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2-DIMENSIONAL COMBINATORIAL CALABI FLOW IN HYPERBOLIC BACKGROUND GEOMETRY 3 1 HUABINGE, XU XU 0 2 n a J Abstract. Fortriangulatedsurfaceslocallyembeddedinthestandard hyperbolic space, we introduce combinatorial Calabi flow as the nega- 8 2 tive gradient flow of combinatorial Calabi energy. We prove that the flowproducessolutionswhichconvergetoZCCP-metric(zerocurvature ] circle packing metric) if the initial energy is small enough. Assuming G the curvature has a uniform upper bound less than 2π, we prove that D combinatorial Calabi flow exists for all time. Moreover, it converges to ZCCP-metric if and only if ZCCP-metric exists. . h t a 1. Introduction m [ Consider a compact surface X with a triangulation T = V,E,F , where 1 thesymbolsV,E,F representthesetofvertices,edgesandfa{cesresp}ectively. v A positive function r : V (0,+ ) defined on the vertices is called a 5 → ∞ circle packing metric and a function Φ : E [0,π/2] is called a weight on 0 → 5 the triangulation. Throughout this paper, a function defined on vertices is 6 regarded as a column vector and N = V# is used to denote the number 1. of vertices. Moreover, all vertices, marked by v1,...,vN, are supposed to be 0 ordered one by one and we often write i instead of v . Thus we may think i 3 of circle packing metrics as points in RN , N times of Cartesian product of 1 >0 (0, ). : ∞ v i Given (X,T,Φ), every circle packing metric r determines a piecewise lin- X earmetriconX byattachingedgee alengthl = r2+r2+2r r cos(Φ ). r ij ij i j i j ij a Thislength structuremakes each triangleinT isomqetric toan Euclideantri- angle. Furthermore, the triangulated surface (X,T) is composed by gluing many Euclidean triangles coherently. This case is called Euclidean background geometry in [1], where the cases of hyperbolic background geometry and spherical background geometry are also studied. In these cases, the length l of the edge is determined by the hyperbolic and spherical cosine ij law respectively, coshl = coshr coshr +sinhr sinhr cosΦ , for H2, ij i j i j ij cosl = cosr cosr sinr sinr cosΦ , for S2. ij i j i j ij − Inspired by [1] and [3], we study the negative gradient flow of discrete Calabi energy on X in hyperbolic background geometry. The flow is called 1 2 HUABINGE,XUXU combinatorial Calabi flow [3]. We use (X,T,Φ,H2) to denote the space we want to study in the following in hyperbolicbackground geometry, where X isaclosedsurface,T isafixedtriangulationofX,Φisafixedweightfunction defined on edges, and H2 represents hyperbolic background geometry . It was proved by Thurston[7]that, whenever i,j,k F, thesethreepositive { } ∈ numbers l ,l ,l satisfy the triangle inequalities. Thus the combinatorial ij ik jk triangle i,j,k with lengths l ,l ,l form a hyperbolic triangle in H2. ij ik jk { } For the hyperbolic triangle v v v , the inner angle of this triangle at v i j k i △ jk is denoted by θ , and the combinatorial curvature K at v is defined as i i i jk K = 2π θ . i − i {i,j,k}∈F X jk jk Notice that θ can be calculated by hyperbolic cosine law, thus θ and K i i i are elementary functions of the circle packing metric r. For the hyperbolic triangle v v v , we have θjk + θik + θij = π △ i j k i j k − Area( v v v ). Using this formula, we have the combinatorial Gauss- i j k △ Bonnet formula [1] N K = 2πχ(X)+Area(X). (1.1) i i=1 X 2. Combinatorial Calabi Flow on Surfaces 2.1. The combinatorial Calabi energy. Similarto[3],thecombinatorial Calabi energy in hyperbolic background geometry is defined as N (r)= K 2 = K2. (2.1) i C k k i=1 X Remark 2.1. In Euclidean background geometry, the average curvature N K = K /N = 2πχ(X)/N is a combinatorial invariant. Then av i=1 i PN N N (2πχ(X))2 (K K )2 = K2 NK2 = K2 . i − av i − av i − N i=1 i=1 i=1 X X X Therefore, it makes no difference to define combinatorial Calabi energy as (r) = N K2 or as (r) = N (K K )2, since they differ only by C i=1 i C i=1 i − av a combinatorial constant. However, in hyperbolic background geometry, the P 2πχ(XP)+Area(X) average curvature K = contains the area term and hence av N is no longer a combinatorial invariant. Therefore, the combinatorial Calabi energy we define here do not contain K term. av Set u = lntanh ri, then the map u= u(r) maps RN to RN homeomor- i 2 >0 <0 phically. In the following we sometimes treat functions in r , and sometimes i in u . i 2-DIMENSIONAL COMBINATORIAL CALABI FLOW IN HYPERBOLIC BACKGROUND GEOMETRY3 The gradient of the combinatorial Calabi energy (u) = N K2 is C i=1 i ∂K1 ∂KN K P ∇u1C ∂u1 ··· ∂u1 1 u =  ...  = 2 ... ... ...  ... = 2LTK, ∇ C ∇uNC ∂∂uKN1 ··· ∂∂KuNNKN      where K = (K , ,K )T and 1 N ··· ∂K1 ∂K1 L = (Lij)N×N = ∂(K1,··· ,KN) =  ∂u...1 ·.·.·. ∂u...N  (2.2) ∂(u , ,u ) 1 ··· N ∂KN ∂KN  ∂u1 ··· ∂uN    is the Jacobian of the curvature map K = K(u). In [3] and [4], we had interpret the Jacobian of the curvature map K to be a type of discrete Laplace operator. Here, (2.2) is also a type of discrete Laplacian, called discrete dual-Laplacian in hyperbolic background geometry and written as H2-DDL for short. 2.2. The Combinatorial Calabi flow. Definition 2.1. Given (X,T,Φ,H2), where X, T, Φ [0,π/2] are defined ∈ as before. The combinatorial Calabi flow in hyperbolic geometry background is defined as u˙(t) = LTK. (2.3) − Proposition 2.1. Combinatorial Calabi flow (2.3) is the negative gradient flow of Combinatorial Calabi energy and the solution of (2.3) exists locally. Moreover, combinatorial Calabi energy is descending along the flow. Proof: BythedefinitionofcombinatorialCalabiflow(2.3),wehaveu˙(t) = LTK = 1 . The flow is a system of ODEs. Therefore, given any − −2∇uC initial metric u(0) RN , or equivalently r(0) RN , the solution of Calabi ∈ <0 ∈ >0 flow exists for the interval t [0,ǫ), where ǫ is a positive number small ∈ enough. By direct computation, we have ′(t) = 1 2 0. Thus the C −2k∇uCk ≤ combinatorial Calabi energy is descending along the flow. Q.E.D. Proposition 2.2. The combinatorial curvature evolves according to K˙ = LLTK. − Proof: K˙ = ∂(K1,···,KN)u˙ =Lu˙ = LLTK. Q.E.D. ∂(u1,···,uN) − Denote Γ(u) = LTK, then the combinatorial Calabi flow equation (2.3) − could bewritten as u˙ = Γ(u), which is an autonomous system. Thefunction Γ(u) (or the matrix L) determines all the properties of the autonomous system. 4 HUABINGE,XUXU 3. Discrete Dual-Laplacian 3.1. Discrete dual-Laplacian. We derive the explicit form of L = ∂Ki ij ∂uj first. As ∂θijk = ∂θjik and ∂Ki = ∂Kj [1], we have L = L . We write j ∼ i ∂uj ∂ui ∂uj ∂ui ij ji in the following if the vertices i and j are adjacent. (1) If j ∼ i, we have ∂K ∂K i i = sinhr j ∂u ∂r j j jk ∂ 2π θ − {i,j,k}∈F i = sinhr (cid:16) P∂rj (cid:17) j jk ∂θ = i sinhr j − ∂r j {i,j,k}∈F X jk jl ∂(θ +θ ) = i i sinhr , j − ∂r j where k,l are the vertices such that i,j,k , i,j,l are adjacent { }{ } ∂θjk ∂θjl faces. Set B = i sinhr + i sinhr , then ij ∂rj j ∂rj j ∂K i = B , j ∼ i. (3.1) ij ∂u − j (2) If j = i, we have ∂K ∂K i i = sinhr i ∂u ∂r i i jk ∂ 2π θ − {i,j,k}∈F i = sinhr (cid:16) P∂ri (cid:17) i jk ∂θ = i sinhr i − ∂r i {i,j,k}∈F X ∂ θik +θij +Area( v v v ) j k △ i j k = sinhr (cid:16) ∂r (cid:17) i i {i,j,k}∈F X jk kj ∂θ ∂θ ∂Area( v v v ) = i sinhr + i sinhr + △ i j k sinhr j k i ∂r ∂r ∂r j k ! i {i,j,k}∈F {i,j,k}∈F X X jk jl ∂θ ∂θ ∂ = i sinhr + i sinhr +sinhr Area( v v v ) . j j i i j k ∂r ∂r ∂r  △  j∼i j j ! i {i,j,k}∈F X X   2-DIMENSIONAL COMBINATORIAL CALABI FLOW IN HYPERBOLIC BACKGROUND GEOMETRY5 Set ∂ A = sinhr Area( v v v ) , i i i j k ∂r  △  i {i,j,k}∈F X then we have   ∂K i = A + B . (3.2) i ij ∂u i j∼i X (3) If j ≁ i and j = i, then 6 ∂K i = 0. (3.3) ∂u j Set A= diag A , ,A and L = (L ) , where 1 ··· N B B ij N×N (cid:8) (cid:9) B , (cid:0)j = i,(cid:1) ik k∼i (LB)ij =  PBij, j i,  −0, j ∼≁ i, j = i, 6 then we have   Theorem 3.1. The matrix L is positive definite and L = A+L , B where A is positive definite and L is semi-positive definite. B Proof: Combining the formulas (3.1), (3.2) and (3.3), we get A + B , j = i, i ik k∼i Lij =  BP, j i, ij  −0, j ∼≁ i, j = i. 6 By Lemma 2.2. in [1], weknow Ai > 0, Bij > 0 for j i. By Lemma 3.10. ∼ in [1], we know the matrix L is semi-positive definite. Thus L is positive B definite. Q.E.D. The classical discrete Laplace operator “∆” is often written as the following form ([2]) ∆f = ω (f f ). i ij j i − j∼i X Notice that the weight ω can be arbitrarily selected for different purpose. ij Using a special weight which comes from dual structure of circle patterns, we established ∆ = L in [3] and [4]. Formally, we also define discrete − Laplace operator ∆ = L in this paper. Both ∆ and L act on function f − by matrix multiplication, that is ∆f = Lf, − 6 HUABINGE,XUXU or in component form ∆f = B (f f ) A f . i ij j i i i − − j∼i X It’s interesting that the combinatorial Calabi flow (2.3) can be written as dr du i i = sinhr ∆K or = ∆K . i i i dt dt du Its matrix form = ∆K is similar to the 2-dimension smooth Calabi flow dt ∂g = ∆K. ∂t Remark 3.1. (Global Rigidity) Consider the combinatorial Ricci potential u N f(u) , K du , u RN , i i ∈ <0 Zu0 i=1 X where u is an arbitrarily point in RN . The integral is well-defined, since 0 <0 N K du is a closed differential form. Moreover, because Hess(f) = i=1 i i ∂(K1,...,KN) = L is positive definite by Theorem 3.1, f = K : RN RN P∂(u1,...,uN) ∇ <0 → is an embedding. Thus the combinatorial curvature determines u and hence the circle packing metric r uniquely. This property is called global rigidity [1, 7]. 3.2. Some estimates of the discrete Laplacian with Φ 0. We will ≡ establish some estimates of the elements of discrete Laplacian if the weight function Φ 0. ≡ Lemma 3.1. For any x,y,z >0, sinh(2x+y+z) sinhxsinhysinhz 0< <cosh1. (3.4) sinh(x+y)sinh(x+z)s sinh(x+y+z) As the proof of the lemma is elementary and tedious, we postpone it to Appendix A. Corollary 3.1. Suppose a hyperbolic triangle ijk is configured by circle △ packing metric r ,r ,r > 0 with zero weight. Then i j k jk ∂θ 1 0 < i sinhr < , j ∂r 2 j jk ∂θ 0 < i sinhr < cosh1, i − ∂r i ∂Area( ijk) 0 < △ sinhr < cosh1. i ∂r i 2-DIMENSIONAL COMBINATORIAL CALABI FLOW IN HYPERBOLIC BACKGROUND GEOMETRY7 Proof: If Φ 0, we can derive the following expressions directly by a ≡ lengthy but ordinary calculation using hyperbolic cosine law: ∂θjk 1 sinhr sinhr sinhr i sinhr = i j k, j ∂rj sinh(ri +rj)s sinh(ri +rj +rk) jk ∂θ sinh(2r +r +r ) sinhr sinhr sinhr i sinhr = i j k i j k. i ∂ri −sinh(ri+rj)sinh(ri +rk)s sinh(ri+rj +rk) Therefore, we have jk ∂θ 1 0 < i sinhr < . j ∂r 2 j By Lemma 3.1, we have jk ∂θ 0 < i sinhr < cosh1. i − ∂r i By the formula θjk +θik +θij = π Area( ijk) and Lemma 3.1, we have i j j − △ ∂Area( ijk) 0< △ sinhr i ∂r i sinh(2r +r +r ) sinh(r +r ) sinh(r +r ) sinhr sinhr sinhr i j k i j i k i j k = − − sinh(ri +rj)sinh(ri +rk) s sinh(ri +rj +rk) < cosh1. Q.E.D. Remark 3.2. For arbitrary weight Φ : E [0,π/2], Bennett Chow and → Luo Feng [1] had got that ∂θjk sinhr cosθki′ cosθik′cosθik i sinhr = j j − j j , ∂rj j sinhlij sinθjik ∂θjk sinhr cosθikcosθjk′ +cosθijcosθkj′ i sinhr = i j i k i . ∂ri i −sinhlij sinθjik By Corollary 3.1, we can get the following proposition easily. Proposition 3.1. For fixed (X,T,Φ 0,H2), we have ≡ 0 < B < 1, ij 0 < A < d cosh1 dcosh1, i i ≤ where d = max d , d is the degree at vertex i. i i 1≤i≤N{ } Corollary 3.2. Given (X,T,Φ 0,H2), then r(t), the solution of the com- ≡ binatorial Calabi flow (2.3), has a positive lower bound in finite time. 8 HUABINGE,XUXU Proof: Note that (2 d)π < K < 2π and∆K = B (K K ) A K , i i ij j i i i − − − j∼i then by combinatorial Calabi flow equation dui = ∆PK and the estimates of dt i B and A in Proposition 3.1, we have ij i u (t) u (0) < c t, i i 2 | − | where c is a constant depending on the combinatorial structure of the tri- 2 angulation. As u = lntanh ri, we have i 2 r (t) 0< c e−c2t < tanh i < c ec2t, (3.5) 1 3 2 wherec = min tanh ri(0) ,c = max tanhri(0) . Noticethatc ec2t 1 1≤i≤N 2 3 1≤i≤N 2 3 → + as t + , wnhile tanhois always uppenr boundedoby 1, the estimate on ∞ → ∞ the right hand side of (3.5) will be of no use if t is big enough. Thus we can only get an estimate of the lower bound of r (t), i 1+c e−c2t 1 r (t) ln > 0. i ≥ 1 c e−c2t 1 − The estimate shows that the solution r(t) of the combinatorial Calabi flow (2.3) has a positive lower bound in finite time. Specially, if t [0,T], then ∈ 1+c e−c2T 1 we have r (t) > c > 0, where c = ln > 0. Q.E.D. i T T 1 c e−c2T 1 − 4. Local Convergence of Combinatorial Calabi Flow 4.1. ZCCP-metric (zero curvature circle packing metric). Zero curvature circle packing metric is a circle packing metric that determines constant zero curvature, written as ZCCP-metric for short. we denote it as r∗ (or u∗ in u-coordinate) if it exists. The ZCCP-metric is important for the combinatorial Calabi flow (2.3), because it is the unique stable point of the autonomous system u˙ = Γ(u) = LK. − There are combinatorial obstructions for the existence of ZCCP-metric, which was found by Thurston [7]. We quote the combinatorial conditions here. Proposition 4.1. For fixed (X,T,Φ,H2), where X, T, Φ [0,π/2] are ∈ defined as before, there exists ZCCP-metric if and only if the following two combinatorial conditions are satisfied simultaneously: (1) For any three edges e ,e ,e forming a null homotopic loop in X, if 1 2 3 3 Φ(e ) π, then e ,e ,e form the boundary of a triangle of T; i=1 i ≥ 1 2 3 (2) For any four edges e ,e ,e ,e forming a null homotopic loop in X, 1 2 3 4 iPf 4 Φ(e ) 2π, then e ,e ,e ,e form the boundary of the union i=1 i ≥ 1 2 3 4 of two adjacent triangles. P 2-DIMENSIONAL COMBINATORIAL CALABI FLOW IN HYPERBOLIC BACKGROUND GEOMETRY9 4.2. Convergence with small initial Calabi energy. Lemma 4.1. Given (X,T,Φ,H2), where X, T, Φ [0,π/2] are defined ∈ as before. If the solution of the combinatorial Calabi flow (2.3) lies in a compact subset of RN , then the solution exists for all t [0,+ ) and the <0 ∈ ∞ solution has exponential convergence rate. Proof: Denote the solution by u(t). u(t) must exists for all t [0,+ ), otherwise u(t) will attain the boundary of RN . This contradict∈s the c∞on- <0 dition u(t) RN . By the same condition, the eigenvalue of matrix { } ⊂⊂ <0 L has a uniform positive lower bound λ , that is, λ L(t) λ along the 1 1 ≥ combinatorial Calabi flow. Hence (cid:0) (cid:1) ′(t)= 2KTK˙(t) = 2KTL2K 2λ2KTK = 2λ2 (t). C − ≤ − 1 − 1C So (t) (0)exp−2λ21t and Ki(t) K(t) = (t) (0)exp−λ21t. As C ≤C ≤ C ≤ C u(t) RN , we know L and sinhr are bounded along the combinato- { } ⊂⊂ <0 ij(cid:12) (cid:12) (cid:12) i (cid:12) p p rial Calabi flow. Hence (cid:12) (cid:12) (cid:12) (cid:12) dui = K = L K c exp−λ21t, i ij j 1 dt △ ≤ (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)Xj (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)dri(cid:12) = (cid:12)sinh(cid:12)r (cid:12)K = sinh(cid:12) r L K c exp−λ21t, i i i ij j 2 dt △ ≤ (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Xj (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) where c1, c(cid:12)2 ar(cid:12)e po(cid:12)sitive consta(cid:12)nts.(cid:12) This implies tha(cid:12)t the solution converges with exponential rate. Q.E.D. Assuming the Calabi flow exists for all time and converges, then we have Proposition 4.2. Given(X,T,Φ,H2), where X, T, Φ [0,π/2] are defined ∈ as before. If the solution of Calabi flow (2.3) exists for t [0,+ ) and ∈ ∞ converges for some initial circle packing metric, then (1) the solution has exponential convergence rate; (2) there exists ZCCP-metric on (X,T,Φ,H2); (3) the Euler characteristic of X is negative, i.e. χ(X) < 0. Proof: As the solution of combinatorial Calabi flow converges for some initial data, the solution u(t) will lie in a compact subset in RN from some <0 time T. Thus the solution has exponentially convergence rate by Lemma 4.1. Suppose r(t),t [0,+ ), is the solution to the combinatorial flow, K(t) ∈ ∞ and L(t) are the corresponding curvature and discrete Laplacian along the flow. Bythehypothesisintheproposition,wehaver(+ )= lim r(t) t→+∞ RN exists, so does K(+ ) = lim K(t) and L(+∞ ) = lim L(t)∈. >0 ∞ t→+∞ ∞ t→+∞ Thus the combinatorial Calabi energy (t) = N K2(t) converges as C i=1 i t . The derivative of the combinatorial Calabi energy → ∞ P ′(t)= 2KTK˙(t) = 2KTL2K C − 10 HUABINGE,XUXU converges also. As ′(t) 0, we have ′(+ ) = lim C′(t) 0. Then t→+∞ (+ ) exists withC(+ ≤) 0 and ′(C+ ∞) exists with ′(+ )≤ 0. This iCmpl∞ies that ′(+ )C= 0∞, w≥hich is eqCuiva∞lent to C ∞ ≤ C ∞ KTL2K(+ )= 0. ∞ By the positivity of L, we have K(+ ) = 0. Thus r(+ ) is a ZCCP- ∞ ∞ metric. By the combinational Gauss-Bonnet formula (1.1), we have χ(X) < 0. Q.E.D. Theorem 4.1. For fixed (X,T,Φ,H2), where X, T, Φ [0,π/2] are defined as before, suppose there exists a circle packing metric∈u∗ that determines constant zero curvature, then the solution of combinatorial Calabi flow (2.3) exists for all t [0,+ ) and converges exponentially fast to u∗ if the initial ∈ ∞ combinatorial Calabi energy is small enough. Proof: We claim that u∗ is the only asymptotically stable point of the combinatorial Calabi flow (2.3). In fact, Du∗( LK)= L2(u∗) < 0. − − So the claim is true by the Lyapunov stability theorem. This implies the solution of the combinatorial Calabi flow (2.3) exists for t [0,+ ) and converges exponentially to u∗ if the initial metric u(0) is cl∈ose to∞u∗. By Remark 3.1, this is equivalent to the initial combinatorial Calabi energy is small enough. Q.E.D. Similarly, we can prove Theorem 4.2. Given any user prescribed combinatorial curvature K¯ on fixed (X,T,Φ,H2), where X, T, Φ [0,π/2] are defined as before. If K¯ is admissible, that is, there exists a c∈ircle packing metric r¯ with K¯ = K(r¯), and the initial modified combinatorial Calabi energy K(0) K¯ 2 is small k − k enough, then the solution of the following modified combinatorial Calabi flow u˙ = L(K¯ K) (4.1) − exists for t [0,+ ) and converges exponentially fast to u¯. ∈ ∞ The modified combinatorial Calabi flow (4.1) provides an algorithm to compute circle packing metrics with user prescribed combinatorial curva- tures. In fact, any algorithm, aiming at minimizing the modified combinatorial Calabi energy K K¯ 2, tends to find circle packing metric r¯ k − k automatically. 5. Global convergence for K < C < 2π i Along the combinatorial Calabi flow, the evolution equation of combinatorial curvature K˙ = ∆2K contains the ∆2 term. It is a fourth order − equation. Therearenotefficient “Maximal Principle”for fourthorder equa- tions, this may be the most important differences between the combinatorial Calabi flow and the combinatorial Ricci flow introduced in [1]. We give