The C∞-convergence of SG circle patterns to the Riemann mapping
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(2) 1352. S.-Y. Lan, D.-Q. Dai / J. Math. Anal. Appl. 332 (2007) 1351–1364. should converge to f after suitable normalization. This conjecture was proved in [10]. Refinements and generalizations were given in [4–8,11,13], etc. For the study of circle configurations, classical circle packings comprised by disjoint open disks are generalized to circle patterns, where the disks may overlap. Schramm [12] introduced circle patterns with the combinatorics of the square grid and showed that SG circle patterns converge to meromorphic functions. In comparison with hexagonal circle packings, the framework of SG circle patterns is more tractable. Doyle has constructed entire immersed hexagonal circle packings analogous to the exponential map and conjectured that these packings (i.e., Doyle spirals) are the only entire immersed hexagonal circle packings (see [3]). For SG circle patterns, this conjecture is false because there is an SG circle pattern analogous to the entire function z √ 2 erf ( iz ), where erf (z) = e−w dw (see [12]). Next, the collection of entire SG circle patterns in the sphere is infinite-dimensional and SG circle patterns can be used to exhibit some explicit finite-dimensional families (see [1,2,12]). In [7] He and Schramm proved that hexagonal disk packings converge in C ∞ (Ω) to the Riemann mapping. Then it is natural to ask whether there is similar C ∞ -convergence for SG circle patterns. Here, we shall give an affirmative answer to this question. Our method involves modifications and generalizations of those in [7]. We define two discrete Schwarzians, and their boundedness is proved directly using the formulas of Laplacian and Taylor for the Schwarzians, while in [7] only one discrete Schwarzian is needed and the proof of its boundedness is equivalent to that of the third order derivatives (see [7, Lemma 6.1]). Moreover, we construct Möbius transformations by means of the three intersection points in each circle, whereas in [7] they are defined by the three tangent points of three circles in the hexagonal circle packings (see [7, §9]). We also note that recent work of Matthes [9] proves C ∞ -convergence of SG circle patterns to the conformal map. However, our approach is different from that of Matthes. First, Matthes constructs the approximating functions by solving a discrete Cauchy problem. While we define directly the approximating mappings using the relationship between two combinatorially equivalent SG circle patterns. Secondly, any order partial derivatives of the discrete functions in [9] are defined as the respective difference quotients. Here the partial derivatives of the discrete functions are defined by the discrete directional derivatives. Thirdly, it is proved [9] that the discrete solutions converge to the solution of analytic Cauchy problem in C ∞ , which lead to C ∞ -convergence of the corresponding circle patterns. Whereas here it is shown the Möbius transformations converge in C ∞ . Using the relation between the approximating mappings and the Möbius transformations, we get that the approximating functions converge to the Riemann mapping in C ∞ . This paper is organized as follows. We shall introduce briefly some terminology and definitions related to SG circle patterns and discrete differential operators in Section 2. In Section 3, using the Möbius invariants for SG circle patterns, we give the definition of discrete Schwarzians of SG circle patterns. Then we construct suitable Möbius transformations that can be expressed in terms of the discrete Schwarzians. In Section 4, we shall show that the discrete Schwarzians are uniformly bounded in C ∞ . We shall show C ∞ convergence in Section 5. 2. SG circle patterns and discrete differential operators In this section we shall introduce briefly some terminology and definitions related to SG circle patterns and discrete differential operators (see [7,12] for more details). For each positive number > 0, let SG be the cell complex whose vertices form the square lattice V = Z + iZ = {v = n + im: (n, m) ∈ Z × Z}, whose edges are the pair [v, v ] such.
(3) S.-Y. Lan, D.-Q. Dai / J. Math. Anal. Appl. 332 (2007) 1351–1364. 1353. Fig. 1. Cell complex SG . V and Vˆ consist of the black points and the white points, respectively.. Fig. 2. (a) Regular SG circle pattern and (b) general SG circle pattern.. that |v − v | = and v, v ∈ V , and whose 2-cells are the squares {v + x + iy: x, y ∈ [0, ]}, v ∈ V (see Fig. 1). ˆ is said An indexed collection P = {Pv : v ∈ V } of oriented circles in the Riemann sphere C to be a circle pattern for SG or a SG circle pattern (see Fig. 2) if the following three conditions hold. (a) Whenever v, v are neighbors in SG , the corresponding circles Pv , Pv intersect orthogonally. (b) If v1 , v2 are neighbors of a vertex v in SG , and they belong to the same square of SG , then the circles Pv1 , Pv2 are distinct and tangent. (c) Whenever the situation is as in (b) and v2 is neighbor of v, which is one step counterclockwise from v1 , the circular order of the triplet of points Pv ∩ Pv1 − Pv2 , Pv1 ∩ Pv2 , Pv ∩ Pv2 − Pv1 agrees with the orientation of Pv . Clearly, the 1-skeleton √ of SG is the graph of regular SG circle pattern Q whose each circle has radius equal to / 2..
(4) 1354. S.-Y. Lan, D.-Q. Dai / J. Math. Anal. Appl. 332 (2007) 1351–1364. Let W be a subset of V . A vertex v ∈ W is said to be an interior vertex of W if all its neighboring vertices are contained in W . Let int W denote the set of the interior vertices of W . Given a function ρ : W → R, we define the discrete directional derivative ∂ρj : int W → R by ∂j ρ(v) = ρ v + i j − ρ(v) /, for each j ∈ Z4 . The discrete Laplacian of a function ρ : W → R is a function in int W defined by the formula 3 ρ v + i j − ρ(v) . Δ ρ(v) = 1/ 4 2 j =0. Suppose that Ω is a simply connected bounded region in the complex plane. For any differentiable function F : Ω → R, let ∂j F denote the directional derivative F (z + i j s) − F (z) , s→0 z. ∂j F (z) = lim. where j = 0, 1, 2, 3. Let f : Ω → C be some function defined in Ω. For each > 0, let f be ⊂ V , with values in Cd . Assume that for each some function defined on some set of vertices V whenever ∈ (0, δ1 ). z ∈ Ω, there are some δ1 , δ2 > 0 such that {v ∈ V : |v − z| < δ2 } ⊂ V If for every z ∈ Ω and every δ > 0, there are some δ1 , δ2 > 0 such that |f (z) − f (v)| < δ, for every ∈ (0, δ1 ) and every v ∈ V with |v − z| < δ2 , then we say f converges to f , locally uniformly in Ω. Let n ∈ N, and suppose that f is C n -smooth. If for every sequence j1 , j2 , . . . , jk ∈ Z4 with k n we have ∂jk ∂jk−1 · · · ∂j1 f → ∂jk ∂jk−1 · · · ∂j1 f locally uniformly in Ω, then we say that f converges to f in C n (Ω). The functions f are said to be uniformly bounded in C n (Ω) provided that for every compact K ⊂ Ω there is some constant C(K, n) such that ∂ ∂ jk jk−1 · · · ∂j1 f K∩V < C(K, n) whenever k n, and is sufficiently small, where · denotes L∞ -norm. The functions f are uniformly bounded in C ∞ (Ω), if they are uniformly bounded in C n (Ω) for every n ∈ N. 3. The Möbius invariants and discrete Schwarzians In this section, using the Möbius invariants for SG circle patterns (also see [12]), we give the definition of discrete Schwarzians of SG circle patterns. Then we construct suitable Möbius transformations that can be expressed in terms of the discrete Schwarzians. For the cell complex SG , let Vˆ denote the set of centers of squares of SG , i.e., Vˆ = {v + /2 + i/2: v ∈ V } (see Fig. 1). For the sake of convenience, let ωj = i j (/2 + i/2) for ˆ any j = 0, 1, 2, 3. Let P = {Pv : v ∈ V } be a circle pattern for SG on the Riemann sphere C. ˆ Note that the intersection of the four circles Pu+ωj (j = 0, 1, 2, 3) is a single point for any u ∈ V . We denote the point by q(u). For any v ∈ V , note that the four points q(v + ωj ) (j = 0, 1, 2, 3) are all on the circle Pv , so the first Möbius invariant function ν(1) : V → R of P is defined by . ν(1) (v) = − cr q(v + ω0 ), q(v + ω2 ); q(v + ω3 ), q(v + ω1 ) (q(v + ω0 ) − q(v + ω3 ))(q(v + ω2 ) − q(v + ω1 )) . (1) =− (q(v + ω0 ) − q(v + ω1 ))(q(v + ω2 ) − q(v + ω3 )).
(5) S.-Y. Lan, D.-Q. Dai / J. Math. Anal. Appl. 332 (2007) 1351–1364. 1355. Fig. 3. The cell complex SGh consists of these small squares with dotted edges. (1). It is clear that ν (v) does not change when a Möbius transformation is applied to all four points (1) and that ν (v) > 0 because q(v + ω0 ), q(v + ω2 ) separates q(v + ω3 ) and q(v + ω1 ). The (2) second Möbius invariant function ν : Vˆ → C of P is defined by . ν(2) (u) = − cr q(u + ), q(u − ); q(u − i), q(u + i) (2) for each u ∈ Vˆ . Note that ν does not change if we apply a Möbius transformation to P . Let m ) (j = 0, 1, 2, 3) be a Möbius transformation that takes q(u) to ∞. Then the four circles m(Pu+ω j are mapped to lines, and together form a rectangle with vertices m(q(u + )), m(q(u + i)), m(q(u − )) and m(q(u − i)). Hence we deduce easily that. m(q(u + )) − m(q(u − i)) 2 |m(q(u + )) − m(q(u − i))|2 (2) = . (2) ν (u) = − m(q(u + )) − m(q(u + i)) |m(q(u + )) − m(q(u + i))|2 (j ). (2). So ν is real and positive. Here it is pointed out that ν (j = 1, 2) is similar to s in [7, §4]. Now we define the two discrete Schwarzians of P as follows: let −2 (1) (3) ν (v) − 1 h(1) (v) = for each vertex v ∈ V and let −2 (2) ν (u) − 1 h(2) (u) = . (4). for every u ∈ Vˆ . It is easy to see that h = h ≡ 0 if P is a regular SG circle pattern, because (j ) ν(1) = ν(2) ≡ 1 for regular SG circle patterns. And the same, h (j = 1, 2) is also similar to h in [7, §4]. Let SGh denote (1/2)SG + (1 + i)/4, and the set of vertices of SGh is denoted by Vh (see Fig. 3). For any w ∈ Vh , let v ∈ V be the unique vertex of SG that is closest to w. Set u1 = v + 2(w − v), u2 = v + 2i(w − v), u3 = v − 2i(w − v), then u1 , u2 , u3 ∈ Vˆ . Let Mw be the Möbius transformation that takes ∞, 0, 1 to q(u1 ), q(u2 ), q(u3 ), respectively. Set (1). (2). M[w1 ,w2 ] = Mw−11 ◦ Mw2 for each directed edge [w1 , w2 ] of SGh . Let ej (v) ˜ = [v˜ + ωj /2, v˜ + ωj +1 /2] for any v˜ ∈ V ∪ Vˆ and for any j ∈ Z4 , then we have.
(6) 1356. S.-Y. Lan, D.-Q. Dai / J. Math. Anal. Appl. 332 (2007) 1351–1364. Lemma 1. There hold for the following equalities: 1/2 z, Me2 (u) (z) = Me0 (u) (z) = 1 − i h(2) (u) + 1 (2) −1/2 Me3 (u) (z) = Me1 (u) (z) = 1 − i(1/) h (u) + 1 z. (5) (6). for each u ∈ Vˆ and. −1 −1 + 1 /(1 − z), Me2 (v) (z) = Me0 (v) (z) = −2 h(1) (v) + 1 −1 Me3 (v) (z) = Me1 (v) (z) = 2 h(1) /(1 − z) (v) + 1 + 1. for every v. (7) (8). ∈ V .. Proof. For any u ∈ Vˆ , we first consider the directed edge e2 (u) = [u + ω2 /2, u + ω3 /2] ∈ SGh . By the definition of M[w1 ,w2 ] , we get that M[w1 ,w2 ] does not change if we apply a Möbius transformation to P . So we may assume that Mu+ω2 /2 is the identity. It follows from the definition of Mw that q(u) = ∞,. q(u − ) = 0,. q(u − i) = 1.. Hence we deduce that the four points q(u ± ), q(u ± i) form the vertices of a rectangle. From (2), we get that. q(u + ) − q(u − i) 2 , ν(2) (u) = − q(u − i) − q(u − ) which implies. 1/2 1/2 q(u + ) = q(u − i) − i ν(2) q(u − i) − q(u − ) = 1 − i ν(2) .. Since that M[u+ω2 /2,u+ω3 /2] takes ∞, 0, 1 to q(u), q(u−i), q(u+i), respectively, we conclude that 1/2 z. Me2 (u) (z) = 1 − i ν(2) (2). It follows from (4) that Me2 (u) (z) = 1 − i(h (u) + 1)1/2 z. Similarly, we deduce that (2) Me0 (u) (z) = 1 − i(h (u) + 1)1/2 z and that (6) holds. Next, we consider the directed edge e0 (v) = [v + ω0 /2, v + ω1 /2] ∈ SGh for any v ∈ V . We assume without loss of generality that Mv+ω0 /2 is the identity. Then we obtain q(v + ω0 ) = ∞,. q(v + ω1 ) = 0,. q(v + ω3 ) = 1.. By (1), we get. q(v + ω2 ) = 1/ ν(1) (v) + 1 .. Note that M[v+ω0 /2,v+ω1 /2] takes ∞, 0, 1 to q(v + ω1 ), q(v + ω2 ), q(v + ω0 ), respectively, combining with (3), we deduce that −1 −1 −1 −1 Me0 (v) = ν(1) (v) + 1 /(1 − z) = −2 h(1) + 1 /(1 − z), (v) + 1 with the same arguments as above, we deduce that −1 −1 Me2 (v) (z) = −2 h(1) + 1 /(1 − z) (v) + 1 and that (8) holds.. 2.
(7) S.-Y. Lan, D.-Q. Dai / J. Math. Anal. Appl. 332 (2007) 1351–1364. 1357. 4. The C ∞ -boundedness of discrete Schwarzians We shall show that the discrete Schwarzians are uniformly bounded in C ∞ (Ω) in this section.
(8) Ω be the union of squares Without loss of generality, we assume 0 ∈ Ω. For any > 0, let SG √ of SG that are contained in Ω and whose distance to ∂Ω is greater than or equal to 2 /2.
(9) Ω that contains 0 (see Let SGΩ be the closure of the connected component of the interior of SG Fig. 4). The set of vertices of SGΩ is denoted by VΩ , and the set of centers of squares of SGΩ by VˆΩ . It is clear that the 1-skeleton of SGΩ is the graph of some sub-pattern of Q contained in Ω. Let f : Ω → U be a Riemann function, where U is the unit disk. Set ν(1) (v) = 1 + 2 Re Sf (v) for each boundary vertex v ∈ ∂VΩ , where Sf denotes the Schwarzian derivative of f . Then (1) SG Dirichlet principle [12, Theorem 6.2] implies that ν is extended to a solution of the SG (2) (1) Dirichlet problem on SGΩ . Let ν be the companion of ν in the solution of the SG − CR equation [12, §5], i.e.,. (1). (2) ν (v + i)−1 + 1 2 ν (v + ω0 ) = (9) (2) (1) ν (v + ω1 ) ν (v)−1 + 1 and. (2). ν (v + ω0 ) (2). ν (v + ω3 ). =. (1). ν (v + ) + 1. 2. (1). ν (v) + 1. (10). for any v ∈ VΩ , such that ν(2) (ωj ) = 1 + 2 Im Sf (ωj ),. (11). where ωj = i j (/2 + i/2) (j = 0, 1, 2, 3). Then we have Lemma 2. (i) The equality 2 h(1) (v) = Re Sf (v) + · O(1). (12). holds for each v ∈ VΩ . (ii) The equality 2 h(2) (u) = Im Sf (u) + δ (u) · O(1). (13). holds for each v ∈ VˆΩ , where δ (u) denotes the combinatorial distance in SGΩ from u to ω0 . Proof. (i) First, for any v ∈ int VΩ , expand Sf in power series about v and note that 3 jk k=0 i = 0 for j = 1, 2, 3, we obtain 3 2 . j Δ Re Sf = Re 1/ 4 Sf v + i − Sf (v) = O 2 . j =0.
(10) 1358. S.-Y. Lan, D.-Q. Dai / J. Math. Anal. Appl. 332 (2007) 1351–1364. Fig. 4. The cell complex SGΩ is the closure of domain bounded by closed curve L.. Consider the function 2 g1 (v) = h(1) (v) − Re Sf + β|v| ,. where β ∈ (0, 2 ) is some function of . Similar to the proof of [12, Lemma 9.2], we deduce from the Taylor’s formula and the properties of Möbius invariant h(1) that g1 has no maxima in int VΩ if β is chosen as β = C 2 with C > 0 a sufficiently large constant. Since that g1 (v) = β|v|2 = O( 4 ) on ∂VΩ , we get 2 h(1) (v) Re Sf (v) + O . (1). On the other hand, if we let g2 (v) = h (v) − Re Sf (v) − β|v|2 , then similar to the above arguments, we deduce that 2 h(1) (v) Re Sf (v) + O . So it follows that. 2 h(1) (v) = Re Sf (v) + O .. (ii) It follows from (9) that. log ν(2) (v + ω0 ) − log ν(2) (v + ω1 ) = 2 log ν(1) (v + i)−1 + 1 − 2 log ν(1) (v)−1 + 1 .. By (3) and Taylor formula, we deduce that log ν(2) (v + ω0 ) − log ν(2) (v + ω1 ) −1 −1 = 2 log 1 + 2 h(1) + 1 − 2 log 1 + 2 h(1) +1 (v + i) (v) 4 2 (1) = 2 log 2 − 2 h(1) (v + i) − 2 log 2 − h (v) + O 4 2 (1) = 2 h(1) (v) − h (v + i) + O . Hence it follows from (12) that.
(11) S.-Y. Lan, D.-Q. Dai / J. Math. Anal. Appl. 332 (2007) 1351–1364. log ν(2) (v + ω0 ) − log ν(2) (v + ω1 ) = 2 Re Sf (v) − Sf (v + i) + O 4 = 2 Re iSf (v) + O 4 = 3 Re Sf (v) + O 4 = 2 Im Sf (v + i) − 2 Im Sf (v) + O 4 .. 1359. (14). Similarly, we deduce from (10) that log ν(2) (v + ω0 ) − log ν(2) (v + ω3 ) = 2 Im Sf (v + ) − 2 Im Sf (v) + O 4 .. (15). By (11), we deduce that log ν(2) (/2 + i/2) = 2 Im Sf (/2 + i/2) + O 4 .. (16). So we conclude from (14)–(16) that. log ν(2) (u) = 2 Im Sf (u) + δ (u)O 2 ,. which, combining with (4), implies that the equality (13) holds.. 2. Lemma 3. Let v ∈ VΩ , u ∈ VˆΩ , and suppose that the distance δ from v (respectively, u) to ∂Ω is greater than 2. Then h(1) (v) C,. h(2) (u) C. for some constant C = C(δ, f ) which depends only on δ and f . Proof. Note that Re Sf and Im Sf are bounded on compact subset K ⊂ Ω and δ = O(1/) on K. By Lemma 2, we conclude that the lemma holds. 2 Similar to the case of regular hexagonal lattice (see [7, Lemma 7.1]), for regular square lattice we have Lemma 4. Suppose (i) W is a subset of V (or Vˆ ); (ii) u ∈ int W ; (iii) δ is the distance from u to V − W (or Vˆ − W ). If ψ : W → R, then the inequality (17) δ ∂j ψ(u) < 5 ψ + (1/2)δ 2 Δ ψ int W holds for j = 0, 1, 2, 3. Theorem 1. Let n be an integer, and let Vδ (respectively, Vˆδ ) be the set of vertices of VΩ (respectively, VˆΩ ) whose distance to ∂Ω is at least δ for each δ > 0. Then there are constants C = C(n, δ) and μ = μ(n, δ) > 0 such that (1) (2) ∂ ∂ ∂ ∂ < C, <C (18) jn jn−1 · · · ∂j1 h jn jn−1 · · · ∂j1 h Vˆ V δ. δ. holds for each < μ, and j0 , j1 , . . . , jn ∈ Z4 . Proof. The proof proceeds by induction on n. The case n = 0, by Lemma 3, we get that (18) holds. So we suppose that n > 0, and that (18) holds for 0, 1, 2, . . . , n − 1. Let ϕ1 = ∂jn−1 · · · ∂j1 h(1) ,. ϕ2 = ∂jn−1 · · · ∂j1 h(2) ..
(12) 1360. S.-Y. Lan, D.-Q. Dai / J. Math. Anal. Appl. 332 (2007) 1351–1364. Note that the operators Δ and ∂j commute with each other, we have (1) Δ ϕ1 = Δ ∂jn−1 · · · ∂j1 h(1) = ∂jn−1 · · · ∂j1 Δ h ,. and (2) Δ ϕ2 = Δ ∂jn−1 · · · ∂j1 h(2) = ∂jn−1 · · · ∂j1 Δ h .. By Lemma 2, it follows that. 2 Δ h(1) = Δ Re Sf + Δ O(1) ,. 2 Δ h(2) = Δ Im Sf + Δ δ O(1) .. Since that Δ Sf = O( 2 ) and δ = O(1/) on compact subset K ⊂ Ω, we have 2 2 Δ h(2) Δ h(1) = O + O(1), = O + Δ O(1) on compact subset K ⊂ Ω. Note that O(1) ∈ C ∞ (Ω), we deduce that there exists a constant C1 = C1 (δ, n) such that Δ ϕ2 (v) ˆ C1 . Δ ϕ1 (v) C1 , V V δ. δ. and Vˆδ , respectively, it follows from Lemma 4 that Since that |ϕ1 | and |ϕ2 | are bounded on |∂jn ϕ1 | and |∂jn ϕ2 | are bounded on Vδ and Vˆδ , respectively, which completes the induction step. So (18) holds for any integer n and any j0 , j1 , . . . , jn ∈ Z4 . 2 Vδ. 5. The C ∞ -convergence of SG circle patterns In this section we shall show that SG circle patterns converge in C ∞ (Ω) to the Riemann mapping function. (1) (2) For ν and ν given in Section 4, based on the local theory of SG circle patterns [12, Theorem 6.1], there exists an SG circle pattern PΩ = {Pv : v ∈ VΩ } for SGΩ in the unit disk U that has (1) (2) ν and ν as its Möbius invariants. Suppose that PΩ is normalized by Möbius transformation so that q(ω0 ) = f (ω0 ), = i j (1 + i)/2. where ωj and let f (v) denote the. q(ω1 ) = f (ω1 ),. q(ω3 ) = f (ω3 ),. (19). (j = 0, 1, 3). For each v ∈ VΩ , let g (v) be the point q(v + (1 + i)/2), center of the disk Pv , and r (v) the radius of Pv . Then we have. Theorem 2. The discrete functions f (v) and g (v) converge to the conformal mapping f (z) : Ω → U in C ∞ (Ω), and r (v)/ converges in C ∞ (Ω) to |f |, where U denotes the unit disk and |f | is the module of the derivative of f . In order to prove the theorem, we need the following lemma. Lemma 5. For each v ∈ int VΩ , let T = T (v) be the Möbius transformation that takes the three points ω0 , ω1 , ω3 to the points q(v + ω0 ), q(v + ω1 ), q(v + ω3 ), respectively. Then (i) the limit T (z) = lim T (v) →0. exists for any → 0, and the convergence is C ∞ (Ω), (ii) T (z)(0) = f (z).. (20).
(13) S.-Y. Lan, D.-Q. Dai / J. Math. Anal. Appl. 332 (2007) 1351–1364. 1361. Proof. (i) Let B = B be the Möbius transformation that takes ∞, 0, 1 to ω0 , ω1 , ω3 , respectively. Then we have B(z) =. z − (1 + i)/2 . (1 − i)z + i. (21). Recalling the definition of Mw and M[w1 ,w2 ] as in Section 3, we deduce that T (v) ◦ B = Mv+(1+i)/4 and T (v)−1 ◦ T (v + ) = B ◦ M[v+ω0 /2,v+ω0 +/2] ◦ M[v+ω0 +/2,v+ω0 +] ◦ B −1 . −1 Using the usual matrix notation for Möbius transformation and the fact that M[w1 ,w2 ] = M[w , 2 ,w1 ] we obtain from (5), (7) and (21) that. . (2) −(1 + i)/2 −i(h (v + ω0 ) + 1)1/2 1 −1 T (v) ◦ T (v + ) = ◦ 0 1 1−i i. (1) −1 0 ( −2 (h (v + ) + 1)−1 + 1)−1 ◦ −1 −1. −(1 + i)/2 −1 ◦ . 1−i 1. By Lemma 2 and noting that Sf is Lipschitz, we deduce that. 1 −1 + O 2 = I + C(v) + O 2 , T (v) ◦ T (v + ) = −Sf (v)/2 1 where I denotes the identity matrix and. 0 1 . C(v) = −Sf (v)/2 0 This implies. T (v + ) = T (v) + T (v)C(v) + T (v)O 2 .. Similarly, we have T v + i j = T (v) + i j T (v)C(v) + T (v)O 2. (22). (23). for j = 1, 2, 3. We assume, with no loss of generality, that f (0) = 0,. f (0) = 1,. f (0) = 0,. because the statement of Theorem 2 is Möbius invariant. Thus we get from Taylor’s formula that f i j (1 + i)/2 = i j (1 + i)/2 + O 3 for j = 0, 1, 2, 3. From (19) and the definition of T (v), it follows that T (0) i j (1 + i)/2 = i j (1 + i)/2 + O 3 for j = 0, 1, 2, 3, which implies T (0) = I + O()..
(14) 1362. S.-Y. Lan, D.-Q. Dai / J. Math. Anal. Appl. 332 (2007) 1351–1364. By (22) and (23), we deduce that the matrices T (v) (v ∈ int VΩ ) are bounded in compact subsets of Ω, independently of . On the other hand, (22) and (23) imply ∂j T (v) = i j T (v) · C(v) + T (v) · O() = T (v) · O(1). (24). (2) for j = 0, 1, 2, 3, where O(1) = i j C(v) + O(). It follows from Theorem 1 that h(1) and h ∞ ∞ are C -bounded, so O(1) is bounded in C (Ω). By repeating differentiation of (24), we conclude that T (v) is bounded in C ∞ (Ω) uniformly in . By the properties of C ∞ -convergence of functions (see [7, Lemma 2.1]), we obtain that (20) holds for some sequence of → 0, and the convergence is C ∞ (Ω). In the following we shall show that (20) is also valid for every sequence of → 0. Indeed, let D(z) be the matrix solution of the differential equation. D (z) = D(z)C(z). (25). with initial condition D(0) = I , then we have D z + i j = D(z) + i j D(z)C(z) + O 2 for j = 0, 1, 2, 3. From (22) and (23), we obtain that T v + i j − D v + i j T (v) − D(v) + T (v) − D(v) C(v) + 1 + T (v) O 2 T (v) − D(v) 1 + O() + 1 + T (v) O 2 . Note that T (0) = I + O() = C(0) + O(), we deduce that T (v) − D(z) = δ(v)O 2 + O() 1 + O() δ(v) , where δ(v) is the combinatorial distance from v to 0 in SG . Hence we deduce that T (v) − D(v) O()eO(1) = O() on a compact subset K ∈ Ω, because δ(v) = O(1/) on K. So T (v) = D(v) + O(), which implies that (20) holds for every → 0. In Eq. (24), taking limit as → 0, we obtain ∂j T (z) = i j T (z)C(z).. (26). Hence, we get ∂j T (z) = −∂j +2 T (z), which implies that T (z) is a matrix-value analytic function of z. In addition, it follows from (26) that the determinant of T (z) is constant in Ω. Note that at z = 0 this determinant is 1. So T (z) is a Möbius transformation for each z ∈ Ω. (ii) Let. a(v) b(v) T (z) = . c(v) d(v) Then T (z) satisfies the differential equation (25). By the definitions of Schwarzian derivative and C(v), we deduce that Sb/d = Sf ..
(15) S.-Y. Lan, D.-Q. Dai / J. Math. Anal. Appl. 332 (2007) 1351–1364. 1363. Note that b/d(0) = f (0) = 0, (b/d) (0) = f (0) = 1, (b/d) (0) = f (0) = 0. So we conclude that T (z)(0) = b(z)/d(z) = f (z). 2 Proof of Theorem 2. First, we write T (v) and T (z) as matrices. . a (v) b (v) a(z) b(z) T (v) = , T (z) = . c (v) d (v) c(z) d(z) Then it follows from Lemma 5(i) that c + d and a + b converge to d and b in C ∞ (Ω), respectively. By Lemma 5(ii), we obtain that b(z)/d(z) = f (z). Since the determinant of T (z) is nonzero, d is nonzero in Ω. By the properties of C ∞ -convergence of functions (see [7, Lemma 2.2]), we get that (a + b )/(c + d ) → b/d in C ∞ (Ω). Note that g (v) = T (), so we conclude that the discrete function g → f . Next, let c be the circle that contains the three points (1 + i)/2, (−1 + i)/2, (1 − i)/2. Since that T (v) maps the three points (1 + i)/2, (−1 + i)/2, (1 − i)/2 to q(v + (1 + i)/2), q(v + (−1 + i)/2), q(v + (1 − i)/2), respectively, and note that the three points q(v + (1 + i)/2), q(v + (−1 + i)/2), and q(v + (1 − i)/2) lie in the circle P (v), so T (v) maps the circle c onto P (v). Hence it follows that T (v)(0) = f (v). Similar to the above arguments, we conclude that f (v) = T (v)(0) converges in C ∞ (Ω) to f as → 0. Finally, by the definition of SG circle pattern, we easily get (27) r (v) + r (v + ) = f (v) − f (v + ) = ∂0 f (v), r (v) + r (v + + i) = f (v) − f (v + + i) = ∂0 f (v) + ∂1 f (v + ) (28) and. . r (v + ) + r (v + + i) = f (v + ) − f (v + + i) = ∂1 f (v + ).. From (27)–(29), we get that 2 2 2r (v)/ = ∂0 f (v) + ∂1 f (v + ) + ∂0 f (v) − ∂1 f (v + ) .. (29). (30). Since that ∂j f (j = 0, 1, 2, 3) converges in C ∞ (Ω) to f as → 0, we obtain from [7, Lemma 2.2] that |∂j f | → |f |. Thus it follows from (30) that r(v)/ converges in C ∞ (Ω) to |f |. This completes the proof of Theorem 2.. 2. Acknowledgment We thank the anonymous referee for valuable suggestions.. References [1] S.I. Agafonov, Imbedded circle patterns with the combinatorics of the square grid and discrete Painlevé equations, Discrete Comput. Geom. 29 (2) (2003) 305–319. [2] S.I. Agafonov, A.I. Bobenko, Discrete Z γ and Painlevé equations, Int. Math. Res. Not. (4) (2000) 165–193. [3] K. Callahan, B. Rodin, Circle packing immersions form regular exhaustible surfaces, Complex Var. Theory Appl. 21 (1993) 171–177. [4] P.G. Doyle, Z.-X. He, B. Rodin, Second derivatives of circle packings and conformal mappings, Discrete Comput. Geom. 11 (1994) 35–49. [5] Z.-X. He, An estimate for hexagonal circle packings, J. Differential Geom. 33 (1991) 395–412. [6] Z.-X. He, O. Schramm, On the convergence of circle packings to the Riemann map, Invent. Math. 125 (1996) 285–305..
(16) 1364. S.-Y. Lan, D.-Q. Dai / J. Math. Anal. Appl. 332 (2007) 1351–1364. [7] Z.-X. He, O. Schramm, The C ∞ -convergence of hexagonal disk packings to the Riemann map, Acta Math. 180 (1998) 219–245. [8] S.Y. Lan, D.Q. Dai, Riemann mapping theorems for Beltrami equations by circle packings, Pacific J. Math. 217 (2004) 139–147. [9] D. Matthes, Convergence in discrete Cauchy problems and applications to circle patterns, Conform. Geom. Dyn. 9 (2005) 1–23. [10] B. Rodin, D. Sullivan, The convergence of circle packings to the Riemann mapping, J. Differential Geom. 26 (1987) 349–360. [11] B. Rodin, Schwarz’s lemma for circle packings, II, J. Differential Geom. 30 (1989) 539–554. [12] O. Schramm, Circle patterns with the combinatorics of the square grid, Duke Math. J. 86 (2) (1997) 347–389. [13] K. Stephenson, A probabilistic proof of Thurston’s conjecture on circle packings, Rend. Sem. Mat. Fis. Milano 66 (1996) 201–291. [14] W. Thurston, The finite Riemann mapping theorem, invited address, in: International Symposium in Celebration of the Proof of the Bieberbach Conjecture, Purdue University, 1985..
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