CIRCLE PATTERNS WITH THE COMBINATORICS OF THE SQUARE GRID. square grid are constructed, and it is shown that the collection of entire, locally

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1 CIRCLE PATTERNS WITH THE COMBINATORICS OF THE SQUARE GRID Oded Schramm The Weizmann Institute Abstract. Explicit families of entire circle patterns with the combinatorics of the square grid are constructed, and it is shown that the collection of entire, locally univalent circle patterns on the sphere is innite dimensional. In Particular, Doyle's conjecture is false in this setting. Mobius invariants of circle patterns are introduced, and turn out to be discrete analogs of the Schwarzian derivative. The invariants satisfy a nonlinear discrete version of the Cauchy-Riemann equations. A global analysis of the solutions of these equations yields a rigidity theorem characterizing the Doyle spirals. It is also shown that by prescribing boundary values for the Mobius invariants, and solving the appropriate Dirichlet problem, a locally univalent meromorphic function can be approximated by circle patterns Mathematics Subject Classication. 30C99, 05B40, 30D30, 31A05, 31C20, 30G25. Key words and phrases. Meromorphic functions, Schwarzian derivative, rigidity, error function, Dirichlet problem. Some of the work exposed here was done while the author visited UCSD, and the author wishes to express thanks to the UCSD mathematics department for its hospitality. This research was partially supported by NSF grant DMS Typeset by AMS-TEX

2 2 ODED SCHRAMM 1. Introduction Some aspects of the relatively new theory of circle patterns (or packings) and their relations to analytic functions can be considered as well understood, while other aspects are still very mysterious and enigmatic. The roots of the topic are in the circle packing theorem [18] which is a uniformization result, and the uniformization theory of circle patterns is now well developed [29], [1], [16]. There has also been steady progress in understanding the convergence of circle packings to conformal maps [22], [12], [7], [17]. Branched packings were considered, and analogues of polynomials and nite Blaschke products have been studied [3], [9], [10]. Also, a satisfying crop of applications and by-products has emerged [21], [4], [20], [24], [15], [13], [2]. However, there is also a darker side: very little is known about circle pattern analogues of entire functions. It seems that Peter Doyle was the rst to look into this area, and constructed entire immersed hexagonal circle patterns analogous to the exponential map. Doyle conjectured that these immersed packings, which came to be known as Doyle spirals, are the only entire immersed hexagonal circle packings. To date, the only other contribution to this topic seems to be Callahan and Rodin's [5]. They showed that entire immersed hexagonal circle packings form regularly exhaustible surfaces, and therefore satisfy Ahlfors' value distribution theory. In particular, Picard's theorem is valid in this setting. We have found that a framework of circle patterns based on the square grid, which we call SG patterns, is more tractable than the traditional hexagonal patterns. It turns out that Doyle's conjecture is false for SG patterns: there is an SG pattern analogous to the entire function erf p iz, where erf(z) = R z e w 2 dw is the error function. We also show that the collection of entire SG patterns in the sphere is innite dimensional, and exhibit some explicit nite dimensional families. \Explicit" means that there is a closed form expression for the radius of any given circle in the pattern. Quite possibly, more explicit nite dimensional families could be found. This would be very interesting. In joint work, yet unpublished, with Rick Kenyon, we have discovered explicit branched SG patterns analogous to polynomials and to the function log z. Much of the theory of the hexagonal packings can be carried over to SG patterns. In particular, Callahan and Rodin's work [5] applies with virtually no modications. On the other hand, many of the results given below for SG patterns are not known for hexagonal packings, and proving (or disproving) them in the hexagonal setting would be a worthy challange. We start with the traditional approach of studying the radius function. In may respects, the `discrete nonlinear Laplace equation' that the radius function of an SG pattern satises is more amenable than the corresponding equation in the hexagonal setting. This is what permits the construction of the erf p iz pattern. We show that the surface detrmined by the erf p iz SG pattern is isometric to the surface determined by the classical error function, up to a constant scaling factor. It is also demonstrated that in a limited but well dened sense, there is no SG pattern analog of the erf function itself, without p i pre-rotation. But the radius function can only take us so far. The analysis of the radius function is the analysis of the circle pattern invariants for the group of isometries

3 SG CIRCLE PATTERNS 3 of the plane. Our main progress is in the analysis of the circle pattern invariants for the group of Mobius transformations. We introduce Mobius invariants and, and discover that they are closely analogous to the real and imaginary parts of the Schwarzian derivative. Naturally, it turns out that and satisfy a discrete nonlinear version of the Cauchy-Riemann equations. After the invariant is eliminated from these equations, one obtains a nonlinear discrete version of the Laplace equation that the invariant satises. Conversely, any that satises this equation has a one parameter family of companion invariants as co-solutions of the nonlinear Cauchy-Riemann system. We show that the nonlinear SG-Laplace equation satises the Dirichlet principle. This gives a parametrization of SG patterns corresponding to nite `simply connected' sub-pieces of the square lattice, in terms of the boundary values of the invariant, and one value of the invariant. That's the local analysis. Two global rigidity results are presented. First, the uniqueness of the embedded SG pattern is established. That is analogous to the rigidity of the hexagonal circle packing, and shows that the Rodin-Sullivan theorem is valid in the SG setting. The second rigidity result is more novel: if the invariant of an entire SG pattern in the sphere is bounded, then the pattern is Mobius equivalent to a Doyle spiral (or to the at pattern). It follows that if the radius function of a planar SG pattern satises c 1 < r(z + 1)=r(z) < c for some constant c, then the pattern is a Doyle spiral. Suppose that f : W! S 2 is a locally injective meromorphic function, where W is simply connected. We show that on compact subsets of W, f can be approximated by an SG pattern, which is obtained by specifying the value of the invariant at the boundary according to Re S f, the real part of the Schwarzian derivative of f, extending to be a solution of the SG-Dirichlet problem, nding an SG-Cauchy-Riemann co-solution that matches the value of Im S f at some interior point, and taking the SG pattern with invariant and. This is a Mobius geometry analogue of a similar theorem by Carter and Rodin [6], with Re S f taking the role of jf 0 j. The last section of the paper contains a collection of several open problems, but we briey describe a few problems now. Suppose that f : C! S 2 is a locally univalent open map. Then the metric on S 2 can be pulled back to C, and the resulting Riemannian surface is called the surface determined by f. Similarly, an entire SG pattern, or an entire immersed hexagonal packing, determines a Riemannian surface S, and the circle pattern is embedded on that surface. An important and fundamental question is: what Riemannian surfaces support entire circle patterns It would also be interesting to understand to what degree the answer depends on the underlying combinatorics. One might conjecture that all the schemes that are based on doubly periodic combinatorics should behave in the same manner, but in our work below there is a hint suggesting that this might not be the case. (Some assumption, like double periodicity, is clearly necessary. Consider, for example, a circle pattern whose combinatorics is described by the hexagonal grid modulo a translation taking the grid to itself. It is not hard to see that such a pattern cannot lie on the universal cover of the once punctured plane. In other words, in that setting Doyle spirals are not present.) The author would like to thank Peter Doyle, Tomasz Dubejko, Alex Eremenko,

4 4 ODED SCHRAMM Zheng-Xu He, Yakar Kannai, Rick Kenyon, Al Marden, and Burt Rodin for valuable discussions and advice. 2. Notations and Terminology Let SG be the cell complex whose vertices are the Gaussian integers, V (SG) = Z+ iz, whose edges are the pairs [z; z 0 ] such that jz z 0 j = 1 and z; z 0 2 V (SG), and whose 2-cells are the squares z + x + iy : x; y 2 [0; 1], z 2 V (SG). When z; z 0 are neighbors in SG, we write z z 0. The vertices z; z 0 are called half-neighbors if they belong to the same square of SG, but are not neighbors. Let G be a subgraph of the 1-skeleton of SG. Suppose that for every z 2 V (G) (the vertices of G) there is associated an oriented circle C z in the Riemann sphere S 2 = ^C. We now formalize what it means for the circles to be combinatorially like the pattern in Figure2.1. Consider the following three conditions. (1) Whenever z z 1 are neighbors in G, the corresponding circles C z ; C z1 intersect orthogonally. (2) If z 1 ; z 2 are neighbors of a vertex z in G, and they belong to the same square of SG, then the circles C z1 ; C z2 are distinct and tangent. It follows from (1) and (2) that the tangency point of C z1 and C z2 lies in C z. Hence there will be three special points on C z, namely, C z1 \ C z2 ; C z \ C z1 C z2 ; C z \ C z2 C z1. (3) Whenever the situation is as in (2) and z 2 is the neighbor of z which is one step counterclockwise from z 1 (that is, z 2 = z + i(z 2 z)), the circular order of the triplet of points C z \C z1 C z2 ; C z1 \C z2 ; C z \C z2 C z1 agrees with the orientation of C z. Figure 2.1. The regular SG pattern. Observe that when the circles in Figure 2.1 are oriented positively with respect to the planar disks they bound, these three conditions are satised. Denitions. A circle pattern for G is an indexed collection C = C z : z 2 V (G) of oriented circles in S 2 that satises (1){(3) above. The pattern is planar, if each C z C and is positively oriented with respect to the bounded component of C C z. The pattern is embedded, provided that whenever z; z 0 2 V (G) are

5 SG CIRCLE PATTERNS 5 not neighbors the open disks determined by C z and C z 0 are disjoint. (The open disk determined by an oriented circle C is the component D of ^C C such = C with agreement of orientation.) If G is the whole 1-skeleton of SG, then a circle pattern for G is called an entire circle pattern. A vertex in G is called an interior vertex, if it has 4 neighbors in G, otherwise, it is a boundary vertex. The set of boundary vertices of G is denoted by (G), and the set of interior vertices is denoted V int (G). 3. The Radius Function Let G be a subgraph of the 1-skeleton of SG. The next proposition allows the studying of a circle pattern through its radius function, the function which assigns to every vertex in G the radius of the corresponding circle. If C = (C z : z 2 V (G)) is a planar circle pattern for a graph G SG, we let r C (z) denote the radius of the circle C z. Sometimes, when no confusion is likely, we will use the notation r(z) in place of r C (z) Proposition. Let Y be a union of squares of SG and let G be the graph which is the intersection of Y and the 1-skeleton of SG. (1) Suppose that C = (C z : z 2 V (G)) is a planar circle pattern for G, Then for every interior vertex z 0 2 V int (G), we have r(z 0 ) = H r(z 0 + 1); r(z 0 + i); r(z 0 1); r(z 0 i) ; (3.1) where r = r C and H(r 1 ; r 2 ; r 3 ; r 4 ) = s (r r r r 1 4 )r 1r 2 r 3 r 4 r 1 + r 2 + r 3 + r 4 : (2) Conversely, if the interior of Y is connected, Y is simply connected, and r : V (G)! (0; 1) satises (3.1) for every z 2 V int (G), then there is a planar circle pattern for G such that radius(c z ) = r(z) for all z 2 V (G). This pattern is unique, up to isometries of C. Proof. Results of this kind are quite standard in the eld. We therefore only give a proof of the existence part of (2), which is the harder claim. Even the nonexpert should have no trouble in supplying the missing arguments after seeing the proof of (2). The following construction is essentially the same as the one in Thurston's notes [28, Chap. 13]. Consider two neighboring z 0 ; z 1 2 V (G). Let A(z 0 ; z 1 ) be the right angled triangle with vertices 0; r(z 0 ); ir(z 1 ). The vertex r(z 0 ) [repectively, ir(z 1 )] will be called the z 0 [respectively, z 1 ] vertex of A(z 0 ; z 1 ). The edge [0; r(z 0 )] of A(z 0 ; z 1 ) will be called its z 0 edge, and the edge [0; ir(z 1 )], its z 1 edge. Each edge of A(z 0 ; z 1 ) gets an orientation as part 0 ; z 1 ). Note that the circle of radius r(z 0 ) with center at the z 0 vertex of A(z 0 ; z 1 ) is orthogonal to the circle of radius r(z 1 ) with center at the z 1 vertex of A(z 0 ; z 1 ). Let X be the disjoint union of the A(z 0 ; z 1 ) over all oriented edges [z 0 ; z 1 ] in G. Consider the space X 0 obtained by pasting together edges of X as follows. The hypothenuse of A(z 0 ; z 1 ) is pasted to the hypothenuse of A(z 1 ; z 0 ), with orientations

6 6 ODED SCHRAMM reversed. If z; z + i j ; z + i j+1 are all in V (G) (j = 0; 1; 2 or 3), then the z edge of A(z; z + i j ) is pasted with orientation reversed to the z edge of A(z + i j+1 ; z). In this way every edge of X is pasted to at most one other edge, which has the same length, and X 0 is an oriented surface with boundary. We choose the gluing maps to be isometries. Hence X 0 has a metric. We now show that the metric on X 0 is locally Euclidean in the interior of X 0 ; that is, in X 0. Let p be a point in X, whose image p 0 2 X 0 is not 0. Suppose rst that p is the vertex of A(z 0 ; z 1 ) where the right angle is. Since p 0 0, it follows that four triangles are glued at p 0. Note that the angles glued at p 0 are all right angles. Hence X 0 is locally Euclidean near p 0. Now suppose that p is the z 0 vertex of A(z 0 ; z 1 ). Since p 0 is interior to X 0, it follows that z 0 is an interior vertex of G. Hence 8 triangles are pasted at p 0, namely A(z 0 ; z 0 + i j ); A(z 0 + i j ; z 0 ); j = 0; 1; 2; 3. The sum of the angles at p 0 is Hence, = 2 X zz 0 arctan r(z) r(z 0 ) : (3.2) e i=2 = Y zz 0 r(z 0 ) + ir(z) r(z0 ) + ir(z) : From (3.1), it now immediately follows that Im e i=2 = 0: (3.3) Because each of the eight angles pasted at p 0 is in the range (0; =2), it follows that =2 2 (0; 2). Hence, (3.3) implies that = 2. So X 0 is locally Euclidean near p 0. It is easy to see that X 0 is locally Euclidean near p 0 if p is not a vertex of any of the triangles. So the interior of X 0 is locally Euclidean. There is some open nonempty set W X 0 that's isometric to a disk in the plane. Let f be such an isometry. Because X 0 is locally Euclidean, given any simple curve with initial point in W, f can be analytically continued along, and the result of this analytic continuation is a local isometry dened in a neighborhood of. Since X 0 is simply connected, the Monodromy Theorem tells us that f has an analytic extension, dened in the interior of X 0. It follows that there is a local isometry F : X 0! C. For every z 2 V (G), we choose C z to be the circle of radius r(z) centered at F (p 0 z ), where p0 z is the image in X0 of the z -vertices of the triangles A(z; z 0 ), with z 0 any neighbor of z. It is easy to check that C = C z : z 2 V (G) is a circle pattern for G. The uniqueness part is left to the reader Denition. The surface X 0 in the proof will be called the surface determined by the circle pattern C. map. The map F : X 0! C is called the developing It is worth while to note, for future use, some properties of the function H from 3.1.

7 SG CIRCLE PATTERNS Properties of H. In the following, r 1 ; r 2 ; r 3 ; r 4 > 0 are arbitrary. (1) Homogeneity: H(tr 1 ; tr 2 ; tr 3 ; tr 4 ) = th(r 1 ; r 2 ; r 3 ; r 4 ) holds for all t > 0. (2) Symmetry: H is invariant under permutations of its arguments. (3) Reciprocity: H r 1 1 ; r1 2 ; r1 3 ; r1 4 = H(r1 ; r 2 ; r 3 ; r 4 ) 1. (4) Monotonicity: H is strictly monotone increasing in each of its arguments. (5) As r 1! 1 and r 3! 0, while r 2 ; r 4 are kept xed, H(r 1 ; r 2 ; r 3 ; r 4 ) tends to p r 2 r 4, the geometric mean of r 2 and r 4. (6) If p r 1 r 3 = p r 2 r 4, then p r 1 r 3 = H(r 1 ; r 2 ; r 3 ; r 4 ). The verication of these properties is left to the reader. They all have a geometric proof, as well as a direct proof from the denition of H. Note that properties (2) and (3) do not hold for the corresponding function for circle packing immersions of the hexagonal lattice. In joint work with R. Kenyon we shall show that property (3) can be used to obtain explicit constructions for circle patterns corresponding to the polynomials z d. 4. Exp and Erf P. Doyle has constructed circle packing immersions based on the hexagonal combinatorics that are analogous to the exponential functions e az, later to be called Doyle spirals, and conjectured that these, and the regular hexagonal packings, are the only entire circle packing immersions with the combinatorics of the innite hexagonal grid. See [5]. We now show that Doyle spirals are present in the setting of SG-circle patterns, and Doyle's conjecture is false in this setting. Suppose that z 0 2 V (SG) and z 1 ; z 2 ; z 3 ; z 4 are its neighbors. Let r : V (SG)! (0; 1). From 3.3.(6) it follows that for every a 2 C nonzero r(z) = je az j satis- es (3.1) at every z 2 V (SG). Consequently, r corresponds to an entire circle pattern. See Figure 4.1. Figure 4.1. An SG Doyle spiral. We shall call these patterns SG Doyle spirals, since they are clearly the analogues of the Doyle spirals for SG circle patterns. The surface determined by an SG Doyle spirals (or by a hexagonal Doyle spiral) is easily seen to be the universal cover of

8 8 ODED SCHRAMM C f0g, which is the same as C with the metric pulled back by the exponential map exp : C! C f0g. This is one sense in which these spirals are analogues of the exponentials. Also, the function je az j is the same as the absolute value of the derivative of an exponential. Now consider the function r(z) = e axy, where z = x + iy and a is a real constant. It is immediate that r(z) is equal to the geometric mean of r(z + 1) and r(z 1), and to the geometric mean of r(z + i) and r(z i). Consequently, by property 3.3.(6), r gives rise to an entire planar SG circle pattern. Note that r(z) = cjf 0 (z)j for f(z) = erf( p iaz) and an appropriate c, where erf is the error function erf(z) = (2=) R e z2 dz. Also note that the picture of the circle pattern is similar to the image of the rotated and scaled square grid under f, as indicated in Figures 4.2.a, and 4.2.b. We shall call these circle patterns the p isg erf patterns. Figure 4.2.a. The erf-image of a square grid, erf p isg. Figure 4.2.b. The p isg erf pattern. The Figures 4.2.a, and 4.2.b are not misleading, the erf circle pattern is indeed geometrically related to the pattern given by the error function. Before we make

9 this statement precise, we need to recall a denition. SG CIRCLE PATTERNS 9 Denition. Let f : C! X be holomorphic, where X is either the sphere or C. Then the metric on X can be pulled back via f to C. The resulting surface is called the surface determined by f : C! X Theorem. The surface S a determined by the p isg erf pattern with radius function r(z) = e axy (a > 0) is isometric to the surface R determined by c erf : C! C, where c is a constant which will be determined below. Proof. To simplify notations, only the case a = 1 will be discussed. Set S = S 1. The circle in S corresponding to a vertex z 2 V (SG) will be denoted Cz ~. Let S denote the completion of S. We now show that S S contains precisely 2 points. Let S(n) denote the part of S covered by circles Cx+iy ~ with xy > n, and their interiors. Then S(n) is clearly complete. The complement S S(n) has two components: let A 1 (n) be the component of S S(n) that contains circles ~C x+iy with y < 0 < x, and let A 2 (n) be the component of S S(n) that contains circles Cx+iy ~ with x < 0 < y. It is easy to see that the diameters of A 1 (n) and A 2 (n) tend to zero as n! 1. So any sequence p n with p n 2 A 1 (n) is a Cauchy sequence. Consequently, there is a point q 1 2 S such that p n! q 1. It is immediate that q 1 =2 S, so q 1 2 S S. Similarly, any sequence p 0 n with p 0 n 2 A 2 (n) tends to a point q 2 2 S S. The points q 1 ; q 2 are distinct, because the distance from A 1 (1) to A 2 (1) is clearly positive. It follows that S = S [ fq 1 ; q 2 g. Let F : S! C denote the developing map. Then F has a continuous extension F : S! C. Set q 0 j = F (q j), for j = 1; 2. We now determine the points q1; 0 q 0 2. Note that the circles C 0 ; C 1 ; C 1 ; C i ; C i all have radius 1. Since we are free to modify the developing map by an isometry of C, we assume without loss of generality that the centers the circles C 0 ; C 1 ; C 1 ; C i ; C i are at 0; p 2; p 2; i p 2; i p 2, respectively. Because of the symmetry r(x + iy) = r(y ix) of the radius function r(z) = xy, the circle C x+iy will be the reection of the circle C yix in the diagonal x = y. Consequently, the centers of the circles C n(1i) will all lie on the diagonal x = y. Since C n(1i) is tangent to C (n+1)(1i) it follows by induction that for n > 0 the center of C n(1i) is at the point Set 1 i j1 ij 1 + 2e e (n1)2 + e n2 : c = X n=1 e n2 : Then it follows that q 0 1 = c(1 i)=j1 ij and similarly q0 2 = c(i 1)=ji 1j. Suppose that p 0 is any point on S and : [0; 1]! C is a path in C starting at (0) = F (p 0 ). Assume that does not pass through the points q 0 1 and q 0 2. Then has a lift ~ : [0; 1]! S with ~(0) = p 0. (That ~ is a lift of means = F ~.) Indeed, let Z be the set of t 2 [0; 1] so that the restriction of to [0; t] has a lift. Because F is a local isometry, the set Z is open in [0; 1], and when has a lift in [0; t] it is unique. The set Z is not empty, since 0 2 Z. Set t 1 = sup Z. Then the restriction of to [0; t 1 ) has a lift : [0; t 1 )! S. Because S is complete, it follows

10 10 ODED SCHRAMM that has a continuous extension : [0; t 1 ]! S. Therefore, F (t 1 ) = (t 1 ). Since avoids q 0 1; q 0 2, we have (t 1 ) 6= q 1 ; q 2, and is a path in S. So has a lift to [0; t 1 ] and sup Z = t 1 2 Z. Because Z is open in [0; 1], it follows that Z = [0; 1]. Now let p 0 be the center of C0 ~ in S, and let l be the lift of the line x = y passing through p 0. Then S l has two components. Let B 1 be the component of S l whose closure in S contains q 1, and let B 2 be the component of S l whose closure in S contains q Lemma. q 0 1 =2 F (B 1) and q 0 2 =2 F (B 2). Proof. Suppose, for example, that q F (B 1). Let p be some point in B 1 with F (p) = q 0 1. Since F is a local isometry, a small disk B(q 0 1 ; ) can be lifted to a small disk B(p; ) in B 1. Every line segment of the form [q1; 0 q] contained in the half plane x > y will then have a lift to a segment in B 1 starting at p. By taking the union of all these lifts, it follows that the identity map from the half plane x > y onto itself has a lift g : fz : x > yg! B 1. The boundary of the image of g is a lift of the line x = y. Suppose rst that this lift is not l. Then g can be extended beyond the line x = y ; that is, there is a lift of the whole plane C to B 1 whose restriction to x > y is g. This means that B 1 contains a subset isometric to C, which implies that B 1 is isometric to C, a contradiction. On the other hand, if the boundary of g fx > yg is l, then it follows that the image of g is all of B 1, which is false, because B 1 [ l is not complete. Now let erf (z) = c(1 i) erf(z)=j1 ij. Then R is isometric to the surface R determined by erf. Using lim z!+1 erf(z) = 1, a discussion entirely analogous to the above shows that the completion R of R consists of R with two additional points, one, say q 1, over q 0 1 and one, say q 2, over q 0 2. Let l be the lift to R that passes through 0 2 R of the line x = y. (Recall that R = C with the metric pulled back via erf.) Let B 1 be the component of R l whose closure in R contains q 1, and let B 2 be the component of R l whose closure in R contains q 2. As in the proof of the lemma, it is easy to see that q 0 1 =2 erf (B1) and q 0 2 =2 erf (B2). Let p be any point in B 1 [ l. Take any path that connects p and 0 in B 1 [ l. Set = erf. We know that does not pass through q 0 1. Hence has a (unique) lift ~ to B 1 [ l that intersects l. Let f 1 (p) be the endpoint of ~ corresponding to p. Since B 1 [ l is simply connected, f 1 (p) does not depend on the choice of. It is clear that f 1 : B 1 [ l! B 1 [ l is an isometry. Similarly, we dene the isometry f 2 : B 2 [ l! B 2 [ l. These two isometries can be glued along l, to give an isometry from R to S. This proves the theorem. One can slightly generalize the p isg erf pattern, by taking r(z) = e axy+bx+cy with real a; b; c. When b; c are integer multiples of a, this corresponds to pretranslating the square grid. When they are not, you get an essentially new pattern, which is very similar to the pattern for the case b = c = 0. Altogether, up to similarity, this gives a 3-dimensional family of p isg erf patterns.

11 SG CIRCLE PATTERNS 11 It is natural to ask for pre-rotated versions of these patterns. To be precise, these patterns are not analogues of the true erf function, but of the functions erf p ai(z + c), with a a real constant, and c a complex constant. One may ask for a circle pattern which would mimick erf(z), without pre-rotation. Here is another view of the situation. The function H can be thought of as a nonlinear version of the discrete Laplacian for log r(z). The linear discrete Laplacian, g(z) = (1=4) g(z + 1) + g(z + i) + g(z 1) + g(z i) g(z). behaves in many respects like the Laplacian in R 2. In particular, the space of discrete harmonic functions on the square grid that have O jz 2 j growth is spanned by the functions 1; x; y; xy; x 2 y 2. We have seen that any linear combination of 1; x; y; xy is log r(z) for some planar circle pattern. Direct inspection shows that r(z) = e x2 y 2 does not satisfy equation (3.1). The question arises if there is some radius function similar to e x2 y 2 which does. The corresponding pattern may be an erf analog, without pre-rotation. It will follow from the following theorem that, in some well dened sense, such a pattern does not exist Theorem. Let G be the intersection of SG with the quadrant fx + iy : x > jyjg, and let r : V (G)! (0; 1) be a function that satises (3.1) at every interior vertex of G. Then r(z) 6 C maxfr(v) : v 2 (G); Re v 6 xg e 3x log x ; holds for every z 2 V (G). Here x = Re z, and C is some absolute constant. Proof. Suppose that R : V (G)! (0; 1) is a function that satises the the inequality R(z) > H 1; R(z + i); R(z 1); R(z i) ; (4.1) for every interior vertex z 2 V int (G). (Here, the expression H(1; r 2 ; r 3 ; r 4 ) stands for the limit of H(r 1 ; r 2 ; r 3 ; r 4 ) as r 1! 1.) Let n be some positive integer. Set V n = fz 2 V (G) : Re z 6 ng, and let M be the maximum of r(z)=r(z) for z 2 V n. Let z 2 V n \ V int (G). Then we have from the properties of H, r(z) = H r(z + 1); r(z + i); r(z 1); r(z i) = MH r(z + 1)=M; r(z + i)=m; r(z 1)=M; r(z i)=m 6 MH r(z + 1)=M; R(z + i); R(z 1); R(z i) < MH 1; R(z + i); R(z 1); R(z i) 6 MR(z): Therefore, the maximum of r(z)=r(z) in V n is attained in V n (G). We now choose the function R, and prove that it satises (4.1). Set a(x) = log(x + 3); and dene b(n) inductively by b(0) = 0 and b(n + 1) = b(n) + a(n);

12 12 ODED SCHRAMM for positive integers n. For negative n, let b(n) = b jnj. Now set R(z) = e 3b(x)b(y) ; for every z = x + iy 2 V (G). We will show that R satises (4.1). Note that (4.1) is the same as R(x + iy) 2 > R x 1 + iy R x + i(y 1) + R x 1 + iy R x + i(y + 1) + For our particular R, this evaluates to e 6b(x)2b(y) > + R x + i(y + 1) R x + i(y 1) : e 3b(x1)b(y)+3b(x)b(y1) + e 3b(x1)b(y)+3b(x)b(y+1) + e 6b(x)b(y+1)b(y1) : (4.2) It is clearly sucient to prove this for 0 6 y 6 x1, since both sides do not change if we replace y by y. Assume rst that y 2 [1; x 1]. Then (4.2) reduces to 1 > e 3a(x1)+a(y1) + e 3a(x1)a(y) + e a(y1)a(y) : We estimate the right hand side, as follows, e 3a(x1)+a(y1) + e 3a(x1)a(y) + e a(y1)a(y) = (x + 2) 3 (y + 2) + (x + 2) 3 (y + 3) 1 + (y + 2)(y + 3) 1 6 (y + 2) 2 + (y + 3) y : So (4.2) holds for y 2 [1; x 1]. For y = 0, (4.2) reduces to which is the same as e 6b(x) > 2e 3b(x1) e 3b(x)b(1) + e 3b(x)b(1) e 3b(x)b(1) ; 1 > 2e 3a(x1)a(0) + e 2a(0) : The right hand side is easily estimated, as follows, 2e 3a(x1)a(0) + e 2a(0) = 2(x + 2) 3 =3 + 1=9 6 1: Hence (4.1) holds throughout V int (G). Now note that b(n) = b jnj 6 Z jnj 0 log(t + 3) dt = (jnj + 3) log jnj + 3 jnj 3 log 3:

13 Therefore, it is easy to conclude that SG CIRCLE PATTERNS 13 R(z) 6 Ce 3x log x ; (4.3) holds in V (G), for an appropriate constant C < 1. On the other hand, we have R(z) > 1: (4.4) Fix any z 2 V (G), and take n = x = Re z. We have seen that the maximum of r(v)=r(v) on V n is achieved on V n \ (G). Hence r(z) 6 R(z) maxfr(w) : w 2 V n \ (G)g minfr(w) : w 2 V n \ (G)g : The theorem now follows from (4.3) and (4.4). Because erf 0 (z) = (2=)e z2, we have erf 0 (z) = 2= on the diagonals j Re zj = j Im zj. At a bounded distance from the diagonals we have erf 0 (z) 6 e c 1 jzj, for an appropriate constant c 1. So it is reasonable to require an SG pattern analogue of erf (without pre-rotation) to have a radius function r satisfying r(z) 6 e c 2jzj on the diagonals. By Theorem 4.3, r would then satisfy r(z) 6 Ce c 2jzj+3jzj log jzj. Hence r would deviate from the behaviour of j erf 0 j on the imaginary axis. It follows that there is no entire SG pattern that mimicks erf in the sense of the growth rate of r(z) along the axis and diagonals. Similarly, Theorem 4.3 is applicable to many other functions in place of erf, for example, R e w4 dw. It is also applicable to non-locally injective functions and branched circle patterns (which we have not dened here), e.g., e z2. Notwithstanding the above, we cannot conclude that there is no entire SG pattern analogue of erf ; it is conceivable that there exists some SG pattern possessing many of the geometric properties of erf. It seems that entire SG circle patterns cannot be expected to parallel many entire locally injective functions. The non-isotropy of the square grid is the underlying reason. There is every reason to expect that the circle packings based on the hexagonal grid would do no better. Perhaps, circle patterns based on more isotropic graphs (e.g., graphs with some randomness) might be better. We end this section with the moral that computer simulations cannot be trusted when searching for entire circle patterns. For example, it is easy to create pictures suggesting the existence of a true SG erf pattern (without pre-rotation). Figure The Mobius Invariants The radius function is a handy tool for studying SG circle patterns in the plane. However, it is not useful for studying SG patterns on the sphere. We now present Mobius invariants for SG pattern in the sphere. Suppose that C = C z : z 2 V (SG) is a circle pattern for SG on the Riemann sphere ^C. The set of centers of squares of SG will be denoted by V (SG); that is, V (SG) = fz + 1=2 + i=2 : z 2 V (SG)g. Let w 2 V (SG). Note that the See

14 14 ODED SCHRAMM Figure 4.3. A fake SG erf pattern intersection of the four circles C w1=2i=2 is a single point. We denote this point by p C (w), or sometimes just p(w), when there can be no confusion about the implied C. Now let z 2 V (SG). The four points p(z 1=2 i=2) are all on the circle C z. Hence their cross ratio is real. Recall that the cross ratio of four points p 1 ; p 2 ; p 3 ; p 4 2 ^C is dened by cr p 1 ; p 2 ; p 3 ; p 4 = (p 1 p 3 )(p 2 p 4 ) (p 1 p 4 )(p 2 p 3 ) ; and does not change when a Mobius transformation is applied to all four points. Let the invariant of C be the function : V (SG)! R dened by C (z) = (z) = cr p(z+1=2+i=2); p(z1=2i=2); p(z+1=2i=2); p(z1=2+i=2) : (The reason for the awkward order is that it makes formulas below a bit simpler.) The invariant does not change if we modify C by a Mobius transformation. Because fp(z + 1=2 + i=2); p(z 1=2 i=2)g separates p(z + 1=2 i=2) and p(z 1=2 + i=2) on C z, it follows that (z) > 0 for z 2 V (G). It turns out that the invariant is not sucient to determine C up to Mobius transformations. Another invariant is necessary. For each w 2 V (SG) we dene C (w) = (w) = cr p(w + 1); p(w 1); p(w i); p(w + i) : This is the invariant for C. We now show that (w) is real and positive. Let m be a Mobius transformation that take p(w) to 1. The four circles m C w1=2i=2 are lines, and together form a rectangle with corners p (w + 1); p (w + i); p (w 1); p (w i), where p = p m(c). See Figure 5.1. Hence p (w + 1) p (w + i) = p (w i) p (w 1),

15 SG CIRCLE PATTERNS 15 p (w 1) p (w + i) = p (w i) p (w + 1), and the ratio p (w + 1) p (w + i) = p (w + 1) p (w i) is pure imaginary. Therefore, p (w + 1) p 2 (w i) p (w + 1) p (w i) 2 C (w) = m(c) (w) = p (w + 1) p = (w + i) p (w + 1) p (w + i) : 2 (5.1) p(w+i) p(w+1) p*(w-i) p*(w+1) p(w-1) p(w-i) p(w) p*(w-1) p*(w+i) Figure 5.1. The conguration determining (w). Just as the radius function for a circle pattern is a discrete analogue to the absolute value of the derivative of an analytic function, we shall see that log and log are analogous to the real and imaginary parts of the Schwarzian derivative, respectively. The Schwarzian derivative of a function f is dened by f 00 0 S f = f 0 1 f f 0 : (5.2) It is left-mobius invariant; that is, S mf = S f for any Mobius transformation m. If f; g are meromorphic locally injective functions dened on a connected open set, and S f = S g, then f = m g for some Mobius transformation m. See [19, xii.1] for a brief discussion of the Schwarzian derivative. The analogies between log + i log and the Schwarzian derivative will be pointed out from time to time, as we progress. For starters, note that when f(z) = g(iz) we have S f (z) = S g (iz), and similarly C (z) = C (iz) 1 ; C (z) = C (iz) 1, when C = C z : z 2 V (SG) and C = C z : z 2 V (SG) satisfy C z = C iz. Examples. Before we proceed further with the general theory, let us calculate the invariants for the patterns described previously. To accomplish this, we rst relate the Mobius invariants to the radius function. Suppose that C is a planar SG pattern. Take w 2 V (SG). The Mobius transformation m(z) = z p(w) 1 takes p(w) to 1. If C is a circle of radius r passing through p(w), then m(c) is a line whose distance from 0 is (2r) 1. Hence (5.1) shows that (w) = r(w + 1=2 + i=2) 1 + r(w 1=2 i=2) 1 r(w + 1=2 i=2) 1 + r(w 1=2 + i=2) 1 2 :

16 16 ODED SCHRAMM To calculate (z), let c denote the center of C z, set q j = p z + i j (1 + i)=2 c, and observe that q j = q j1 r(z) + ir z + i j r(z) ir (z + i j ) : Hence, (z) = cr q 0 ; q 2 ; q 3 ; q 1 = (q 0 q 3 )(q 2 q 1 ) (q 0 q 1 )(q 2 q 3 ) = = = q r(z)+ir(z+1) 3 1 q r(z)+ir(z1) r(z)ir(z+1) 1 1 r(z)ir(z1) q 1 r(z)+ir(z+i) r(z)ir(z+i) r(z)+ir(z+1) 1 1 r(z)ir(z+i) r(z)+ir(z+i) 1 r(z)ir(zi) r(z)ir(z+1) 1 r(z)+ir(z1) q 3 r(z)+ir(zi) r(z)ir(zi) 1 r(z)ir(z1) 1 r(z)+ir(zi) 1 1 r(z) r(z + i) + i r(z) r(z i) + i r(z) r(z + 1) i 1 r(z) r(z 1) i 1 : Because is positive, taking absolute values in both sides gives (z) = s r(z)2 + r(z + i) 2 r(z) 2 + r(z i) 2 r(z)2 + r(z + 1) 2 r(z) 2 + r(z 1) 2: (5.3) We are now ready to compute and for the known examples of planar SG patterns. For the at SG pattern, where all circles have the same radius, (and its Mobius images), we have = 1; = 1. Take some a 2 C, and let C(a) denote the SG Doyle spiral with radius function r(z) = je az j = e a 1 Re za 2 Im z, where a 1 = Re a, a 2 = Im a. Using the above formula for and, we get C(a) (z) = ea 2 + e a 2 e a ; 1 + e a 1 e a 2 + e a 1 2 C(a) (w) = e a 2+a 1 : + 1 Let exp a (z) = e az. Then, using the denition (5.2) of the Schwarzian derivative, S expa = a 2 =2 = a 2 2 =2a2 1 =2ia 1a 2. Up to Mobius transformations, the functions exp a are the only functions with constant Schwarzian derivative. Similarly, it will follow from results below that the SG Doyle spirals are the only SG patterns with constant and. Moreover, note the following analogies between S expa and log C(a) + i log C(a). Both do not change when a is replaced by a, Re S expa and log C(a) are both monotone increasing functions of ja 2 j and ja 1 j, and vanish when ja 2 j = ja 1 j, Im S expa and log C(a) both vanish when a 1 a 2 = 0, and do not change when a 1 and a 2 are exchanged. It is also interesting to note that S expa = log C(a) + i log C(a) + O(a 4 );

17 SG CIRCLE PATTERNS 17 near a = 0. For the p isg erf pattern with radius function r(z) = e xy (z = x+iy), we have (z) = ex + e x e y + e y ; (z) = 1 e e x+y + 1 e x + e y 2 : (5.4) If f is meromorphic, the real and imaginary parts of S f satisfy together the Cauchy-Riemann equations. So if log + i log is indeed analogous to S f, we would expect and to be related in a similar way. As will be seen below, this is indeed the case. In the following, we shall determine the necessary and sucient conditions for a pair of functions : V (SG)! (0; 1), : V (SG)! (0; 1) to be the invariants of a circle pattern for SG. (Theorem 5.1.) As usual, it is convenient to start with `necessity'. Let C be some circle pattern for SG. We shall nd equations relating its invariants and. Let HG denote 1 1+i SG + ; that is, HG is a square grid 2 4 with edge-size 1, translated so that 0 is at a center of a square of HG. Take 2 v 2 V (HG), and let z 2 V (SG) be the unique vertex of SG that's closest to v. Set w 1 = z + 2(v z); w 2 = z + 2i(v z); w 3 = z 2i(v z), and note that w 1 ; w 2 ; w 3 2 V (SG). Let m v be the Mobius transformation that takes 1; 0; 1 to p(w 1 ); p(w 2 ); p(w 3 ), respectively. For any directed edge [v 1 ; v 2 ] of HG, set m [v1 ;v 2 ] = m 1 v 1 m v2 : (5.5) Note that m [v1 ;v 2 ] does not change if we apply a Mobius transformation to C ; it is a Mobius invariant of C. We shall now compute the Mobius transformations m [v1 ;v 2 ] for the various types of edges [v 1 ; v 2 ], in terms of the invariants and. First take v 1 to be of the form w 1(1 + i), where w 2 V (SG), and v 4 2 = w + i(v 1 w) = v When calculating m 2 [v 1 ;v 2 ], we shall assume that m v1 is the identity. This involves no loss of generality, because C can be replaced by its Mobius image m 1 v 1 (C). With this assumption, we have p(w) = 1; p(w 1) = 0; p(w i) = 1: Because p(w) = 1, the four points p(w 1); p(w i) form the vertices of a rectangle, and 2 p(w + 1) p(w i) (w) = : p(w i) p(w 1) This gives p(w + 1) = p(w i) i p (w) p(w i) p(w 1) = 1 i p (w): (The above correct choice of sign for i = p 1 follows from an inspection of the order of the corners p(w 1); p(w i) of the rectangle.) Now, m [v1 ;v 2 ] takes 1; 0; 1 to p(w); p(w i); p(w + 1), respectively. Hence m [v1 ;v 2 ] is easily computed: m [v1 ;v 2 ]() = 1 i p (w):

18 18 ODED SCHRAMM Similar computations yield expressions for m [v1 ;v 2 ] when v 1 = w 1 1 i and 4 4 v 2 = w + i(v 1 w), where w 2 V (SG). We record the results as follows m [w i;w i] () = m [w i;w i] () =1 i p (w): m [w i;w i] () = m [w i;w i] () =1 i p (w) 1 : (5.6) These formula are valid for all w 2 V (SG). Now consider an edge [v 1 ; v 2 ] in HG, where v 1 = z i; v 2 = z i, and z 2 V (SG). We assume with no loss of generality that p(z + 1=2 + i=2) = 1; p(z 1=2 + i=2) = 0; p(z + 1=2 i=2) = 1: From the denition of (z), we get, p(z 1=2 i=2) = 1 (z) : The transformation m [v1 ;v 2 ] takes 1; 0; 1 to p(z 1=2+i=2); p(z 1=2i=2); p(z + 1=2 + i=2), respectively. Hence, m [v1 ;v 2 ]() = (z) : Similar computations yield expressions for m [v1 ;v 2 ] v 2 = z + i(v 1 z). The results are, when v 1 = z i and m [z i;z i] () = m [z i;z i] () = m [z i;z i] () = m [z i;z i] () = (z) ((z) + 1)1 ; 1 ; (5.7) and are valid for all z 2 V (SG). From the denition of the Mobius transformations m [v1 ;v 2 ], it is clear that for any closed path v 0 ; v 1 ; : : : ; v n = v 0 in V (HG) the composition of the Mobius transformations corresponding the edges of the path is the identity; that is, In particular, m [v0 ;v 1 ] m [v1 ;v 2 ] m [vn1 ;v n ] = identity: (5.8) m [v1 ;v 2 ] 1 = m [v2 ;v 1 ]; (5.9) for every directed edge [v 1 ; v 2 ] in HG. Hence the equations (5.6) and (5.7) allow us to compute m [v1 ;v 2 ] for every directed edge [v 1 ; v 2 ] of HG. Let a be of the form a = n + (m + 1=2)i, where n; m 2 Z, and let v j = a + i j (1 + i)=4, for j = 0; 1; 2; : : :. Then v 0 ; v 1 ; v 2 ; v 3 ; v 4 = v 0 is a closed path in HG. Consequently, we get m [v4 ;v 3 ] m [v3 ;v 2 ] m [v2 ;v 1 ] m [v1 ;v 0 ] = identity: (5.10)

19 SG CIRCLE PATTERNS 19 After using the appropriate formulas from (5.6) and (5.7), this becomes p 0 1 i (a (a i=2) p i (a 1=2) 1 ((a+i=2) 1 +1) 1 1 A = ; and simplication yields, (a + 1=2) (a + i=2) 1 2 (a 1=2) = + 1 ; for a 2 V (SG) + i=2: (5.11) (a i=2) Choosing v j = b + i j (1 + i)=4, where b has the form b = m + ni + 1=2, gives, after entirely similar computations, 2 (b + i=2) (b + 1=2) + 1 (b i=2) = ; for b 2 V (SG) + 1=2: (5.12) (b 1=2) + 1 Note that the equations (5.11) and (5.12) are nonlinear discrete versions of the Cauchy-Riemann equations. We shall call them the SG-CR equations. We have established the rst part of the following 5.1. Theorem. (1) Let C be a circle pattern for SG in the sphere, then its and invariants satisfy the SG-CR equations. (2) Conversely, suppose that and are positive functions on V (SG) and V (SG), respectively, and suppose they satisfy the SG-CR equations. Then there is a circle pattern C for SG such that = C and = C. (3) Suppose that C and C are circle patterns for SG and C = C, C = C. Then C is a Mobius image of C. Proof. Part (1) has been proven above. So consider part (2). Use equations (5.6), (5.7) and (5.9) to dene Mobius transformations m [v1 ;v 2 ] for the edges [v 1 ; v 2 ] of HG. We now want to verify that (5.8) holds for every closed path v 0 ; v 1 ; : : : ; v n in HG. Since (5.9) holds, it is enough to demonstrate this for paths that form the boundary of a single square tile of HG. If the path is the boundary of a tile of HG whose center a is in V (SG) + i=2, then (5.8) is equivalent to (5.11), as we have seen. Similarly, if the path is the boundary of a tile of HG whose center b is in V (SG) + 1=2, then (5.8) is equivalent to (5.12). Consider the situation where v j = w + i j (1 + i)=4, j = 0; 1; 2; 3; 4, where w 2 V (SG). Then (5.8) becomes 1 i p (w) 1 i p (w) 1 1 i p (w) p 1 i (w) 1 = ; which indeed holds. Similarly, if v j = z + i j (1 + i)=4, j = 0; 1; 2; 3; 4, where z 2 V (SG), then (5.8) is easily veried. (In fact, in these two cases one can also argue without any computations.)

20 20 ODED SCHRAMM The cases we have checked are sucient to guarantee (5.8) for any closed path v 0 ; v 1 ; : : : ; v n. Now take an arbitrary vertex v 0 in HG, and set m v0 = identity. For every other v 2 V (HG) set m v = m [v0 ;v 1 ] m [v1 ;v 2 ] m [vn1 ;v n ]; where v 0 ; v 1 ; : : : ; v n is an arbitrary path in HG from v 0 to v = v n. Because (5.8) is valid for closed paths in HG, the transformation m v does not depend on the choice of path from v 0 to v. For any w 2 V (SG), dene p(w) = m v (1), where v is one of the vertices of HG closest to w. It does not matter which of these v is chosen, because the transformations in (5.6) preserve 1. For any z 2 V (SG), let C z be the image of R [ f1g under m v, where v is one of the vertices of HG closest to z. It does not matter which of these v is chosen, because the transformations in (5.7) preserve R [ f1g. The orientation of the circle C z is taken as the image of the orientation of R[f1g. It is immediate to verify that for any z 2 V (SG) the points p(z 1=2i=2) appear on C z in the correct cyclic order. One easily shows that the collection C = C z : z 2 V (SG) is a circle pattern for SG, which has invariants and. The details of this, as well as part (3), are left to the reader. An interesting and useful property of the SG-CR equations is that the invariant can be eliminated from them. More precisely, we have the following, 5.2. Theorem. (1) Suppose that : V (SG)! (0; 1) and : V (SG)! (0; 1) satisfy the SG-CR equations. Then satises the equation (z) 2 = ((z + 1) + 1) ((z 1) + 1) ((z + i) 1 + 1) ((z i) 1 + 1) for every z 2 V (SG): (5.13) (2) Conversely, suppose that : V (SG)! (0; 1) satises (5.13). Then there is a : V (SG)! (0; 1) such that and together satisfy the SG- CR equations. Moveover, is unique, up to multiplication by a positive constant. This is similar to the situation with the Cauchy-Riemann y v x x v y u, where v may be eliminated, and the result is the Laplace equation for u. Given a solution u of the laplace equation, a companion solution to the Cauchy-Riemann equation exists, and is unique up to an additive scalar. So the SG situations seems analogous. In fact, equation (5.13) is a nonlinear discrete version of the Laplace equation. Hence, we call it the SG-Laplace equation. Unfortunately, it seems there is no way to eliminate from the SG-CR equations and get an equation involving only. Here is an application of the theorem. Recall the formula (5.4) for the invariants of the p isg erf pattern. From the theorem it follows that if we multiply the invariant by a positive constant b, then the pair ; b still corresponds to an SG immersion on the sphere. In this way, we get a new 2-parameter family of Mobius inequivalent entire SG patterns in the sphere, with invariants given by, (z) = ax + a x a y + a y ; (z) = b a x+y + 1 a x + a y 2 ; (5.14)

21 SG CIRCLE PATTERNS 21 where a; b are arbitrary positive numbers. (In fact, this family embeds in a 4- parameter family, since z may be translated by an arbitrary complex constant.) Let us call these circle patterns erf-like patterns, for lack of a better name. To guess what meromorphic functions the erf-like patterns might correspond to, the Schwarzian derivative should R be examined. Recall that our p isg erf pattern is analogous to the function f = e iz2. The Schwarzian derivative of this function is f 00 0 S f = f 0 1 f f 0 = 2i + 2z 2 : It is then reasonable to guess that the erf-like patterns would correspond to the functions that have Schwarzian derivative S = z 2 + i, where > 0 and 2 R is arbitrary. Figures 5.2.a and 5.2.b show the erf-like pattern with a = 1:152 and b = 1 and the corresponding meromorphic function with Schwarzian equal to z 2 (which is a ratio of two Bessel functions). Figure 5.2.a. Figure 5.2.b. One striking feature of these gures is the =2 rotational symmetry. This symmetry can be read o from the Schwarzian. Indeed, let f be some meromorphic function, and set ^f(z) = f(!z), where j!j = 1. Then S ^f (z) =!2 S f (!z). Consequently, ^f = m f for some Mobius transformation m i! 2 S f (!z) = S f (z). This holds when! = i and S f (z) = z 2. Similarly, the symmetry of the erf-like pattern with b = 1 can be read from the relations (iz) = (z) 1 and (iz) = (z) 1, which are easy to verify. Proof of 5.2. We start with (1). For any z 2 V (SG), 1 = (z + 1=2 + i=2) (z + 1=2 i=2) (z 1=2 i=2) (z 1=2 + i=2) (z + 1=2 i=2) (z 1=2 i=2) (z 1=2 + i=2) (z + 1=2 + i=2) : Taking square roots, and using the SG-CR equations gives 1 = (z + 1) + 1 (z) + 1 (z) (z 1) + 1 (z i) (z) + 1 (z) (z + i) ; (5.15)

22 22 ODED SCHRAMM which simplies to (5.13). The proof of (2) is straightforward. One chooses (1=2 + i=2) arbitrarily. If v; v 0 2 V (SG) are neighbors and (v) is dened, then one of the SG-CR equations denes (v 0 ). So given any v 2 V (SG), we dene (v) by walking along a path from 1=2 + i=2 to v and using either of these equations for every edge. As the computation of the previous paragraph shows, equation (5.13) shows that the choice of path does not inuence the value of (v). This then implies that the SG-CR equations are valid everywhere. The remaining details are left to the reader. For convenience, we introduce the function eh( 1 ; 2 ; 3 ; 4 ) = s ( 1 + 1) ( 3 + 1) Using this notation, equation equation (5.13) can be written as (z) = e H (z + 1); (z + i); (z 1); (z i) : : (5.16) 5.3. Properties of e H. In the following, 1 ; 2 ; 3 ; 4 > 0 are arbitrary. (1) 1 = H e 1 ; 1 ; 1 ; 1. (2) Monotonicity: H e is strictly monotone increasing in each of its arguments. (3) Partial symmetry: H(1 e ; 2 ; 3 ; 4 ) = H(1 e ; 4 ; 3 ; 2 ) = H(3 e ; 2 ; 1 ; 4 ). (4) Rotation: H(4 e ; 1 ; 2 ; 3 ) = H e 1 1 ; 1 2 ; 1 3 ; (5) H(1 e ; 1 ; 3 ; 3 ) = p 1 3. (6) H e 1 ; 1 1 ; 3; 1 = 1. 3 (7) lim 1!0 e H( 1 ; 2 ; 3 ; 4 ) > 0 and lim 2!1 e H( 1 ; 2 ; 3 ; 4 ) < 1. The proof is left to the reader. Property (4) is related to the fact that when two circle patterns C = C z : z 2 V (SG) and C = C z : z 2 V (SG) satisfy C z = C iz, then C (z) = C (iz) 1. One important consequence of (1) and (2) is the validity of the following maximum principle. Suppose that G is a nite subgraph of SG and : V (G)! (0; 1) satises (5.13) at every interior vertex of G. Then attains its maximum and its minimum on (G). 6. The Local Theory The goal of this section is to understand the collection of all circle patterns for a nite `simply connected' subgraphs of SG. First, a `local' version of Theorem 5.1 will be presented Theorem. Let S be a non-empty union of squares of SG so that the interior of S is connected, and the closure simply connected. Let G be the intersection of the 1-skeleton of SG with S, let V S denote the vertices of G that have at least three neighbors in G, and let V (S) be the centers of the squares of S. (1) Let C be a circle pattern for G in the sphere. Then its invariants : V S! (0; 1) and : V (S)! (0; 1) satisfy the SG-CR equations across every edge of SG that borders two squares of S.