On a combinatorial curvature for surfaces with inversive distance circle packing metrics 7 1 Huabin Ge, Xu Xu 0 2 January 10, 2017 n a J 7 Abstract ] Inthispaper,weintroduceanewcombinatorialcurvatureontriangulatedsurfaces T G with inversive distance circle packing metrics. Then we prove that this combinatorial curvature has global rigidity. To study the Yamabe problem of the new curvature, . h we introduce a combinatorial Ricci flow, along which the curvature evolves almost t a in the same way as that of scalar curvature along the surface Ricci flow obtained m by Hamilton [19]. Then we study the long time behavior of the combinatorial Ricci [ flow and obtain that the existence of a constant curvature metric is equivalent to the 1 convergence of the flow on triangulated surfaces with nonpositive Euler number. We v furthergeneralizethecombinatorialcurvaturetoα-curvatureandprovethatitisalso 5 globallyrigid,whichisinfactageneralizedBower-Stephensonconjecture[6]. Wealso 9 7 use the combinatorial Ricci flow to study the corresponding α-Yamabe problem. 1 0 Mathematics Subject Classification (2010). 52C25, 52C26, 53C44. . 1 0 7 1 1 Introduction : v i X This is a continuation of our work on combinatorial curvature in [15]. This paper gen- r a eralizes our results in [15] to triangulated surfaces with inversive distance circle packing metrics. Circle packing is a powerful tool in the study of differential geometry and geo- metric topology and there are lots of research on this topic. In his work on constructing hyperbolic structure on 3-manifolds, Thurston ([26], Chapter 13) introduced the notion of Euclidean and hyperbolic circle packing metric on triangulated surfaces with prescribed intersection angles. The requirement of prescribed intersection angles corresponds to the fact that the intersection angle of two circles is invariant under the M¨obius transforma- tions. For triangulated surfaces with Thurston’s circle packing metrics, there will be singularities at the vertices. The classical combinatorial Gauss curvature K is introduced i 1 to describe the singularity at the vertex v , which is defined as the angle deficit at v . i i Thurston’s work generalized Andreev’s work on circle packing metrics on a sphere [1, 2]. Andreev and Thurston’s work together gave a complete characterization of the space of the classical combinatorial Gauss curvature. As a corollary, they got the combinatorial- topological obstacle for the existence of a constant curvature circle packing metric, which could be written as combinatorial-topological inequalities. Chow and Luo [7] first intro- duced a combinatorial Ricci flow, a combinatorial analogue of the smooth surface Ricci flow, for triangulated surfaces with Thurston’s circle packing metrics and got the equiv- alence between the existence of a constant curvature metric and the convergence of the combinatorial Ricci flow. This work is the cornerstone of applications of combinatorial surface Ricci flow in engineering up to now, see for example [27, 29] and the references therein. Luo [21] once introduced a combinatorial Yamabe flow on triangulated surfaces with piecewise linear metrics to study the corresponding constant curvature problem. The combinatorial surface Ricci flow in [7] and the combinatorial Yamabe flow in [21] are re- cently written in a unified form in [28]. The first author [10] introduced a combinatorial Calabi flow on triangulated surfaces with Thurston’s Euclidean circle packing metrics and proved the equivalence between the existence of constant circle packing metric and the convergence of the combinatorial Calabi flow. The authors [16] further generalized the combinatorial Calabi flow to circle packing metrics with hyperbolic circle packing metrics and got similar results. However, there are some disadvantages for the classical discrete Gauss curvature as statedin[15]. ThefirstisthattheclassicalEuclideandiscreteGausscurvatureisinvariant under scaling, i.e., K (λr) = K (r) for any positive constant λ. The second is that the i i classical discrete Gauss curvature tends to zero, not the Gauss curvature of the smooth surface, as triangulated surfaces approximate a smooth surface. Motivated by the two disadvantages, the authors [15] introduced a new combinatorial curvature defined as R = i Ki for triangulated surfaces with Thurston’s Euclidean circle packing metrics. If we take r2 i g = r2 astheanalogueoftheRiemannianmetric, thenwehaveR (λg) = λ−1R (g), which i i i i hasthesameformasthatofthesmoothGausscurvature. Furthermore,thereareexamples showingthatthiscurvatureactuallyapproximatesthesmoothGausscurvatureonsurfaces as the triangulated surfaces approximate the smooth surface [15]. Then we introduce a combinatorial Ricci flow and a combinatorial Calabi flow to study the corresponding Yamabe problem and got a complete characterization of the existence of a constant R- curvature circle packing metric using these flows. The results in [15] generalized the previous work of the authors in [14]. The authors then further generalized the curvatures to triangulated surfaces with hyperbolic background geometry [16]. We also consider the α-curvature and α-flows on low dimensional triangulated manifolds in [15, 17]. Thurston introduced the intersection angle in the definition of circle packing metric 2 as it is invariant under the M¨obius transformations. The notion of inversive distance of two circles in a M¨obius plane was introduced by H.S.M. Coxeter [8], which generalizes the notionoftheintersectionangleoftwocircles. Furthermore,thenotionofinversivedistance is invariant under the M¨obius transformations, as it is actually defined by the cross ratio [4]. Bowers and Stephenson [6] introduced the inversive distance circle packing metric for triangulated surfaces, which generalizes Thurston’s circle packing metric defined using intersection angle. The inversive distance circle packing metric has some applications in medicalscience,seeforexample[20]. BowersandStephenson[6]onceconjecturedthatthe inversive distance circle packings are rigid. This conjecture was proved to be locally right by Guo [18] and then finally proved by Luo [22] for Euclidean and hyperbolic background geometry. The local rigidity [18] comes from the convexity of an energy function defined on the admissible space and the global rigidity [22] comes from a convex extension of the energy function to the whole space, the idea of which comes from Bobenko, Pinkall and Springborn [5]. Ma and Schlenker [23] gave a counterexample showing that there is no rigidity for the spherical background geometry and John C. Bowers and Philip L. Bowers [3] recently presented a new construction of their counterexample using only the inversive geometry of the 2-sphere. The combinatorial Ricci flow for triangulated surfaces with inversive distance circle packing metrics was introduced in [27, 28], using the classical combinatorial Gauss curvature. Recently, the first author and Jiang [11, 12]studied the longtimebehavioroftheflowusingtheextendingmethodandgotsomeinterestingresults. In this paper, we generalize our definition of combinatorial curvature in [15] for Thurston’s circle packing metrics to inversive distance circle packing metrics. Given a weighted triangulated surface (M,T,I) with inversive distance I ≥ 0 for every edge ij {ij} ∈ E, for any function r : V → [0,+∞), the edge length l is defined to be ij (cid:113) l = r2+r2+2r r I ij i j i j ij for Euclidean background geometry and l = cosh−1(coshr coshr +I sinhr sinhr ) ij i j ij i j for hyperbolic background geometry. If the lengths {l ,l ,l } satisfy the triangle in- ij jk ik equalities for any face (cid:52)ijk, r is called an inversive distance circle packing metric. Denote Ω = {(r ,r ,r )|l < l +l ,l < l +l ,l < l +l } ijk i j k ij ik kj ik ij jk jk ji ik and Ω = ∩ Ω = {r ∈ RN |l < l +l ,l < l +l ,l < l +l ,∀∆ijk ∈ F}. ∆ijk∈F ijk >0 ij ik kj ik ij jk jk ji ik 3 If r ∈ Ω, the triangulated surface could be taken as obtained by gluing many triangles along their edges coherently and there will be singularities at the vertices. The classical discrete Gauss curvature K is introduced to describe the singularity, which is defined as the angle deficit at a vertex, i.e. K = 2π−(cid:80) θjk, where θjk is the inner angle of i (cid:52)ijk∈F i i (cid:52)ijk at i. In this paper, we introduce a new combinatorial curvature, which is defined as K i R = i r2 i for Euclidean background geometry and K i R = i tanh2 ri 2 for hyperbolic background geometry. For this combinatorial curvature, we prove the fol- lowing main theorem on global rigidity. Theorem 1.1. Given a weighted closed triangulated surface (M,T,I) with I ≥ 0 and χ(M) ≤ 0. R ∈ C(V) is a given function on M. (1) If R ≡ 0, then there exists at most one Euclidean inversive distance circle packing metric r ∈ Ω up to scaling such that its R-curvature is 0. If R ≤ 0 and R (cid:54)≡ 0, then there exists at most one Euclidean inversive distance circle packing metric r ∈ Ω such that its R-curvature is the given function R. (2) If R ≤ 0, then there exists at most one hyperbolic inversive distance circle packing metric r ∈ Ω such that its R-curvature is the given function R. Wefurthergiveacounterexample,i.e. Example1,whichshowsthatthereisnorigidity fortriangulatedsurfaceswithpositiveEulernumber. SotheresultinTheorem1.1issharp. Following [15], we generalize the combinatorial curvature R to the α-curvature R , i α which is defined as R = Ki for Euclidean background geometry and R = Ki for α,i rα α,i tanhα ri i 2 hyperbolic background geometry. For the α-curvature R , we have the following general- α ized global rigidity. Theorem 1.2. Given a weighted closed triangulated surface (M,T,I) with I ≥ 0 and αχ(M) ≤ 0. R ∈ C(V) is a given function on M. (1) If αR = 0, then there exists at most one Euclidean inversive distance circle packing metric r ∈ Ω up to scaling with α-curvature R; If αR ≤ 0 and αR (cid:54)≡ 0, then there exists at most one Euclidean inversive distance circle packing metric r ∈ Ω with α-curvature R. 4 (2) If αR ≤ 0, then there exists at most one hyperbolic inversive distance circle packing metric r ∈ Ω with α-curvature R. The result in Theorem 1.2 is also sharp. In the special case of α = 0, this is Bowers and Stephenson’s conjecture on the global rigidity of the classical combinatorial Gauss curvature K for inversive distance circle packing metrics, which was proved by Guo [18] and Luo [22]. So Theorem 1.2 in fact partially solves a generalized Bowers-Stephenson conjecture. To study the constant curvature problem of the curvature R for the Euclidean back- ground geometry, we introduce a combinatorial Ricci flow, defined as dg i = (R −R )g , (1.1) av i i dt where g = r2 and R = 2πχ(M). We have the following main result on the the combina- i i av ||r||2 2 torial Euclidean Ricci flow (1.1) and existence of constant R-curvature metrics. Theorem 1.3. Given a weighted closed triangulated surface (M,T,I) with I ≥ 0. (1) Along the Euclidean Ricci flow (1.1), the curvature R evolves according to dR i = ∆R +R (R −R ), i i i av dt where the Laplace operator is defined to be N (cid:18) (cid:19) (cid:18) (cid:19) 1 (cid:88) ∂Ki 1 (cid:88) ∂Ki ∆f = − f = − (f −f ) i r2 ∂u j r2 ∂u j i i j=1 j i j∼i j with u = lnr2 for f ∈ C(V). i i (2) IfthesolutionoftheEuclideancombinatorialRicciflow(1.1)staysinΩandconverges to an inversive distance circle packing metric r∗ ∈ Ω, then there exists a constant R-curvature metric in Ω and r∗ is such one. (3) If χ(M) ≤ 0 and there exists a Euclidean inversive distance circle packing metric r∗ ∈ Ω with constant R-curvature, then the solution of the Euclidean combinatorial Ricci flow (1.1) develops no essential singularities in finite time and at time infinity; Furthermore, if the solution of (1.1) develops no removable singularities in finite time, then the solution of (1.1) exists for all time, converges exponentially fast to the constant R-curvature metric r∗ and does not develop removable singularities at time infinity. 5 (4) If χ(M) ≤ 0 and there exists a Euclidean inversive distance circle packing metric r∗ ∈ Ω with constant R-curvature, then the solution of the extended combinatorial Ricci flow dr i = (Rav −R(cid:101)i)gi (1.2) dt exists for all time and converges exponentially fast to r∗. ThedefinitionsofessentialsingularityandremovablesingularityaregiveninDefinition 4.4andR(cid:101)isanextensionoftheR-curvature. TheexistenceofconstantR-curvaturemetric is in fact equivalent to the convergence of the extended Ricci flow (1.2). Forthehyperbolicbackgroundgeometry, themostimportantconstantcurvatureprob- lemisthezerocurvatureproblem,whichfrequentlyappearsinengineering. Following[16], we treat it as a special case of the prescribing curvature problem. Interestingly, we found that the results of the prescribing curvature problems for Euclidean and hyperbolic back- ground geometry are similar, so we state them together in a unified form here. Theorem 1.4. Given a closed triangulated surface (M,T) with inversive distance I ≥ 0. Given a function R ∈ C(V), the modified combinatorial Ricci flow is defined to be dg i = (R −R )g , (1.3) i i i dt where g = r2 for the Euclidean background geometry and g = tanh2 ri for the hyperbolic i i i 2 background geometry. (1) If the solution of the modified Ricci flow (1.3) stays in Ω and converges to r ∈ Ω, then we have R is admissible and R(r) = R. (2) Suppose R ≤ 0 and R is admissible with R(r) = R, then the modified Ricci flow (1.3) develops no essential singularity in finite time and at time infinity. Furthermore, if the solution of the Ricci flow (1.3) develops no removable singularities at finite time, then the solution of (1.3) exists for all time, converges exponentially fast to r and does not develop removable singularities at time infinity. (3) If R ∈ C(V) is admissible, R(r) = R with r ∈ Ω and R ≤ 0, then any solution of the extended combinatorial Ricci flow dg i = (Ri−R(cid:101)i)gi dt exists for all time and converges exponentially fast to r. 6 Wecanalso introduceacombinatorialα-Ricciflowtostudy theconstant and prescrib- ing curvature problem for the α-curvature R . The results are parallel to Theorem 1.3 α and Theorem 1.4 respectively. The precise statements of the results are given in Theorem 5.4 and Theorem 5.5 respectively, so we do not list them here. The paper is organized as follows. In Section 2, we establish the framework and recall some basic facts on inversive distance circle packing metrics and the classical Gauss curvature,thenweintroducethenewdefinitionofcombinatorialcurvaturefortriangulated surfaces with inversive distance circle packing metrics. In Section 3, we give a detailed proof of Theorem 1.1. We first prove the local rigidity of the new combinatorial curvature R with respect to the inversive distance circle packing metricsbyvariationalprinciples,i.e. Theorem3.2. Thenweprovetheglobalrigidityofthe newcombinatorialcurvaturerwithrespecttotheinversivedistancecirclepackingmetrics, i.e. Theorem 3.3, using the extension of convex functions by constants in [5, 22]. Theorem 3.2 and Theorem 3.3 together generate Theorem 1.1. We further give a counterexample, i.e. Example 1, which shows that the global rigidity in Theorem 3.3 is sharp. In Section 4, we first pose the combinatorial Yamabe problem of the combinatorial curvature R with respect to the inversive distance circle packing metrics and then we introduce the combinatorial Ricci flow to study the combinatorial Yamabe problem. In subsection 4.1, we study the behavior of the Euclidean combinatorial Ricci flow. We first derive the evolution of R along the Euclidean combinatorial Ricci flow (1.1) in Lemma 4.2, then we study the necessary condition for the convergence of the flow (1.1) and obtain Proposition 4.3. To study the long time behavior of the flow , we introduce the notion of singularities in Definition 4.4. Then we get Theorem 4.8, Theorem 4.9 and Theorem 4.10 with the aid of Ricci potential function and extended combinatorial Ricci flow. Lemma 4.2, Proposition 4.3, Theorem 4.8, 4.9 and 4.10 together generate Theorem 1.3. We also usetheEuclideancombinatorialRicciflowtostudytheprescribingcurvatureproblemand obtain Theorem 4.11. In subsection 4.2, we study the hyperbolic prescribing curvature problem using the hyperbolic combinatorial Ricci flow. We obtain the necessary condition for the convergence of the hyperbolic flow in Proposition 4.14 and the main result on hyperbolic prescribing curvature problem in Theorem 4.18. Theorem 4.11, Proposition 4.14 and Theorem 4.18 together generate Theorem 1.4. In Section 5, we generalize the definition of R-curvature to α-curvature and then study therigidityofthecurvature,theconstantcurvatureproblemandtheprescribingcurvature problem. We obtain Theorem 1.2, Theorem 5.4 and Theorem 5.5 in this section. 7 2 Definition of combinatorial Gauss curvature In this section, we give the preliminaries that are needed in the paper, including the notations, the definition of inversive distance circle packing metrics. Then we introduce the new combinatorial Gauss curvature for inversive distance circle packing metrics. 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 I : E → [0,+∞) be a function assigning each edge {ij} a weight I ∈ [0,+∞), which is denoted as I ≥ 0 in the ij paper. The triple (M,T,I) will be referred to as a weighted triangulation of M in the following. All the vertices are ordered one by one, marked by v ,··· ,v , where N = V(cid:93) 1 N is the number of vertices, and we often use i to denote the vertex v for simplicity in the i following. We use i ∼ j to denote that 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 RN and f is the value of f at i. And we use C(V) to i denote the set of functions defined on V. Each map r : V → (0,+∞) is a circle packing, which could be taken as the radius r i of a circle attached to the vertex i. Given (M,T,I), we attach each edge {ij} the length (cid:113) l = r2+r2+2r r I (2.1) ij i j i j ij for Euclidean background geometry and l = cosh−1(cosh(r )cosh(r )+I sinh(r )sinh(r )) (2.2) ij i j ij i j for hyperbolic background geometry. Then I is the inversive distance of the two circles ij centered at v and v with radii r and r respectively. If I ∈ [0,1], the two circles i j i j ij intersect at an acute angle and we can take I = cosΦ with Φ ∈ [0, π] and then the ij ij ij 2 inversive distance circle packing is reduced to Thurston’s circle packing. However, for general weight I ∈ [0,+∞), in order that the lengths l ,l ,l for a face ∆ijk ∈ F ij ij jk ik satisfy the triangle inequalities, the admissible space of the radius Ω = {(r ,r ,r )|l < l +l ,l < l +l ,l < l +l } (2.3) ijk i j k ij ik kj ik ij jk jk ji ik is not R3 . In fact, it is proved [18] that the admissible space Ω is a simply connected >0 ijk open subset of R3 . Note that the set Ω maybe not convex. Set >0 ijk Ω = ∩ Ω = {r ∈ RN |l < l +l ,l < l +l ,l < l +l ,∀∆ijk ∈ F} (2.4) ∆ijk∈F ijk >0 ij ik kj ik ij jk jk ji ik tobethespaceofadmissibleradiusfunction. ΩisobviouslyanopensubsetofRN . Every >0 r ∈ Ω is called a circle packing metric. For more information on inversive distance circle 8 packing metrics, the readers can refer to Stephenson [25], Bowers and Hurdal [4] and Guo [18]. Now suppose that for each face ∆ijk ∈ F, the triangle inequalities are satisfied, i.e. r ∈ Ω, then the weighted triangulated surface (M,T,I) could be taken as gluing many triangles along the edges coherently, which produces a cone metric on the triangulated surface with singularities at the vertices. To describe the singularity at the vertex i, the classical discrete Gauss curvature is introduced, which is defined as K = 2π− (cid:88) θjk, (2.5) i i (cid:52)ijk∈F wherethesumistakenoverallthetriangleswithiasoneofitsvertices. DiscreteGaussian curvature K satisfies the following discrete version of Gauss-Bonnet formula [7] i (cid:88) K = 2πχ(M)−λArea(M), (2.6) i i∈V whereλ = 0,−1inthecaseofEuclideanandhyperbolicbackgroundgeometryrespectively. Guo [18] further proved the following property for the classical combinatorial Gauss cur- vature. Lemma 2.1. (Guo [18]) Given (M,T,I) with inversive distance I ≥ 0. Set u = lnr2 i i for the Euclidean background geometry, u = lntanh2 ri for the hyperbolic background i 2 geometry and L = ∂(K1,···,KN). Then ∂(u1,···,uN) (1) L is symmetric and positive semi-definite with rank N −1 and kernel {t1|t ∈ R} on Ω for the Euclidean background geometry; (2) L is symmetric and positive definite on Ω for the hyperbolic background geometry. Lemma 2.1 implies the local rigidity of the inversive distance circle packing metric obtained in [18]. Based on Guo’s work on local rigidity [18] and Bobenko, Pinkall and Springborn’s work [5] on convex extension of functions, Luo [22] proved the following global rigidity for inversive distance circle packing metrics. Theorem 2.2. (Luo [22]) Given a weighted closed triangulated surface (M,T,I) with inversive distance I ≥ 0. (1) A Euclidean inversive distance circle packing metric on (M,T,I) is uniquely deter- mined by its combinatorial Gauss curvature K up to scaling. (2) A hyperbolic inversive distance circle packing metric on (M,T,I) is uniquely deter- mined by its combinatorial Gauss curvature K. 9 As noted in [15], the classical definition of combinatorial Gauss curvature K with i Euclidean background geometry in (2.5) has two disadvantages. The first is that the classical combinatorial Gauss curvature is scaling invariant, i.e. K (λr) = K (r) for any i i λ > 0; The second is that, as the triangulated surfaces approximate a smooth surface, the classical combinatorial curvature K could not approximate the smooth Gauss curvature, i as we obviously have K tends zero. Motivated by the two disadvantages, we introduce a i newcombinatorialGausscurvaturefortriangulatedsurfaceswithThurston’scirclepacking metrics in [15] and we can generalize the curvature to the case of inversive distance circle packing metrics here. For the following applications, we always set (cid:40) r , Euclidean background geometry i s (r) = (2.7) i tanh ri, hyperbolic background geometry 2 in this paper. Then we can generalize the definition of combinatorial Gauss curvature to triangulated surfaces with inversive distance circle packing metrics as follows. Definition 2.3. Given a triangulated surface (M,T) with inversive distance I ≥ 0 and a circle packing metric r ∈ Ω, the combinatorial Gauss curvature at the vertex i is defined to be K i R = , (2.8) i s2 i where K is classical combinatorial Gauss curvature at i given by (2.5) and s is given by i i (2.7). AstheinversivedistancegeneralizesThurston’sintersectionangle,theDefinition2.3of combinatorial Gauss curvature naturally generalizes the definition of combinatorial Gauss curvature in [15, 16]. It is obvious that the curvature R is an elementary function of r i and then obviously a smooth function of the radius function r. In the following, we often refer to the combinatorial Gauss curvature in Definition 2.3 as R-curvature for short. As the Riemannian metric tensor is a positive definite symmetric 2-tensor in Rieman- nian geometry, we can take g = r2 as the analogue of the Riemannian metric in the i i combinatorial setting with Euclidean background geometry. Then we have R (λg) = λ−1R (g) i i for any constant λ > 0. Furthermore, Example 1 in [15] shows that the combinatorial Gauss curvature R could approximate the smooth Gauss curvature up to a uniform con- i stant π for the Euclidean background geometry. Both facts indicate that R is a good i candidate as the combinatorial Gauss curvature on triangulated surfaces with inversive distance circle packing metrics. 10