppose ( M, T , Φ) is a weighted triangulated surface.(1) There exists a negative constant curvature metric if and only if Y ∩ R N< (cid:54) = φ .(2) There exists a zero curvature metric if and only if 0 ∈ Y .(3) If there exists a positive constant curvature metric, then Y ∩ R N> (cid:54) = φ .We include the case of χ ( M ) > R -curvature metric, then there exists a circle packing metric r ∗ satisfying (3.23),which implies Y ∩ R N> (cid:54) = φ .Similar to Thurston’s condition (1.4), conditions in Theorem (3.26) also show that thecombinatorial structure of the triangulation and the topology of surface, which have norelation to the circle packing metrics, contain some information of R -curvature.For the surfaces with χ ( M ) >
0, the property of constant R -curvature metric seemscomplicated and confusing as exhibited in Example 2 and Example 3. We do not knowwhether the condition Y ∩ R N> (cid:54) = φ is sufficient for the existence of constant R -curvaturemetric. Even through, we can get some sufficient conditions for the existence of constant R -curvature metric by the discrete maximum principle. Theorem 3.27.
Suppose ( M, T , Φ) is a weighted triangulated surface. If there is a circlepacking metric r satisfying R i ≥ i ∈ V , and − (cid:88) ( e,v ) ∈ Lk ( I ) ( π − Φ( e )) + 2 πχ ( F I ) < I of V , then there exists a non-negative constant R -curvaturemetric on ( M, T , Φ).
Proof.
By the maximum principle, i.e. Theorem 3.5, R i ≥ (cid:4) Given a compact complex manifold admitting at least one K¨ahler metric, to find theextreme metric which minimizes the L norm of the curvature tensor in a given principalcohomology class, Calabi [7] introduced the Calabi flow, which could be written as ∂g∂t = ∆ g K · g (4.1)on a Riemannian surface. Chru´sciel [16] proved that the Calabi flow (4.1) exists for alltime and converges to a constant Gaussian curvature metric on closed surfaces using theBondi mass estimate, assuming the existence of the constant Gaussian curvature in thebackground which is ensured by the the uniformization theorem. Chang [10] pointedout that Chru´sciel’s results still hold for arbitrary initial metric, which implies the uni-formization theorem on closed surfaces with genus greater than one. Chen [11] gave ageometrical proof of the long-time existence and the convergence of the Calabi flow onclosed surfaces. The first author [20] first introduced the notion of combinatorial Calabiflow in Euclidean background geometry and proved that the convergence of the flow isequivalent to the existence of constant classical discrete Gauss curvature. Then we [22]studied the combinatorial Calabi flow in hyperbolic background geometry. We also usethe combinatorial Calabi flow to study some constant combinatorial curvature problemon 3-dimensional triangulated manifolds [23]. In this section, we study the R -curvatureproblem by combinatorial Calabi flow. Definition 4.1.
For a weighted triangulated surface ( M, T , Φ) with circle packing metric r , the combinatorial Calabi flow is defined as dg i dt = ∆ R i · g i , (4.2)where g i = r i and ∆ is the Laplacian operator given by (3.11).It is easy to check that the total measure µ ( M ) = (cid:107) r (cid:107) of M is invariant along thecombinatorial Calabi flow (4.2). Interestingly, the combinatorial curvature R i evolvesaccording to dR i dt = − R i ∆ R i − ∆ R i , dr i dt = − r i (cid:88) j ∂K i ∂u j R j = − r i (cid:88) j ∂K i ∂u j ( R j − R av ) . Set Γ( r ) i = − r i (cid:88) j ∂K i ∂u j ( R j − R av ) . If there exists a constant curvature metric r ∗ , then we have D j Γ i | r ∗ = − r i r j (cid:32)(cid:88) k r k ∂K i ∂u k ∂K j ∂u k − R av ∂K i ∂u j (cid:33) . If we further assume that the condition (3.14), i.e. λ ( − ∆) > R ∗ av = πχ ( M ) || r ∗ || , is satisfied,then D Γ | r ∗ is negative semi-definite with rank N − { t | t ∈ R } . From thisfact we can get local convergence results similar to Theorem 3.13, Corollary 3.14. We canalso get results similar to Lemma 3.15, Corollary 3.16, Theorem 3.18 and Corollary 3.19.We just state the following main theorem for this section here. Theorem 4.2.
Suppose ( M, T , Φ) is a weighted triangulated surface with χ ( M ) ≤ R -curvature metric r ∗ . Proof.
If the solution of (4.2) converges, u (+ ∞ ) must be a critical point of this ODEsystem, i.e. ∆ R (+ ∞ ) = − Σ − LR (+ ∞ ) = 0, which implies that R (+ ∞ ) belongs to thekernel of L and hence r (+ ∞ ) is a constant curvature metric.Assume there exists a constant curvature metric r ∗ . Write the flow (4.2) in the matrixform ˙ u = ∆ R = − Σ − LR or ˙ r = − LR , where r = ( r , · · · , r N ) T . It is easy to see that ddt F ( u ( t )) = − ( K − R av r ) T Σ − LR = − R T LR ≤ . Using the properties of the Ricci potential, we can get the convergence result by similararguments in the proof of Theorem 3.18. (cid:4)
Analogous to the combinatorial Ricci flow, we can also use the combinatorial Calabiflow to study the combinatorial prescribing curvature problem. In fact, we have thefollowing result. 32 heorem 4.3.
Suppose ( M, T , Φ) is a weighted triangulated surface, R ∈ C ( V ) is afunction defined on M with R i ≤ i . Then R is admissible if and only if thesolution of the modified combinatorial Calabi flow dg i dt = ∆( R − R ) i g i (4.3)exists for all time and converges. (cid:4) The proof is just a combination of that of Theorem 3.23, Remark 7 and Theorem 4.2,we omit it here.Following [20], we can also introduce a notion of energy to study the combinatorialcurvature R i . Definition 4.4.
For a weighted triangulated surface ( M, T , Φ) with circle packing metric r , the combinatorial Calabi energy is defined as (cid:101) C ( r ) = N (cid:88) i =1 ϕ i , (4.4)where ϕ i = K i − πχ M || r || r i .Consider the combinatorial Calabi energy (cid:101) C as a function of u , we have ∇ u (cid:101) C = 2 A T ϕ ,where A = ∂ ( ϕ , · · · , ϕ N ) ∂ ( u , · · · , u N ) = ∂ϕ ∂u · · · ∂ϕ ∂u N ... . . . ... ∂ϕ N ∂u · · · ∂ϕ N ∂u N . Note that A is in fact given by (3.16). Thus, if χ ( M ) ≤ A is symmetric, positive semi-definite with rank N − { t | t ∈ R } . Using this fact, we can define a new flow,which is the gradient flow of (cid:101) C . We call it the modified combinatorial Calabi flow. Definition 4.5.
For a weighted triangulated surface ( M, T , Φ) with circle packing metric r , the modified combinatorial Calabi flow is defined as˙ u = − ∇ u (cid:101) C, (4.5)or equivalently, ˙ u = − A T ϕ, (4.6)where ϕ = ( ϕ , · · · , ϕ N ) T . 33ote that (cid:80) i ϕ i = 0, so in the case of χ ( M ) ≤
0, if the flow (4.5) converges, itconverges to the constant curvature metric. Along the flow (4.5), we have˙ ϕ = − AA T ϕ, ˙ (cid:101) C = − || A T ϕ || ≤ ddt F ( u ( t )) = ( ∇ u F ) T · ˙ u = ϕ T · ˙ u = − ϕ T A T ϕ ≤ , which implies that F ( u ( t )) is decreasing along the flow (4.5).Following the arguments in the proof of Theorem 3.18 and 4.2, we can derive thefollowing result. Theorem 4.6.
Suppose ( M, T , Φ) is a weighted triangulated surface with χ ( M ) ≤ r ∗ is equivalent to the convergence of themodified combinatorial Calabi flow (4.5). (cid:4) α -curvature and combinatorial α -flows In the previous sections, we investigated the properties of the new discrete Gauss curvature R i = K i /r i . Since the area of the disk packed at i is just πr i , the denominator ofthe curvature r i (omitting the efficient π ) may also be considered as an “area element”attached to vertex i . We want to know whether there exists other types of “area element”.Suppose A i is a general “area element”, which is an analogy of the volume element in thesmooth case, the combinatorial Gauss curvature could be defined as R i = K i /A i . Theaverage curvature should be 2 πχ ( M ) / (cid:80) A i , which is an analogy of (cid:82) Rdµ/ (cid:82) dµ in thesmooth case. If we expect the functional F ( u ) = (cid:90) uu N (cid:88) i =1 ( K i − πχ ( M ) (cid:80) A i A i ) du i to be well defined, the following formula ∂∂u i (cid:18) A j (cid:80) k A k (cid:19) = ∂∂u j (cid:18) A i (cid:80) k A k (cid:19) (5.1)should be satisfied for all i and j . It is a trivial observation that A i = r αi always satisfies(5.1) for all α ∈ R .When χ ( M ) ≤
0, we have already seen in the previous sections that the combinatorialRicci flow and Calabi flow are good enough to evolve circle packing metrics to a constant R -curvature metric. However, in the case of χ ( M ) >
0, we do not know how to evolveit. Maybe this is because that the growth behavior of F is uncertain (in integral level),34r equivalently, Hess u F is not positive semi-definite (in differential level), even for thesimplest triangulation of S (see Example 2).The above two reasons motivate us to consider a new “area element” A i = r αi . Wefind that, for χ ( M ) >
0, if α <
0, combinatorial flow methods are good enough to evolvethe curvature to a constant. For χ ( M ) ≤
0, if α >
0, the properties of combinatorial flowsare almost the same with previous sections. It is very interesting that [14] and [20] can beincluded into the case of α = 0. Definition 5.1.
For a weighted triangulated surface ( M, T , Φ) with a circle packing metric r , the combinatorial α -curvature is defined as R α,i = K i r αi , (5.2)where α is a real number.For this type of curvature, we can also consider the corresponding constant curvatureproblem and prescribing curvature problem. As the methods are all the same as that wedealt with R -curvature, we will give only the outline and skip the details in the following.The measure defined on M is now µ α ( i ) = r αi and the average curvature is now R α,av = (cid:82) M R α dµ α (cid:82) M dµ α = 2 πχ ( M ) || r || αα , where || r || α = ( (cid:80) Ni =1 r αi ) α and || r || αα is defined to be N in the case of α = 0. In thissection, we set u = ln r i and define the Ricci potential as F α ( u ) = (cid:90) uu N (cid:88) i =1 ( K i − R α,av r αi ) du i , where u ∈ R N is an arbitrary point. By direct calculations, it is easy to check that theRicci potential F α is well-defined. We further have ∇ u F α = K − R α,av r α and Hess u F α = (cid:101) L − αR α,av Σ α (cid:32) I − r α · ( r α ) T || r || αα (cid:33) Σ α = r α . . . r α (cid:32) Λ α − αR α,av (cid:32) I − r α · ( r α ) T || r || αα (cid:33)(cid:33) r α . . . r α , where r α = ( r α , · · · , r αN ) T , Λ α = Σ − α (cid:101) L Σ − α and (cid:101) L ij = ∂K i ∂u j with u j = ln r j . Notethat the matrix I − r α · ( r α ) T || r || αα has eigenvalues 1 ( N − cr α | c ∈ R } . Following the arguments in the proof of Lemma 3.17, we have, if the firstpositive eigenvalue of Λ α satisfies λ (Λ α ) > αR α,av ,Hess u F α is positive semi-definite with rank N − { c | c ∈ R } . Especially, if αχ ( M ) ≤ Hess u F α is positive semi-definite with rank N − { c | c ∈ R } .We can also define the following modification of the combinatorial Ricci flow, which iscalled α -Ricci flow. Definition 5.2.
For a weighted triangulated surface ( M, T , Φ) with circle packing metric r , the α -Ricci flow is defined to be dr i dt = − R α,i r i . (5.3)The normalization of the α -Ricci flow is dr i dt = ( R α,av − R α,i ) r i . (5.4)Notice that, when α = 0, the flow (5.3) and the normalized flow (5.4) are just Chow andLuo’s combinatorial Ricci flows [14]. In this case, Π Ni =1 r i (or (cid:80) Ni =1 u i ) is invariant along thenormalized flow (5.4). When α (cid:54) = 0, (cid:107) r (cid:107) α (or (cid:80) Ni =1 e αu i ) is invariant along the normalizedflow (5.4). Along the normalized α -Ricci flow, the α -curvature evolves according to dR α,i dt = 1 r αi N (cid:88) j =1 (cid:18) − ∂K i ∂u j (cid:19) R α,j + αR α,i ( R α,i − R α,av ) . (5.5)For α = 2, this is just the evolution equation (3.13) derived in Lemma 3.4. As theevolution equation (5.5) is still a heat-type equation, the discrete maximum principle, i.e.Theorem 3.5, could be applied to this equation. The evolution of curvature (5.5) underthe normalized α -Ricci flow suggests us to define the α -Laplacian as∆ α f i = 1 r αi N (cid:88) j =1 (cid:18) − ∂K i ∂u j (cid:19) f j = 1 r αi (cid:88) j ∼ i (cid:18) − ∂K i ∂u j (cid:19) ( f j − f i ) , (5.6)where f ∈ C ( V ) and u i = ln r i . Using this Laplacian, we can define the combinatorial α -Calabi flow as dr i dt = (∆ α R α ) i r i . (5.7)Similarly, denote ϕ α,i = K i − πχ ( M ) || r || αα r αi and (cid:101) C α ( r ) = (cid:80) Ni =1 ϕ α,i , we can define a modified α -Calabi flow as ˙ u = − ∇ u (cid:101) C α . (5.8)36pplying the discrete maximal principle to α -Ricci flow (5.4), we can get similar exis-tence results for constant α -curvature metric. Theorem 5.3.
Suppose ( M, T , Φ) is a weighted triangulated surface. If the initial metric r (0) satisfying αR α,i (0) < i ∈ V , then the normalized α -Ricci flow (5.4) convergesto a constant α -curvature metric. Proof.
Note that the α -curvature R i,α evolves according to (5.5) along the normalized α -Ricci flow (5.4). The maximum principle, i.e. Theorem 3.5, is valid for this equation.By the maximum principle, if α > R α,i < i ∈ V , we have( R α, min (0) − R α,av ) e αR α,av t ≤ R α,i − R α,av ≤ R α,av (1 − R α,av R α, max (0) ) e αR α,av t . If α < R α,i > i ∈ V , we have R α,av R α, min (0) ( R α, min (0) − R α,av ) e αR α,av t ≤ R α,i − R α,av ≤ ( R α, max (0) − R α,av ) e αR α,av t . In summary, if αR α,i < i ∈ V , there exists constants c and c such that c e αR α,av t ≤ R α,i − R α,av ≤ c e αR α,av t , which implies the exponential convergence of the normalized α -Ricci flow (5.4). (cid:4) Using the normalized α -Ricci flow (5.4), the combinatorial α -Calabi flow (5.7) and themodified α -Calabi flow (5.8), we can give a characterization of the existence of constant α -curvature metric. Theorem 5.4.
Suppose ( M, T , Φ) is a weighted triangulated surface with αχ ( M ) ≤ α -curvature circle packing metric, the convergence of α -Ricci flow (5.4), the convergence of α -Calabi flow (5.7) and the convergence of modified α -Calabi flow (5.8) are mutually equivalent. Furthermore, if the solutions of the flowsconverge, then they all converge exponentially fast to a constant R α -curvature metric. Proof.
Assume there exists a constant curvature metric r ∗ , u ∗ is the corresponding u -coordinate. Consider the α -Ricci potential F α ( u ) = (cid:90) uu ∗ N (cid:88) i =1 ( K i − R α,av r αi ) du i . (5.9)It has similar properties as stated in Lemma 3.17. Restricted to the hypersurface U = { u ∈ R N | (cid:80) i e αu i = (cid:80) i e αu i (0) } , we havelim u ∈U ,u →∞ F α ( u ) = + ∞ F α | U is also proper. The rest proof is similar to that of Theorem 3.18 and Theorem4.2, so we omit it here. (cid:4) We can also use the combinatorial α -Ricci flow, combinatorial α -Calabi flow and mod-ified combinatorial α -Calabi flow to study the prescribing α -curvature problem. Specifi-cally, we have the following result. Theorem 5.5.
Suppose ( M, T , Φ) is a weighted triangulated surface, α ∈ R is a givenreal number, R α ∈ C ( V ) is a function defined on M with αR α,i ≤ i = 1 , · · · , N .Then R α is an admissible α -curvature if and only if the combinatorial α -Ricci flow withtarget dr i dt = ( R α,i − R α,i ) r i , exists for all time and converges, if and only if the combinatorial α -Calabi flow with target dr i dt = ∆ α ( R α − R α ) i r i exists for all time and converges, if and only if the modified combinatorial α -Calabi flowwith target ˙ u = − ∇ u C exists for all time and converges, where C = (cid:80) i ( K i − R α,i r i ) . Remark 8.
For α = 0, the condition αχ ( M ) ≤ χ ( M ) and the equivalences in Theorem 5.4 and Theorem 5.5 are alwaysvalid. For α = 0, the equivalent condition given by combinatorial Ricci flow is obtained in[14], and the equivalent condition given by combinatorial Calabi flow is obtained in [20].Now we consider the uniqueness and existence of constant α -curvature metrics. As theresults are parallel to that of R -curvature, we just give the statements of the results andomit the proofs. Theorem 5.6.
Suppose ( M, T , Φ) is a weighted triangulated surface with αχ ( M ) ≤ α -curvature metric is unique if it exists. Specificly, if αχ ( M ) = 0, thenthere exists at most one constant α -curvature metric up to scaling. If αχ ( M ) <
0, thenfor any c ∗ , there exists at most one metric with α -curvature R α,i ≡ c ∗ . Remark 9.
For α = 0, this is the result obtained by Thurston [48] and Chow and Luo[14]. 38 heorem 5.7. Suppose ( M, T , Φ) is a weighted triangulated surface with αχ ( M ) ≤ α -curvature metric if and only if there exists a circle packingmetric r ∗ such that, for any nonempty proper subset I of V ,2 πχ ( M ) (cid:80) i ∈ I r ∗ αi (cid:107) r ∗ (cid:107) αα > − (cid:88) ( e,v ) ∈ Lk ( I ) ( π − Φ( e )) + 2 πχ ( F I ) . (5.10) Theorem 5.8.
Suppose ( M, T , Φ) is a weighted triangulated surface. If αχ ( M ) <
0, thenthere exists a constant α -curvature metric if and only if Y ∩ R N< (cid:54) = φ when α > Y ∩ R N> (cid:54) = φ when α < Corollary 5.9.
Suppose ( M, T , Φ) is a weighted triangulated surface with αχ ( M ) > α -curvature metric, then there exists a circle packing metric r ∗ such that (5.10) is valid, which implies that Y ∩ R N> (cid:54) = φ in the case of α > χ ( M ) >
0, and Y ∩ R N< (cid:54) = φ in the case of α < χ ( M ) < Theorem 5.10.
Suppose ( M, T , Φ) is a weighted triangulated surface. If there is a circlepacking metric r satisfying R α,i ≥ i ∈ V , and − (cid:88) ( e,v ) ∈ Lk ( I ) ( π − Φ( e )) + 2 πχ ( F I ) < I of V , then there exists a non-negative constant α -curvaturemetric. Remark 10.
Theorem 3.12 and Theorem 3.20 are now special cases of Theorem 5.3 andTheorem 5.10.
Suppose M is a 3-dimensional compact manifold with a triangulation T = { V, E, F, T } ,where the symbols V, E, F, T represent the sets of vertices, edges, faces and tetrahedronsrespectively. A sphere packing metric is a map r : V → (0 , + ∞ ) such that the lengthbetween vertices i and j is l ij = r i + r j for each edge { i, j } ∈ E , and the lengths l ij , l ik , l il , l jk , l jl , l kl determines a Euclidean tetrahedron for each tetrahedron { i, j, k, l } ∈ T .We can take sphere packing metrics as points in R N> , N times Cartesian product of (0 , ∞ ),where N = V denotes the number of vertices. It is pointed out [24] that a tetrahedron { i, j, k, l } ∈ T generated by four positive radii r i , r j , r k , r l can be realized as a Euclideantetrahedron if and only if Q ijkl = (cid:18) r i + 1 r j + 1 r k + 1 r l (cid:19) − (cid:32) r i + 1 r j + 1 r k + 1 r l (cid:33) > . (6.1)39hus the space of admissible Euclidean sphere packing metrics is M T = (cid:8) r ∈ R N> (cid:12)(cid:12) Q ijkl > , ∀{ i, j, k, l } ∈ T (cid:9) . Cooper and Rivin [18] called the tetrahedrons generated in this way conformal and provedthat a tetrahedron is conformal if and only if there exists a unique sphere tangent to allof the edges of the tetrahedron. Moreover, the point of tangency with the edge { i, j } isof distance r i to v i . They further proved that M T is a simply connected open subset of R N> , but not convex.For a triangulated 3-manifold ( M, T ) with sphere packing metric r , there is also thenotion of combinatorial scalar curvature. Cooper and Rivin [18] defined the combinatorialscalar curvature K i at a vertex i as angle deficit of solid angles K i = 4 π − (cid:88) { i,j,k,l }∈ T α ijkl , (6.2)where α ijkl is the solid angle at the vertex i of the Euclidean tetrahedron { i, j, k, l } ∈ T and the sum is taken over all tetrahedrons with i as one of its vertices. K i locally measuresthe difference between the volume growth rate of a small ball centered at vertex v i in M and a Euclidean ball of the same radius. Cooper and Rivin’s definition of combinatorialscalar curvature is motivated by the fact that, in the smooth case, the scalar curvatureat a point p locally measures the difference of the volume growth rate of the geodesic ballwith center p to the Euclidean ball [34, 3].Similar to the two dimensional case, Cooper and Rivin’s definition of combinatorialscalar curvature K i is scaling invariant, which is not so satisfactory. The authors [23]once defined a new combinatorial scalar curvature as K i r i on 3-dimensional triangulatedmanifold ( M, T ) with sphere packing metric r . Motivated by the analysis in Section 2, wefind that it is more natural to define the combinatorial scalar curvature in the followingway. Definition 6.1.
For a triangulated 3-manifold ( M, T ) with a sphere packing metric r ,the combinatorial scalar curvature at the vertex i is defined as R i = K i r i , (6.3)where K i is given by (6.2).As R i differs from K i only by a factor r i , R i still locally measures the difference betweenthe volume growth rate of a small ball centered at vertex v i in M and a Euclidean ball ofthe same radius. Furthermore, according to the analysis in Section 2, r i is the analogueof the smooth Riemannian metric. If (cid:101) r i = cr i for some positive constant c , we have40 R i = c − R i . This is similar to the transformation of scalar curvature in the smooth caseunder scaling.Analogous to the two dimensional case, we can define a measure on the vertices. As r i is the analogue of the Riemannian metric, we can take the measure as µ i = r i , whichcorresponds to the volume element. Then the total combinatorial scalar curvature is S = (cid:90) M Rdµ = N (cid:88) i =1 R i r i = N (cid:88) i =1 K i r i . (6.4)Note that S is just the functional introduced by Cooper and Rivin in [18]. For the totalcombinatorial scalar curvature S , we have the following important property. Lemma 6.2. ([18, 45, 26]) Suppose ( M, T ) is a triangulated 3-manifold with spherepacking metric r , S is the total combinatorial scalar curvature. Then we have ∇ r S = K. (6.5)If we set Λ = Hess r S = ∂ ( K , · · · , K N ) ∂ ( r , · · · , r N ) = ∂K ∂r · · · ∂K ∂r N · · · · ·· · · · ·· · · · · ∂K N ∂r · · · ∂K N ∂r N , then Λ is positive semi-definite with rank N − r .It should be emphasized that , as pointed out by Glickenstein [24], the element ∂K i ∂r j for i ∼ j maybe negative, which is different from that of the two dimensional case.By the definition of measure µ , the total measure of M is µ ( M ) = (cid:80) Ni =1 r i . We willdenote the total measure of M by V for simplicity in the following, if there is no confusion.The average combinatorial scalar curvature is R av = (cid:82) M Rdµ (cid:82) M dµ = S V = (cid:80) Ni =1 K i r i (cid:80) Ni =1 r i . (6.6) For the curvature R i , it is natural to consider the corresponding constant curvature prob-lem. Suppose R i = λ, ∀ i ∈ V, for some constant λ , then we have K i = λr i , which impliesthat λ = S V . We can take S V as a functional of the sphere packing metric r . However, this41unctional is not scaling invariant in r . So we can modified the functional as S V / . Thisrecalls us of the smooth Yamabe problem.The classical Yamabe problem aims at solving the problem of existence of constantscalar curvature metric on a closed manifold. In order to study the constant scalar cur-vature problem on a closed Riemannian manifold ( M, g ), Yamabe [50] introduced theso-called Yamabe functional Q ( g ) and the Yamabe invariant Y M, [ g ] , which are defined as Q ( g ) = (cid:82) M Rdv g ( (cid:82) M dv g ) − n ,Y M, [ g ] = inf g ∈ [ g ] Q ( g ) , where [ g ] is the conformal class of Riemannian metric g . Trudinger [49] and Aubin [2]made lots of contributions to this problem, and Schoen [46] finally gave the solution of theYamabe Problem. We refer the readers to [34] for this problem.For piecewise flat manifolds, one can introduce similar functionals and invariants.Champion, Glickenstein and Young [8] studied Einstein-Hilbert-Regge functionals andrelated invariants on triangulated manifolds with piecewise linear metrics. The authors[23] also introduced a type of combinatorial Yamabe functional and studied its proper-ties. To study the curvature defined by (6.3), we can introduce the following definition ofcombinatorial Yamabe functional and Yamabe invariant on triangulated 3-manifolds withsphere packing metrics. Definition 6.3.
Suppose ( M, T ) is a triangulated 3-manifold with a fixed triangulation T . The combinatorial Yamabe functional is defined as Q ( r ) = S V = (cid:80) Ni =1 K i r i ( (cid:80) Ni =1 r i ) / , r ∈ M T . (6.7)The combinatorial Yamabe invariant with respect to T is defined as Y M, T = inf r ∈ M T Q ( r ) . The admissible sphere packing metric space M T for a given triangulated manifold( M, T ) is an analogue of the conformal class [ g ] of a Riemannian manifold ( M, g ), asevery admissible sphere packing metric could be taken to be conformal to the metric withall r i = 1. We call M T the combinatorial conformal class for ( M, T ). It is uniquelydetermined by the triangulation T of M . Y M, T is referred to as the Yamabe constant for( M, T ). 42ote that the combinatorial Yamabe invariant Y M, T is an invariant of the conformalclass M T and is well defined, as we have | Q ( r ) | = (cid:12)(cid:12)(cid:12)(cid:12) S V / (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Σ K i r i (cid:0) Σ r i (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:16) Σ K / i (cid:17) = (cid:107) K (cid:107) / , (6.8)and K i satisfies (4 − d ) π ≤ K i < π , where d is the maximal degree and d ≤ E . If theequality in (6.8) is achieved, the corresponding sphere packing metric must be a constantcurvature metric.By direct computation, we have ∇ r i Q = 1 V / ( K i − R av r i ) , (6.9)which implies that r is a constant combinatorial scalar curvature metric if and only if itis a critical point of combinatorial Yamabe functional Q ( r ).Analogue to the smooth Yamabe problem, we can raise the following combinatorialYamabe problem on 3-dimensional triangulated manifold. The Combinatorial Yamabe Problem.
Given a 3-dimensional manifold M with atriangulation T , find a sphere packing metric with constant combinatorial scalar curva-ture in the combinatorial conformal class M T .We could further consider finding a suitable triangulation for M which admits a con-stant combinatorial scalar curvature metric.It is easy to see that the Platonic solids with tetrahedral cells all admit constantcombinatorial scalar curvature metric, including the 5-cell, the 16-cell, the 600-cell, etc.In these cases, the constant combinatorial scalar curvature metrics arise from symmetryand taking the radii equal. In 1980s, Hamilton [30, 31] proposed the Yamabe flow to the Yamabe problem. For a closedn-dimensional Riemannian manifold ( M n , g ) with n ≥
3, the Yamabe flow is defined to be ∂∂t g ij = − Rg ij (6.10)with normalization ∂∂t g ij = ( r − R ) g ij , (6.11)43here R is the scalar curvature of g and r is the average of the scalar curvature. Along thenormalized Yamabe flow, the volume is invariant, the total scalar curvature is decreased,and the scalar curvature evolves according to ∂R∂t = ( n − R + R ( R − r ) . (6.12)It is proved by Hamilton [31] that the solution to the Yamabe flow (6.11) exists for all timeand the solution converges exponentially fast to a metric with constant scalar curvatureif R < ≤ n ≤
5. He [6] further handled the case of n ≥ R i , we introduce the following combinatorial Yamabeflow. Definition 6.4.
Given a 3-dimensional triangulated manifold ( M, T ) with sphere packingmetric r , the combinatorial Yamabe flow is defined to be dg i dt = − R i g i , (6.13)with normalization dg i dt = ( R av − R i ) g i , (6.14)where g i = r i and R av is the average of the combinatorial scalar curvature given by (6.6).Following the 2-dimensional case, it is easy to check that the solutions of (6.13) and(6.14) can be transformed to each other by a scaling procedure. And it is easy to see that,if the solution of (6.14) converges to a sphere packing metric r (+ ∞ ), then r (+ ∞ ) is ametric with constant combinatorial scalar curvature. Analogous to the smooth Yamabeflow, we have the following properties. Proposition 6.5.
Suppose r ( t ) is a solution of (6.14) on a triangulated 3-manifold ( M, T ).Along the flow (6.14), the total measure µ ( M ) = V = (cid:80) Ni =1 r i is invariant and the totalcombinatorial scalar curvature S is decreased.44 roof. The normalized combinatorial Yamabe flow could be written as dr i dt = 12 ( R av − R i ) r i . Using this equation, we have dVdt = 3 N (cid:88) i =1 r i dr i dt = 32 N (cid:88) i =1 r i ( R av − R i ) = 0 . For the total combinatorial scalar curvature S , we have d S dt = N (cid:88) i =1 dK i dt r i + N (cid:88) i =1 K i dr i dt = 12 N (cid:88) i,j =1 r i ∂K i ∂r j ( R av − R j ) r j + 12 N (cid:88) i =1 K i r i ( R av − R i )= − N (cid:88) i =1 K i ( K i − R av r i ) r i = − N (cid:88) i =1 ( K i − R av r i ) r i , (6.15)which implies that S is decreased along the flow (6.14). Note that Lemma 6.2 is used inthe third step. (cid:4) Remark 11.
As the total measure of M is invariant along (6.14), we will focus on theproperties of (6.14) in the following. Furthermore, we will assume r (0) ∈ S N − (cid:44) { r ∈ R N ; || r || = ( (cid:80) r i ) = 1 } in the following.By direct calculations, we find that the curvature R i evolves according to the followingequation along the normalized combinatorial Yamabe flow (6.14) dR i dt = − r i N (cid:88) j =1 ∂K i ∂r j r j R j + R i ( R i − R av ) . (6.16)If we define the Laplacian as∆ f i = − r i N (cid:88) j =1 ∂K i ∂r j r j f j = 1 r i (cid:88) j ∼ i ( − ∂K i ∂r j r j )( f j − f i ) (6.17)45or f ∈ C ( V ), then the equation (6.16) could be written as dR i dt = 12 ∆ R i + R i ( R i − R av ) , (6.18)which has almost the same form as the evolution equation (6.12) of scalar curvature alongthe Yamabe flow (6.11) on three dimensional smooth manifolds. The Laplacian definedby (6.17) satisfies (cid:82) M ∆ f dµ = 0 and ∆ c = 0 for any f ∈ C ( V ) and constant c ∈ R .Furthermore, by the calculations in [24], we have∆ f i = 1 r i (cid:88) j ∼ i l ∗ ij l ij ( f j − f i ) , (6.19)where l ∗ ij is the area dual to the edge { i, j } . Note that this form of Laplacian is verysimilar to Hirani’s definition [32] of Laplace-Beltrami operator∆ f i = 1 V ∗ i (cid:88) j ∼ i l ∗ ij l ij ( f j − f i ) , except the first factor. It should be mentioned that r i is a type of volume.Though the definition (6.17) of Laplacian has lots of good properties, it is not a Lapla-cian on graphs in the usual sense, as l ∗ ij may be negative, which makes the maximumprinciple for (6.18) not so good. This was founded by Glickenstein and he studied theproperties of such Laplacian in [24, 25].To study the long time behavior of (6.14), we need to classify the solutions of the flow. Definition 6.6.
A solution r ( t ) of the combinatorial Yamabe flow (6.14) is nonsingularif the solution r ( t ) exists for t ∈ [0 , + ∞ ) and { r ( t ) } ⊂⊂ M T ∩ S N − .In fact, the condition { r ( t ) } ⊂⊂ M T ∩ S N − ensures the long time existence of theflow (6.14). Furthermore, we have the following property for nonsingular solutions. Theorem 6.7.
If there exists a nonsingular solution for the flow (6.14), then there existsat least one sphere packing metric with constant combinatorial scalar curvature R i on( M, T ). Proof.
By (6.15), we know that Q ( r ) is decreasing along the flow (6.14). As Q ( r )is uniformly bounded by (6.8), the limit lim t → + ∞ Q ( r ( t )) exists. Then there exists asequence t n ↑ + ∞ such that( Q ( r )) (cid:48) ( t n ) = − N (cid:88) i =1 ( K i − R av r i ) r i → . (6.20)46s { r ( t ) } ⊂⊂ M T ∩ S N − , there exists a subsequence, denoted as r n , of r ( t n ) such that r n → r ∗ ∈ M T ∩ S N − . Then (6.20) implies that r ∗ satisfies K ∗ i = R av ( r ∗ i ) and r ∗ is asphere packing metric with constant combinatorial scalar curvature R i . (cid:4) In fact, under some conditions, we find that the sphere packing metrics with constantcombinatorial scalar curvature are isolated on S N − . Theorem 6.8.
The sphere packing metrics with nonpositive constant combinatorial scalarcurvature are isolated in M T ∩ S N − . Proof.
Define G : M T → R N r (cid:55)→ (cid:18) r ( K − λr ) , · · · , r N ( K N − λr N ) (cid:19) , where λ = S V . It is easy to see that the zero point of G corresponds to the metric withconstant combinatorial scalar curvature. By direct calculations, the Jacobian of G at theconstant curvature metric is DG = (cid:32) Λ − λ Σ (cid:32) I − r / · ( r / ) T V (cid:33) Σ (cid:33) Σ − , (6.21)where r / = ( r / , · · · , r / N ) T . If r ∗ is a sphere packing metric with nonpositive constantcombinatorial scalar curvature, then λ ≤ DG is a positive semi-definite matrixwith rank N − { cr | c ∈ R } , which is normal to S N − . Restricted to S N − , DG is positive definite and then nondegenerate, which implies that the zero points of G with nonpositive curvature are isolated in M T ∩ S N − . (cid:4) Suppose the solution of the flow (6.14) exists for t ∈ [0 , T ) with 0 < T ≤ + ∞ , then wehave { r ( t ) } ⊆ M T ∩ S N − . However, if { r ( t ) } ∩ ∂ ( M T ∩ S N − ) (cid:54) = ∅ , the flow will raise singularities. They could be separated into two types as follows. Essential Singularity
There is a vertex i ∈ V such that there exists a sequence of time t n ↑ T such that r i ( t n ) → n → + ∞ ; Removable Singularity
There exists a sequence of time t n ↑ T such that r i ( t n ) ⊂⊂ R > for all vertices i ∈ V , but there exists a tetrahedron { ijkl } ∈ T such that Q ijkl ( t n ) → n → + ∞ . 47t seems that the following conjectures are likely to hold for the normalized combina-torial Yamabe flow (6.14) on 3-dimensional manifolds. Conjecture 1.
The normalized combinatorial Yamabe flow (6.14) will not develop essen-tial singularity in finite time.
Conjecture 2.
If no singularity develops along the normalized flow (6.14), the solutionconverges to a sphere packing metric with constant combinatorial scalar curvature as timeapproaches infinity.It is interesting to note that Glickenstein [25] made a small amount of progress onConjecture 1. He proved that, for some special class of complexes, the flow he introducedwould not develop singularities in finite time.We have mentioned above that the existence of sphere packing metric with constantcombinatorial scalar curvature is necessary for the convergence of the normalized com-binatorial Yamabe flow (6.14). In fact, it is almost sufficient and we have the followingresult.
Theorem 6.9.
Suppose r ∗ is a sphere packing metric on ( M, T ) with nonpositive con-stant combinatorial scalar curvature. If || r (0) − r ∗ || is small enough, the solution of thenormalized combinatorial Yamabe flow (6.14) exists for all time and converges to r ∗ . Proof.
We can rewrite the normalized combinatorial Yamabe flow (6.14) as dr i dt = 12 ( R av − R i ) r i . Set Γ i ( r ) = ( R av − R i ) r i , then by the calculations in the proof of Theorem 6.8, we have D Γ | r ∗ = − Z · r . . . r N , where Z = Λ − R av r / . . . r / N (cid:32) I − r / · ( r / ) T V (cid:33) r / . . . r / N . If R av ≤
0, then D Γ | r ∗ is a matrix with rank N − { cr | c ∈ R } . Furthermore,the nonzero eigenvalues of D Γ | r ∗ are all negative. Note that, along the normalized flow(6.14), || r || is invariant and thus the kernel { cr | c ∈ R } is transversal to the flow. This48mplies that D Γ | r ∗ is negative definite on S N − and r ∗ is a local attractor of the normal-ized combinatorial Yamabe flow (6.14). Then the conclusion follows from the LyapunovStability Theorem. (cid:4) This theorem has the following interesting corollary.
Corollary 6.10.
Given a 3-dimensional triangulated manifold ( M , T ). If the initial totalcombinatorial scalar curvature functional S (0) ≤
0, and no singularity develops along thenormalized combinatorial Yamabe flow (6.14), then the solution converges to a spherepacking metric r ∗ with nonpositive constant combinatorial scalar curvature. Proof.
By the proof of Theorem 6.7, if the solution r ( t ) of (6.14) is nonsingular, thereexists a sequence r n → r ∗ , where r ∗ is a sphere packing metric with constant combinatorialscalar curvature. As S is decreasing along the flow (6.14) and S (0) ≤ r ∗ has nonpositivecombinatorial scalar curvature. Then Theorem 6.9 implies the conclusion of the corollary. (cid:4) Remark 12.
Especially, if R i (0) ≤
0, Corollary 6.10 is still valid.
There are several questions which we find interesting relating to the results in the paper.1. From the results in Section 3, we see that the surfaces with positive Euler char-acteristic are particular for the constant curvature problem on surfaces. Example2 shows that combinatorial Ricci flow (1.12) may not converge to a constant R -curvature metric, while Example 3 shows that constant curvature metric may notbe unique. If we want to approximate a smooth geometric object by correspondingdiscrete objects, the quantities on these objects should obey similar laws or exhibitsimilar properties. However, it has been proved [30, 13] that the Ricci flow on sur-faces converges to a constant curvature metric for any initial metric. The differencesbetween discrete case and the smooth case are very interesting. It deserves deeperstudying. Maybe these differences are caused by the triangulation. The tetrahedrontriangulation used in Example 2 and 3 maybe too rough to approximate the sphere.We expect these differences will disappear as the triangulation becomes finer andfiner. We want to know whether we can find a triangulation for the sphere suchthat the constant curvature metric uniquely exists. We further want to know howto evolve discrete curvature R i = K i /r i along discrete curvature flows to constantcurvature or, more generally, evolve discrete α -curvature to constant curvature when αχ ( M ) >
0. 49. Whether there are similar topological and combinatorial obstructions, similar toThurston’s criterion (3.22), for the existence of constant α -curvature metric? Forthe negative constant curvature metric, we derived an existence result (Theorem3.12) by the discrete maximum principe, which is different from the other results.We want to further develop these methods to derive more existence results. We wantto know whether we can get similar convergence results as that of the smooth surfaceRicci flow and then give a discrete uniformization theorem for R i . We believe thatthe results in [4] will play an important role in the procedure if it is feasible. Wealso want to know the relationship between our conditions and Thurston’s criterion(3.22).3. The first positive eigenvalue of discrete Laplace operator plays an important role inthe proof of the main results in two dimension. It is interesting to estimate the firstpositive eigenvalue of discrete Laplace operator and use it to derive some existenceresults.4. Investigate the prescribing curvature problem when the target curvature R i > αR α,i > i .5. Study the singularities of three dimensional combinatorial Yamabe flow (1.19), andfind topological and combinatorial obstructions for the existence of sphere packingmetric with constant combinatorial scalar curvature.6. Study the discrete Yamabe functional (6.3) and solve the combinatorial Yamabeproblem in three dimension.7. Study the combinatorial Yamebe problem in higher dimensions. Acknowledgements
The first author would like to show his greatest respect to Professor Gang Tian whobrought him to the area of combinatorial curvature flows. The research of the secondauthor is partially supported by National Natural Science Foundation of China undergrant no. 11301402 and 11301399. He would also like to thank Professor Guofang Wangfor the invitation to the Institute of Mathematics of the University of Freiburg and for hisencouragement and many useful conversations during the work. Both authors would alsolike to thank Dr. Dongfang Li for many helpful conversations.
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