shi-yi lan and dao-qing dai- c^∞- convergence of circle patterns to minimal surfaces

8/3/2019 Shi-Yi Lan and Dao-Qing Dai- C^- Convergence of Circle Patterns to Minimal Surfaces

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S.-Y. Lan and D.-Q. Dai

Nagoya Math. J.

Vol. 194 (2009), 149167

C

-CONVERGENCE OF CIRCLE PATTERNS TOMINIMAL SURFACES

SHI-YI LAN and DAO-QING DAI

Abstract. Given a smooth minimal surface F : R3 defined on a simply

connected region in the complex plane C, there is a regular SG circle pattern

Q. By the Weierstrass representation ofF and the existence theorem of SG

circle patterns, there exists an associated SG circle pattern P in C with the

combinatoric ofQ. Based on the relationship between the circle pattern P

and the corresponding discrete minimal surface F : V R3 defined on the

vertex set V of the graph ofQ, we show that there exists a family of discrete

minimal surface : V R3, which converges in C() to the minimal

surface F : R3 as 0.

1. Introduction

The theory of discrete differential geometry is presently emerging on the

border of differential and discrete geometry, which studies geometric shapes

with a finite number of elements (polyhedra) and aims at a development of

discrete equivalents of the geometric notions and methods of surface theory

(see [1], [2], [3], [4], [8], [12], etc.). A smooth geometric shape (such as

surface) appears then as a limit of the refinement of the discretization. One

of the central problems of discrete differential geometry is to find proper

discrete analogues of special classes of surfaces, such as minimal, constant

mean curvature, isothermic, etc. In [2], a new discrete model was intro-

duced to investigate conformal discretizations of minimal surface, i.e., the

analogous discrete minimal surfaces consisting of touching spheres, and of

circles which intersect the spheres orthogonally in their points of touch. It is

proved that the discrete minimal surfaces converge to the smooth ones. Theadvantages of the discretizations are that they respect conformal properties

of surfaces, possess a maximum principle, etc. Here, we are concerned with

the C-convergence of discrete minimal surfaces given in terms of circles

Received June 2, 2006.Revised March 19, 2008.Accepted September 7, 2008.2000 Mathematics Sub ject Classification: 52C26, 53A10, 53C42.


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150 S.-Y. LAN AND D.-Q. DAI

and spheres, that is, the convergence of discrete minimal surface discussed

in [2] is extended to C-convergence.

For each > 0, let SG

denote the square grid with mesh > 0. Thevertices of SG form the square lattice V = {n + mi : (n, m) Z Z},and an edge connects any two vertices of SG at distance . The 1-skeleton

of SG is the graph of regular SG circle pattern Q each circle of which has

radius equal to /

2. Aussume that is a simply-connected domain in C

with = C. Let Q be the largest sub-pattern of Q that is containedin , and let SG be the sub-complex of SG

whose 1-skeleton is equal to

the graph of Q. The vertex set of SG is denoted by V

. Suppose that

F : R3 is a minimal immersion without umbilic points in conformal

curvature line coordinates. First, by the Weierstrass representation ofF andthe local theory of SG circle patterns (see [12, 6]), there is an associatedSG circle pattern P in C with the combinatoric of Q

. In the meantime,

one gets a discrete minimal surface F : V R3 corresponding to F, whichconsists of spheres and circles. Secondly, in terms of Mobius invariants of

circle pattern P, we define the discrete Schwarzians of P and show that

they are uniformly bounded in C(). Thirdly, we construct a Mobius

transformation T through circle pattern P such that they can be expressed

by the discrete Schwarzians. By the C-boundedness of the Schwarzians,

we will prove that T converges in C() to some Mobius transformation

T as 0. Lastly, using the relation between the discrete minimal surfaceF and circle pattern P, and combining with the definition and the C

-

convergence of T, we will show that there exists a family of discrete surface

: V R3, obtained by scaling appropriately the centers of spheres andcircles in F, which converges in C() to the minimal surface F : R3as 0.

This paper is organized as follows. For a given smooth minimal surface

F : R3, we will give the associated SG circle patterns P, the cor-responding discrete minimal surfaces F and the relation between them in

Section 2. In Section 3, we first give the definitions of C-convergence andC-boundedness for discrete functions. Then by using Mobius invariants

of P, we define the discrete Schwarzians of P and show that they are

uniformly bounded in C(). In Section 4, we construct the Mobius trans-

formations T through circle pattern P such that they can be expressed

by the discrete Schwarzians. Then it is proved that T converges in C()

to some Mobius transformation T as 0. In Section 5, by the relationbetween the discrete minimal surface F and the circle pattern P, we will


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CONVERGENCE OF CIRCLE PATTERNS TO MINIMAL SURFACE 151

show that there exists a family of discrete minimal surface : V R3which converges in C() to F : R3 as 0.

2. Circle patterns and discrete minimal surfaces

In this section, for any smooth minimal surface, we apply its Weierstrass

representation to yield associated circle patterns with the combinatorics of

square grid and define corresponding discrete minimal surfaces. Moreover,

we give the relationship between the circle patterns and the discrete minimal

surfaces (also see [12], [2] 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 that |v v

| = and v, v

V

,and whose 2-cells are the squares {v + x + iy : x, y [0, ]}, v V.An indexed collection P = {P(v) : v V} of oriented circles in the

Riemann sphere C is said to be a circle pattern with the combinatorics of

square grid SG (or a SG circle pattern) if the following three conditions

hold.

(a) Whenever v, v are neighbors in SG, the corresponding circles

P(v), P(v) 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 P(v1), P(v2) are distinct and

tangent.

(c) Whenever the situation is as in (b) and v2 is neighbor ofv, which is

one step counterclockwise from v1, the circular order of the triplet of points

P(v) P(v1) P(v2), P(v1) P(v2), P(v) P(v2) P(v1) agreeswith the orientation of P(v).

Clearly, the 1-skeleton of SG is the graph of regular SG circle pattern

Q each circle of which has radius equal to /

2.

Suppose that is a simply connected domain in the complex plane C.

Without loss of generality, we may assume 0

. Let Q be the largest sub-

pattern of Q that is contained in . Let Q be the connected componentof Q that contains 0, and let SG

be the cell complex whose 1-skeleton is

equal to the graph of Q. The set of vertices of SG is denoted by V

, and

the set of centers of squares of SG by V.

A smooth immersed surface in R3 is said to be isothermic if it admits a

conformal curvature line parametrization in a neighborhood of every non-

umbilic. An isothermic immersion is a minimal surface if and only if the

dual immersion is contained in a sphere. In that case the dual immersion is


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in fact the Gauss map of the minimal surface, up to scale and translation.

We first give the following lemma about the Weierstrass representation of

minimal surfaces, which follows from [2].

Lemma 1. Suppose that F : R3 is a minimal immersion withoutumbilic points in conformal curvature line coordinates. Then

(1) F =

Re

1 f(z)2

f(z)dz, Re

i(1 + f(z)2)

f(z)dz, Re

2f(z)

f(z)dz

,

where f : C is a locally injective meromorphic function.

For the locally injective meromorphic function f in Lemma 1, set

(2) (1) (v) = 1 + 2 Re Sf(v)

for each boundary vertex v V, where Sf denotes the Schwarzian deriva-tive of f. Then SG Dirichlet principle [12, Theorem 6.2] implies that

(1)

is extended to a solution of the SG-Dirichlet problem on SG. Let (2) be

the companion of (1) in the solution of the SG

CR equation [12, 5],i.e.,

(3) (2) (v + 0)

(2) (v + 1)

=

(1) (v + i)1 + 1

(1) (v)1 + 1

2

and

(4)(2) (v + 0)

(2) (v + 3)

=

(1) (v + ) + 1

(1) (v) + 1

2

for any v V, such that

(5) (2) (0) = 1 + 2 Im Sf(0),

where j = ij(/2 + i/2) (j = 0, 1, 2, 3).

Based on the local theory of SG circle patterns [12, Theorem 6.1], there

exists an SG circle pattern P = {P(v) : v V} for SG in the com-plex C that has

(1) and

(2) as its Mobius invariants. That is,

(1) (v) =

cr[q0(v), q2(v); q3(v), q1(v)] for any v V and (2) (u) = cr[qj(v +), qj(v ); qj(v i), qj(v + i) for any u = v + j V, where qj(v)


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(j = 0, 1, 2, 3) denotes the point of contact with circles P(v) and P(v+2j)

and cr[q1, q2, q3, q4] denotes the cross ratio of the four points q1, q2, q3, q4 C.On the other hand, we further suppose that there are two kinds of

vertices vs and vc in V such that each edge of SG

has vertices of different

kinds. Then a discrete isothermic surface is a mapping

F : V R3

such that images F(vs) and F(vc) of vs and vc are spheres and circles

respectively. Spheres F(vs) and circles F(vc) intersect orthogonally if vs

and vc belong to the same edge of SG, and spheres F

(v(1)s ) and F(v

(2)s )

(respectively, the circles F(v(1)c ) and F(v

(2)c )) are distinct and tangent if

v(1)s and v(2)s (respectively, v

(1)c and v

(2)c ) belong to the same face of SG.

Let F(v) be the center of sphere (or circle) F(v) for each v V,pj(vs) be the intersection points of spheres F

(vs) and F(vs + j) (j =

0, 1, 2, 3), and let pj(vs) = F(ws) + bj . Then a discrete isothermic surface

F : V R3 is called a discrete minimal surface if it satisfies any one ofthe equivalent conditions below.

(a) The points F(vs) + (1)jbj lie on a circle.(b) There is a d R3 such that (1)j(bj , d) is the same for j = 0, 1, 2, 3.(c) There is a plane through F(vs) such that the points

{pj(vs)

|j =

0, 2} and the points {pj(vs) | j = 1, 3} lie in planes which are parallel to itat the same distance on opposite sides.

From the definition above, it is easy to see that a discrete isothermic

surface is a discrete minimal surface, if and only if the dual discrete isother-

mic surface corresponds to a Koebe polyhedron (see [2, 4]). Let A(v)denote the center of circle P(v) in the circle pattern P for each v V.Then the following lemma, which can follow from [2, 5], gives the relationbetween the discrete minimal surfaces F and the circle patterns P.

Lemma 2. For any vertices v(1)s andv(2)s that belong to the same squareof SG, let F

(v(1)s ) and F(v

(2)s ) denote the centers of spheres F(v

(1)s ) and

F(v(2)s ) in the discrete minimal surface F, respectively. Then

F(v(1)s ) F(v(2)s )(6)

= Re

R(v

(1)s ) + R(v

(2)s )

1 + |q|2A(v

(1)s ) A(v(2)s )

|A(v(1)s ) A(v(2)s )|(1 q2, i(1 + q2), 2q)

,


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where q denotes the point of contact with circles P(v(1)s ) and P(v

(2)s ) in

the circle pattern P and the radii R(v(j)s ) of the sphere F(v

(j)s ) (j = 1, 2)

are

R(v(j)s ) =

1 + |A(v(j)s )|2 |A(v(j)s q|22|A(v(j)s q|

.(7)

For the discrete minimal surface F, we define a discrete surface F :

V R3 by F(v) = F(v) for each v V. Then F is called a dis-crete minimal surface comprised of points and F is called one consisting

of spheres and circles for distinction. Next, we will show that after scalingappropriately, F converges in C() to F : R3.

3. The C-boundedness of discrete Schwarzians

In this section we first give some definitions and notations related to

discrete differential operators (also see [8]). Then using the Mobius invari-

ants (1) and

(2) given in Section 2, we define the discrete Schwarzians of

circle pattern P and show that they are uniformly bounded in C().

Let W be a subset of V. A vertex v W is said to be an interiorvertex of W if all its neighboring vertices are contained in W. Let intWdenote the set of the interior vertices of W. Given a function : W R,we define the discrete directional derivative j : intW R by

j(v) = ((v + ij) (v))/,

for each j Z4. The discrete Laplacian of a function : W R is afunction in intW defined by the formula

(v) = 1/(4

2

)

3j=0((v + i

j

) (v)).

For any differentiable function H : R, let jH denote the direc-tional derivate

jH(z) = lims0

H(z + ijs) H(z)s

,

where j = 0, 1, 2, 3. Let f : Cd be a function defined in . For each > 0, let f be a function defined on some set of vertices V V, with


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values in Cd. Assume that for each z , there are some 1, 2 > 0 suchthat {v V : |v z| < 2} V whenever (0, 1).

If for every z and every > 0, there are some 1, 2 > 0 such that|f(z) f(v)| < , for every (0, 1) and every v V with |v z| < 2,then we say f converges to f, locally uniformly in .

Let n N, and suppose that f is Cn-smooth. If for every sequencej1, j2, . . . , jk Z4 with k n we have jkjk1 j1 f jkjk1 j1 flocally uniformly in , then we say that f converges to f in Cn(). If that

holds for all n N, then the convergence is C. The functions f are saidto be uniformly bounded in Cn() provided that for every compact K there is some constant C(K, n) such that

jkjk1 j1 fKV < C(K, n)whenever k n, and is sufficiently small, where denotes the L-norm.The functions f are uniformly bounded in C(), if they are uniformly

bounded in Cn() for every n N.For a smooth vector function F : R3 and a discrete vector function

F : V R3, it is said that F converges in C() to F if every componentof F converges in C() to the corresponding one of F, and that F are

uniformly bounded in C() if all components ofF are uniformly bounded

in C().

For (1) and (2) given in Section 2, we define the two discrete Schwarz-ians of P as follows: let

(8) h(1) (v) = 2((1) (v) 1)

for each vertex v V and let(9) h(2) (u) =

2((2) (u) 1)for every u V. It is easy to see that h(1) = h(2) 0 if P is a regular SGcircle pattern, because

(1) =

(2)

1 for regular SG circle patterns.

Let SGh denote (1/2)SG + (1 + i)/4, and the set of vertices of SGhis denoted by Vh . For any w Vh , let v V be the unique vertex of SGthat is closest to w. Let Mw be the Mobius transformation that takes ,0, 1 to q0(v), q1(v), q3(v), respectively. Set

M[w1,w2] = M1w1 Mw2

for each directed edge [w1, w2] of SGh. Write ej(v) = [v + j/2, v + j+1/2]

for any v V V and for any j Z4, then we have


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Lemma 3. There hold the following equalities

Me0(u)(z) = Me2(u)(z) = 1

i(h(2) (u) + 1)1/2z,(10)

Me1(u)(z) = Me3(u)(z) = 1 i(1/)(h(2) (u) + 1)1/2z(11)

for each u V andMe0(v)(z) = Me2(v)(z) = (

2(h(1) (v) + 1)1 + 1)1/(1 z),(12)

Me1(v)(z) = Me3(v)(z) = (2(h(1) (v) + 1) + 1)

1/(1 z)(13) for everyv V.

Proof. For any u V

, we first consider the directed edge e2(u) =[u + 2/2, u + 3/2] SGh. By the definition of M[w1,w2], we get thatM[w1,w2] does not change if we apply a Mobius transformation to P

. So we

may assume that Mu+2/2 is the identity. It follows from the definition of

Mw that

q1(v) = , q1(v ) = 0, q1(v i) = 1,where v V is the unique vertex of SG that is closest to u + 3/2. Hencewe deduce that the four points q1(v ), q1(v i) form the vertices of arectangle. From the definition of

(2) in Section 2, we get

(2) (u) =

q1(v + ) q1(v i)q1(v i) q1(v )

2,

which implies

q1(v + ) = q1(v i) i((2) )1/2(q1(v i) q1(v )) = 1 i((2) )1/2.Since Me2(u) takes , 0, 1 to q1(v), q1(v i), q1(v + i), respectively, weconclude from (9) that

Me2(u)(z) = 1 i((2)

)1/2

z = 1 i(h(2)

(u) + 1)1/2

z.

Similarly, we deduce that Me0(u)(z) = 1 i(h(2) (u)+1)1/2z. So (10) holds.Identical to the above arguments, we conclude that (11) holds.

Next, we consider the directed edge e0(v) = [v + 0/2, v + 1/2] SGhfor any v V. We assume with no loss of generality that Mv+0/2 is theidentity. Then we obtain

q0(v) = , q1(v) = 0, q1(v) = 1.


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that g1(v) = |v|2 = O(2) on V. Hence we obtain g1(z) O(2) in V,which implies

h(1)

(v) Re Sf(v) + O(2

).

On the other hand, if we let g2(v) = h(1) (v) Re Sf(v) |v|2, then similar

to the above arguments, we conclude that

h(1) (v) Re Sf(v) + O(2).

So it follows that

h(1) (v) = Re Sf(v) + O(2),

which implies (i).

Next, by (3), the relationship between (1) and h(1) and Taylors for-mula, we get

log (2) (v + 0) log (2) (v + 1)= 2 log((1) (v + i)

1 + 1) 2log((1) (v)1 + 1)= 2 log((1 + 2h(1) (v + i))

1 + 1) 2 log((1 + 2h(1) (v))1 + 1)= 2 log(2 2h(1) (v + i)) 2log(2 2h(1) (v)) + O(4)= 2h(1) (v)

2h(1) (v + i) + O(

4).

Hence it follows from (14) and Taylors formula

log (2) (v + 0) log (2) (v + 1)(16)= 2 Re(Sf(v) Sf(v + i)) + O(4)= 2 Re(iSf(v)) + O(

4)

= 3 Im Sf(v) + O(4)

= 2 Im Sf(v + i) 2 Im Sf(v) + O(4).

Similarly, we conclude from (4) that

log (2) (v + 0) log (2) (v + 3)(17)= 2 Im Sf(v + ) 2 Im Sf(v) + O(4).

By (5) and Taylors formula, we deduce that

log (2) (/2 + i/2) = 2 Im Sf(/2 + i/2) + O(

4).(18)


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So we conclude from (16), (17) and (18) that

log (2) (u) = 2 Im Sf(u) + (u)O(

2),

By the relationship between (2) and h

(2) and Taylors formula, we obtain

h(2) (u) = Im Sf(u) + (u)2 O(1),

which implies (ii). This completes the proof of the lemma.

Lemma 5. Let v V, u V, and suppose that the distance from v(respectively, u) to is greater than 2. Then

h

(1)

(v) C, h(2)

(u) Cfor some constant C = C(, f) which depends only on and f.

Proof. Note that Re Sf and Im Sf are bounded on compact subset

K and = O(1/) on K. By Lemma 4, we conclude that the lemmaholds.

Similar to the case of regular hexagonal lattice (see [8, 7]), for regularsquare lattice we have

Lemma 6. Suppose that(i) W is a subset of V (orV); (ii) u intW;(iii) is the distance from u to V W (or V W). If : W R is anyfunction, then the inequality

(19) |j(u)| < 5 + (1/2)2intWholds for any j = 0, 1, 2, 3.

Proof. The proof of the Lemma is similar to that of [8, Lemma 7.1]

where regular hexagonal lattices were investigated.

Theorem 1. Let n be an integer, and let V (respectively, V ) be the

set of vertices of V (respectively, V) whose distance to is at least for

each > 0. Then there are constants C = C(n, ) and = (n, ) > 0 such

that

jnjn1 j1 h(1) V < C, jnjn1 j1h(2) V < C(20)

hold for each < , and j0, j1, . . . , jn Z4.


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Proof. The proof proceeds by induction on n. In the case n = 0, by

Lemma 5, we get that (20) holds. So we suppose that n > 0, and that (20)

holds for n = 0, 1, 2, . . . , n 1. Let1 = jn1 j1 h(1) , 2 = jn1 j1 h(2) .

Note that the operators and j can commute with each other, so we

have

1 = jn1 j1 h(1) = jn1 j1 h(1) ,

and

2 = jn1 j1 h(2) = jn1 j1 h(2) .

By Lemma 4, it follows that

h(1) = Re Sf +

(2O(1)), h(2) = Im Sf +

(2O(1)).

Since Sf = O(2) and = O(1/) on compact subset K , we have

h(1) = O(2) + O(1), h(2) = O(

2) + (O(1))

on compact subset K . Note that O(1) C(), we deduce that thereexists a constant C1 = C1(, n) such that

1(v)

V

C1,

2(v)

V

C1.

Since |1| and |2| are bounded on V and V , respectively, it follows fromLemma 4 that |1| and |2| are bounded on V and V , respectively,which completes the induction step. So (20) holds for any integer n and

any j0, j1, . . . , jn Z4.4. The Mobius transformations of circle patterns

In this section, we construct the Mobius transformations T through cir-

cle pattern P such that they can be expressed by the discrete Schwarzians

h(1) and h

(2) . By the boundedness of h

(1) and h

(2) , we will prove that T

converges in C() to some Mobius transformation T as 0 and obtainthe relation between T and f.

Note that 0 V, so we may suppose that P is normalized by Mobiustransformations so that

(21) q0(0) = f(0), q1(0) = f(1), q3(0) = f(3),

where j = ij(1 + i)/2 (j = 0, 1, 2, 3). Then we have


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Theorem 2. For each v intV, let T = T(v) be the Mobius transformation that takes the three points 0, 1, 3 to the points q0(v), q1(v),

q3(v), respectively. Then(i) the limit

(22) T(z) = lim0

T(v)

exists for any 0, and the convergence is in C().(ii) T(z)(0) = f(z).

Proof. (i) Let B = B be the Mobius transformation that takes , 0,1 to 0, 1, 3, respectively. Then we have

(23) B(z) = z (1 + i)/2(1 i)z + i .

Recall 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+0/2,v+0/2+/2] M[v+0+/2,v+0+] B1.

Using the usual matrix notation for Mobius transformations and the factthat M[w1,w2] = M

1[w2,w1]

, we obtain from (10), (12) and (23) that

T(v)1 T(v + )

=

(1 + i)/2

1 i i

i(h(2) (v + 0) + 1)1/2 10 1

0 (2(h(1) (v + ) + 1)1 + 1)1

1 1

1

(1 + i)/21 i 1

1.

By Lemma 4 and noting that Sf is Lipschitz, we deduce that

T(v)1 T(v + ) =

1 Sf(v)/2 1

+ O(2) = I + C(v) + O(2),

where I denotes the identity matrix and

C(v) =

0 1

Sf(v)/2 0

.


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This implies

(24) T(v + ) = T(v) + T(v)C(v) + T(v)O(2).

Similarly, we have

(25) T(v + ij) = T(v) + ijT(v)C(v) + T(v)O(2)

for j = 1, 2, 3. We assume, without loss of generality, that

f(0) = 0, f(0) = 1, f(0) = 0,

because the statement of Theorem 2 is Mobius invariant. Thus we get from

Taylors formula that

f(ij(1 + i)/2) = ij(1 + i)/2 + O(3)

for j = 0, 1, 2, 3. From (21) and the definition of T(v), it follows that

T(0)(ij(1 + i)/2) = ij(1 + i)/2 + O(3)

for j = 0, 1, 3, which implies

T(0) = I + O().

By (24) and (25), we deduce that the matrices T(v) (v intV) arebounded in compact subsets of , independently of . On the other hand,

(24) and (25) imply

(26) jT(v) = ijT(v) C(v) + T(v) O() = T(v) O(1)

for j = 0, 1, 2, 3, where O(1) = ijC(v) + O(). It follows from Theorem 1

that h(1) and h

(2) are C-bounded, so O(1) is bounded in C(). By re-

peating differentiation of (26), we conclude that T(v) is bounded in C()

uniformly in . By the properties of C-convergence of functions (see [8,

Lemma 2.1]), we obtain that (22) holds for some subsequence of 0, andthe convergence is C().

In the following we will show that (22) is also valid for every sequence of

0. Indeed, let D(v) be the matrix solution of the differential equation

(27) D(v) = D(v)C(v)


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with initial condition D(0) = I, then we have

D(v + ij) = D(v) + ijD(v)C(v) + O(2)

for j = 0, 1, 2, 3. From (24) and (25), we obtain

|T(v + ij) D(v + ij)| |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

|T

(v) D(v)| = ((v)O(2

) + O())(1 + O())

(v)

,

where (v) is the combinatorial distance from v to 0 in SG. Hence we

deduce

|T(v) D(v)| O()eO(1) = O()on a compact subset K , because (v) = O(1/) on K. So T(v) =D(v) + O(), which implies that (22) holds for every 0.

In equation (26), taking a limit as 0, we obtain

(28) jT(z) = ijT(z)C(z).

Hence, we get

jT(z) = j+2T(z),which implies that T(z) is a matrix-valued analytic function of z. In addi-

tion, it follows from (28) that the determinant ofT(z) is constant in . Note

that at z = 0 this determinant is 1. So T(z) is a Mobius transformation for

each z .(ii) Let

T(z) =a(v) b(v)

c(v) d(v)

.

Then T(z) satisfies the differential equation (27). By the definitions of

Schwarzian derivative and C(v), we deduce that

Sb/d = Sf.

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).


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5. C-convergence to minimal surfaces

We will show that there exists a family of discrete minimal surface

: V R3 which converges in C() to the smooth minimal surfaceF : R3 in this section.

We first give some properties of C-convergence of discrete functions,

that is

Lemma 7. Suppose that f, g converge in C() to functions f, g :

C, respectively. Then the following statements hold in C(): (i)f + g f + g; (ii) fg f g; (iii) 1/f 1/f if f = 0 in ; (iv)

f f if f > 0; (v) |f| |f|.

Proof. Using the definition of discrete directional derivative as well as

the definition ofC-convergence of discrete function, we deduce easily that

the lemma holds.

For each v V, let f(v) and r(v) denote the center and radius ofcircle P(v) in the circle pattern P, respectively, and let g

(v) denote the

intersection point q0(v) of circles P(v) and P(v + 20) in P

. Then we

have

Lemma 8. The discrete functions f(v) and g(v) converge in C()to the locally injective meromorphic function f : C, and r(v)/ con-verges in C() to |f| as 0.

Proof. First, we write T(v) and T(z) as matrices

T(v) =

a(v) b(v)c(v) d(v)

, T(z) =

a(z) b(z)c(z) d(z)

.

Then it follows from Theorem 2(i) that 0c+ d and 0a

+ b converge to d

and b in C(), respectively. By Theorem 2(ii), we obtain that b(z)/d(z) =f(z). Since the determinant of T(z) is nonzero, d is nonzero in . By

Lemma 7, we get that (0a + b)/(0c

+ d) b/d in C(). Note thatg(v) = T(v)(0), so we conclude that the discrete function g

f inC() as 0.

Next, let c be the circle that contains the three points 0, 1, 3. Since

T(v) maps the three points 0, 1, 3 to q0(v), q1(v), q3(v), respectively,

and since the three points q0(v), q1(v), and q3(v) lie in the circle P(v),


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18/19


for any v(1)s , v

(2)s lying in the same square of SG, where

1 = R(v

(1)

s ) + R(v

(2)

s )1 + |g|2 |f

(v

(1)

s ) f

(v

(2)

s )|f(v

(1)s ) f(v(2)s )

(1 (g)2),

2 =R(v

(1)s ) + R(v

(2)s )

1 + |g|2|f(v(1)s ) f(v(2)s )|f(v

(1)s ) f(v(2)s )

i(1 + (g)2),

3 =R(v

(1)s ) + R(v

(2)s )

1 + |g|2|f(v(1)s ) f(v(2)s )|f(v

(1)s ) f(v(2)s )

(2g).

By Lemma 8, we get that f and g converge in C() to f, and that

r

(v)/ converges in C

() to |f

| as 0. Moreover, it is easy to see that(f(v(1)s ) f(v(2)s ))/ converges in C() to 2f. On the other hand, weget from (7) that

R(v(k)s ) =

1 + |f(v

(k)s )|2 |f(v(k)s ) g|2

2|f(v(k)s ) g|

for k = 1, 2. So we deduce from the definitions of l (l = 1, 2, 3) and Lemma

7 that 1, 2 and

3 converge in C

() to (1 f2)/f, i(1 + f2)/f and2f /f, respectively.

Set (v) = 22F(v) for each v V, then we conclude from (29) thatthe discrete minimal surface converges in C() to the smooth minimal

surface F : R3. This completes the proof of Theorem 3.

Acknowledgments. This work is supported in part by NSF of China

(60575004, 10771220), the Ministry of Education of China (SRFDP-

20070558043), NSF of Guangxi (0991081), the Department of Education

of Guangxi (200707MS043), and the Grant for Guangxi University for Na-

tionalities (2008ZD009).

References

[1] S. I. Agafonov, Imbedded circle patterns with the combinatorics of the square grid and

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[3] A. I. Bobenko and U. Pinkall, Disrete isothermic surfaces, J. Reine Angew. Math.,

475 (1996), 187208.


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[4] A. I. Bokenko and Y. B. Suris, Discrete Differnetial Geometry. Consistency as In-

tergrabilty, arXiv:math.DG/0504358 v1, 18 Apr. 2005.

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Riemann map, Acta Math., 180 (1998), 219245.

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University Notes, Princeton, N.J., 1980.

Shi-Yi Lan

Department of Mathematics

Guangxi University for Nationalities

Nanning 530006P. R. China

Dao-Qing Dai

Department of Mathematics

Sun Yat-Sen (Zhongshan) University

Guangzhou 510275

P. R. China

[email protected]

shi-yi lan and dao-qing dai- c^∞- convergence of circle patterns to minimal surfaces

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