Discrete Isothermic Surfaces �Alexander Bobenko yUlrich Pinkall zFachbereich Mathematik, Technische Universit�at Berlin,Strasse des 17. Juni 136, 10623 Berlin, GermanyNovember 10, 1994AbstractDiscrete isothermic surfaces are de�ned. It is shown that these dis-crete surfaces possess properties, which are characteristic for smoothisothermic surfaces (M�obius invariance, dual surface). Quaternioniczero-curvature loop group representations for smooth and discrete iso-thermic surfaces are presented. A Weierstrass type representation fordiscrete minimal isothermic surfaces is obtained and the discrete ca-tenoid and the Enneper surface are constructed.1 IntroductionSurfaces in an Euclidean 3-space studied by the theory of integrable systemsare usually described as immersions of R2 or of some domain in R2F : R2 ! R3: (1)The list includes surfaces with constant negative Gaussian and mean cur-vature and some others (see the survey [2]). In this case R2 is treated as�partially supported by the Sonderforschungsbereich 288ye-mail: bobenko@sfb288.math.tu-berlin.deze-mail: pinkall@sfb288.math.tu-berlin.de1

a space-time of the corresponding integrable system. One can describe to-pological planes, cylinders and tori in this way. To study surfaces of morecomplicated topology one should deal with immersionsF : R ! R3; (2)where R is some Riemann surface. The traditional theory of integrable sy-stems needs essential improvements to be applied to this case and up to nowthere exists no real progress.This paper is part of our general program on the discretization of surfacesdescribed by integrable systems. By such a surface we mean a mapF : G! R3; (3)where G is some graph, which is treated as a discrete space-time of thecorresponding discrete integrable system. When the size of the edges tendsto zero such a surface approximates a corresponding smooth surface (2). Thecase closest to (1) F : Z2! R3 (4)is relatively well understood. In [3], [4] the discrete surfaces (1) with constantGaussian and mean curvature were de�ned and their geometrical propertieswere investigated (for the loop group factorization interpretation of thesesurfaces see [9]). We hope that the integrable discretization (3) may becomea method to understand the structure of its corresponding smooth limit (2).Besides this, the elementary geometry of the case (4) itself is very nice andworth studying.The present paper was initiated by the recent preprint of J. Cie�sli�nski,P. Goldshtein and A. Sym [7]. These authors, based on classical results ofP. Calapso [6] and W. Blaschke [1], give a characterization of isothermicsurfaces as "soliton surfaces" by introducing a spectral parameter. The geo-metrical meaning of this spectral parameter was clari�ed in [5], where itwas shown how pairs of isothermic surfaces are given by curved ats in apseudo-Riemannian symmetric space and vice versa.Here, using the spinor representation of SO(4; 1), in Section 3 we rewritethe zero-curvature representation for the associated family of isothermic sur-faces in terms of 2 � 2 matrices with quaternionic coe�cients. A naturaldiscretization of this representation yields the following de�nition of discreteisothermic surfaces. 2

De�nition 1 A discrete isothermic surface is a map F : Z2 ! R3 such thatthe cross-ratios qn;m = q(Fn;m; Fn+1;m; Fn+1;m+1; Fn;m+1) of all elementaryquadrilaterals are real qn;m 2 R and satisfy the factorization conditionqn;mqn+1;m+1 = qn+1;mqn;m+1: (5)This de�nition coincides with the de�nition (35) in the Remark after De�ni-tion 6 in Section 4.Geometrical properties of discrete isothermic surfaces as well as the ana-lytical description of the corresponding associated family are studied in Sec-tions 4, 5. In Section 7 we de�ne discrete isothermic minimal surfaces, derivea Weierstrass type representation for them and construct some simple ex-amples.2 Isothermic surfaces and their propertiesLet F be a smooth surface in a 3-dimensional Euclidean space and F (x; y)F = (F1; F2; F3) : R ! R3;a conformal parametrization of F< Fx; Fx >=< Fy; Fy >= eu; < Fx; Fy >= 0: (6)Here the brackets denote the scalar product< a; b >= a1b1 + a2b2 + a3b3;and Fx and Fy are the partial derivatives @F=@x and @F=@y.The vectors Fx; Fy as well as the normal N ,< Fx; N >=< Fy; N >= 0; < N;N >= 1; (7)de�ne a moving frame on the surface. We consider a local theory: R is adomain in R2.If the second fundamental form is diagonal� < dF; dN >= eu(k1dx2 + k2dy2) (8)with respect to the induced metric< dF; dF >= eu(dx2 + dy2); (9)3

then the parametrization is called isothermic. In this case k1 and k2 are theprincipal curvatures and the preimages of the curvature lines are the linesx = const and y = const in the parameter domain.De�nition 2 Surfaces which admit isothermic coordinates are called iso-thermic.Surfaces of revolution, quadrics and constant mean curvature surfaceswithout umbilics are examples of isothermic surfaces. We consider isothermicimmersions, i.e. suppose that u(x; y) is �nite.Due to (6, 7) the moving frame� = (Fx; Fy; N)Tsatis�es the following Gauss-Weingarten equations:�x = 0B@ ux=2 �uy=2 k1euuy=2 ux=2 0�k1 0 0 1CA�; �y = 0B@ uy=2 ux=2 0�ux=2 uy=2 k2eu0 �k2 0 1CA �: (10)The compatibility conditions of this system are the Gauss-Codazzi equationsuxx + uyy + 2k1k2eu = 0;2k2 x � (k1 � k2)ux = 0; (11)2k1 y + (k1 � k2)uy = 0:Our goal is to �nd a proper discrete version of isothermic surfaces. Let usmention two important geometrical properties of isothermic surfaces whichwe want to retain in the discrete case.Property 1 (M�obius invariance). Let F : R ! R3 be an isothermic im-mersion and M a M�obius transformation of Euclidean 3-space. Then ~F �M� F : R ! R3 is also isothermic.Proof. Isothermic coordinates (8),(9) can be characterized in terms of F onlyjjFxjj = jjFyjj; < Fx; Fy >= 0; Fxy 2 spanfFx; Fyg: (12)Clearly, if is enough to prove Property 1 for the case of an inversionM~F = F< F;F >:4

Calculating ~Fx; ~Fy we see the conformality of ~F . A direct calculation provesthe third property of (12) of ~F~Fxy = (� � 2< F;Fy >< F;F > ) ~Fx + (� � 2< F;Fx >< F;F > ) ~Fy;where � and � are de�ned byFxy = �Fx + �Fy:Property 2 (Dual surface). Let F : R ! R3 be an isothermic immersion.Then the immersion F � : R ! R3 de�ned by the formulasF �x = e�uFx; F �y = �e�uFy (13)is isothermic. The Gauss maps of F and F � are antipodalN = �N�:The fundamental forms of F � are as follows< dF �; dF � > = e�u(dx2 + dy2);� < dF �; dN� > = �k1dx2 + k2dy2: (14)De�nition 3 The immersion F � : R ! R3 de�ned above is called dual toF .Proof of Property 2. The de�nition (13) of F � makes sense since the equalityF �xy = F �yx is equivalent to (e�uFx)y = 12e�u(uxFy�uyFx) = (�e�uFy)x. Herewe use (13) and the Gauss-Weingarten equation for Fxy. The conformalityof F � is evident. The expressions (14) are obtained by straightforward cal-culation.Remark F �� = F .To discretize isothermic surfaces (see Sect.5) and for investigation by analy-tical methods it is more convenient to use 2 � 2 matrices instead of 3 � 3matrices (10), therefore �rst we rewrite equations (10) for the moving framein terms of quaternions. 5

Let us denote the algebra of quaternions byH, the multiplicative quater-nion group by H� =H n f0g and the standard basis of H by f1; i; j;kgij = k; jk = i; ki = j: (15)Using the standard matrix representation of H the Pauli matrices �� arerelated to this basis as follows:�1 = 0 11 0 ! = i i; �2 = 0 �ii 0 ! = i j;�3 = 1 00 �1 ! = i k; 1 = 1 00 1 ! : (16)We identify 3-dimensional Euclidean space with the space of imaginary qua-ternions Im HX = �i 3X�=1X��� 2 Im H ! X = (X1;X2;X3) 2 R3: (17)The scalar product of vectors in terms of matrices is then< X;Y >= �12trXY: (18)We also denote by F and N the matrices obtained in this way from thevectors F and N .Let us take a unitary quaternion� 2 H�; det� = 1; (19)which transforms the basis i; j;k into the basis Fx; Fy; N :Fx = eu=2��1i�; Fy = eu=2��1j�; N = ��1k�: (20)A simple calculation proves the following form of the Gauss-Weingartensystem.Theorem 1 The moving frame Fx; Fy; N of an isothermic surface is des-cribed by the formulas (20), where the unitary quaternion � satis�es theequations �x = A�; �y = B� (21)with A;B of the formA = uy4 k+ k12 eu=2j; B = �ux4 k� k22 eu=2i: (22)6

The compatibility condition of (21)Ay �Bx + [A;B] = 0 (23)is the Gauss-Codazzi system (11).3 Lax representation for smooth isothermicsurfacesTo put a class of surfaces into frames of the theory of integrable equationsone should "insert" a spectral parameter � into the Gauss-Codazzi systemAy(�)�Bx(�) + [A(�); B(�)] = 0:Geometrically � describes some deformation of surfaces preserving their pro-perties (one can �nd many examples in [2]). In some sense, the case ofisothermic surfaces is more complicated. A characterization of isothermicsurfaces as "soliton surfaces" by introducing a spectral parameter was givenin a short note by J. Cie�sli�nski, P. Goldstein and A. Sym [7]. They descri-bed a Lax pair with coe�cients in so(4; 1). The Lax representation (24, 25)we present below results by using a 4-dimensional spinor representation ofso(4; 1) (see, for example, the Appendix in [8]). By a direct calculation onecan check the following statementTheorem 2 The systemx = U(�); y = V (�); (24)U(�) = A �ieu=2�ie�u=2 A ! ; V (�) = B �jeu=2��je�u=2 B ! ; (25)where A;B are the matrices (22), is compatible if and only if the Gauss-Codazzi equations of isothermic surfaces (11) are satis�ed.Whereas for � = 0 the system (24, 25) is just a doubled Gauss-Weingartensystem (21, 22), the geometrical meaning of the Lax representation with� 6= 0 is more complicated, and we do not discuss it in this paper. For thisinterpretation we refer to the recent preprint [5] by F. Burstall, U. Hertrich-Jeromin, F. Pedit and U. Pinkall, where they showed how pairs of isothermic7

surfaces are given by curved ats in a pseudo-Riemannian symmetric spaceand vice versa.Let us note the following symmetries of the system (24, 25). If (x; y; �)is a solution of the Cauchy problem with (x0; y0; �) = I, then(��) = I 00 �I !(�) I 00 �I ! (26)and (� = 0) = � 00 � ! ; (27)where � 2 H�, solves (21, 22).Theorem 3 (Sym formula) Let (x; y; �) be a solution of (24, 25) satisfying(26, 27), then the isothermic immersion F and its dual F � are given by theformula �1@@� j�=0 � F̂ = 0 FF � 0 ! : (28)The corresponding Gauss maps are�1 0 k�k 0 !����=0 � N̂ = 0 NN� 0 ! :Proof. For F̂x formulas (27, 28) implyF̂x = �1@U@�����=0 = 0 eu=2��1i�e�u=2��1i� 0 ! = 0 FxF �x 0 ! ;where one should use (20, 13). The proof for F̂y and N̂ is the same.4 Discrete isothermic surfacesTo de�ne discrete isothermic surfaces we need a notion of a cross-ratio ofa quadrilateral (X1;X2;X3;X4) in 3-dimensional Euclidean space. One caneasily extend the notion of the cross-ratio of complex numbers to points inR3 identifying a sphere1 S, passing through X1;X2;X3;X4 with the Riemannsphere CP1.It turns out that there is a nice quaternionic description of the cross-ratioif one uses the isomorphism (17).1a plane is a special case 8

De�nition 4 Let X1;X2;X3;X4 2 ImH be 4 points in R3 (see Fig. 1). Theunordered pair fq; �qg of eigenvalues of the quaternionQ = (X1 �X2)(X2 �X3)�1(X3 �X4)(X4 �X1)�1 (29)is called the cross-ratio of the quadrilateral (X1;X2;X3;X4).a

a’

b b’

X1 X2

X4 X3Figure 1: Vertices and edges of a quadrilateralQ is invariant with respect to translations and dilatation of R3; rotationsof R3 act on Q as Q! RQR�1 and therefore preserve fq; �qg.We usually will just speak of the "cross-ratio q 2 C". One has to keepin mind that q is only well de�ned up to complex conjugation. Denoting thequaternions corresponding to the vector-edges as in Fig.1 one getsQ = aa0�1b0�1b:If the quadrilateral (X1;X2;X3;X4) is planar, it can be put to the i; j planeand its edges a1i + a2j = (a11+ a2k)ican be identi�ed with complex numbersa = a1 + ia2 2 C: (30)For the cross-ratio this impliesq = aa0bb0 2 C (31)Lemma 1 The cross-ratio is invariant with respect to the M�obius transfor-mations of R3. 9

Proof. Only for inversions the proof needs some calculation. Describing theinversion by the inverse matrixX ! ~X = X< X;X > = �X�1 (32)we get~Q = (X�11 �X�12 )(X�12 �X�13 )�1(X�13 �X�14 )(X�14 �X�11 )�1 = X�11 QX1:The eigenvalues of Q and ~Q coincide.De�nition 5 Let F : R ! R3 be an immersion and F;Fx; :::; Fyy the valuesof the immersion function and its derivatives at some point (x; y) 2 R. Aone-parametric family of quadrilaterals F � = (F1; F2; F3; F4) with verticesF1 = F + �(�Fx � Fy) + �22 (Fxx + Fyy + 2Fxy);F2 = F + �(Fx � Fy) + �22 (Fxx + Fyy � 2Fxy);F3 = F + �(Fx + Fy) + �22 (Fxx + Fyy + 2Fxy);F4 = F + �(�Fx + Fy) + �22 (Fxx + Fyy � 2Fxy)is called an in�nitesimal quadrilateral at (x; y).Up to terms of order O(�3) the vertices F1; F2; F3; F4 of the in�nitesimalquadrilateral coincide with F (x��; y��); F (x+�; y��); F (x+�; y+�); F (x��; y + �) respectively. The following remark is trivial:Lemma 21) q(F �) = �1 +O(�)() Q(F �) = �I +O(�);2) q(F �) = �1 +O(�2)() Q(F �) = �I +O(�2):Theorem 4 Conformal and isothermic immersions F are characterized interms of in�nitesimal quadrilateral as follows:1) Q(F �) = �I +O(�)() F is conformal,10

2) Q(F �) = �I +O(�2)() F is isothermic.Proof. To calculate the cross-ratio of the in�nitesimal quadrilateral we note,that F � up to scaling is a translation of the quadrilateral with vertices atX1 = 0; X2 = Fx � �Fxy; X3 = Fx + Fy; X4 = Fy � �Fxy:Inverting it by the transformation (32) we send one of the points to in�nity~X1 =1; ~X2 = Fx � �FxykFx � �Fxyk2 ; ~X3 = Fx + FykFx + Fyk2 ; ~X4 = Fy � �FxykFx � �Fxyk2 :The condition Q( ~X1; ~X2; ~X3; ~X4) = �I +O(�2)is equivalent to the equality of vectors�!~X2 ~X3= �!~X3 ~X4 +O(�2);or equivalently2 Fx + FykFx + Fyk2 = Fx � �FxykFx � �Fxyk2 + Fy � �FxykFy � �Fxyk2 +O(�2) = (33)= Fx � �FxykFxk2 1 + 2�< Fx; Fxy >kFxk2 !+ Fy � �FxykFyk2 1 + 2�< Fy; Fxy >kFyk2 !+O(�2):The zero oder term in (33) yields (6) and the term of oder � implies that Fxylies in the tangential plane, i.e. the third condition in (12).The following geometrical properties of the cross-ratio are standard andcan be easily checked:Lemma 3 If the cross-ratio of a quadrilateral (X1;X2;X3;X4) is real q 2R; Q = qI, then X1;X2;X3;X4 lie on a circle. The cross-ratio of a square isq = �1.Now we are in a position to give a de�nition of discrete isothermic surfa-ces. By a discrete surface we mean a map F : Z2 ! R3. We use the followingnotations for the elements of discrete surfaces (n;m are integer labels):Fn;m - for the vertices,[Fn+1;m; Fn;m]; [Fn;m+1; Fn;m] - for the edges,(Fn;m; Fn+1;m; Fn+1;m+1; Fn;m+1) - for the elementary quadrilaterals.Motivated by Theorem 4 above we de�ne discrete isothermic surfaces asfollows: 11

De�nition 6 A discrete isothermic surface is a map F : Z2 ! R3, forwhich all elementary quadrilaterals have cross-ratio �1:Q(Fn;m; Fn+1;m; Fn+1;m+1; Fn;m+1) = �I for all n;m 2 Z: (34)Remark. More generally discrete isothermic surfaces can be de�ned by theproperty that Qn;m is a product of two factorsQ(Fn;m; Fn+1;m; Fn+1;m+1; Fn;m+1) = ��m�n I for all n;m 2 Z; (35)where �n does not depend on m and �m not on n. This condition is refor-mulated in another way in De�nition 1. We restrict ourself in this paper tothe simplest symmetric case (34).Lemma 1 impliesTheorem 5 (M�obius invariance). Let F : Z2 ! R3 be a discrete isothermicsurface and M a M�obius transformation of Euclidean 3-space. Then ~F �M� F : Z! R3 is also isothermic.Property 2 of smooth isothermic surfaces also persists in the discrete case:Theorem 6 (Dual surface). Let F : Z2 ! R3 be a discrete isothermicsurface. Then the discrete surface F � : Z2 ! R3 de�ned (up to translation)by the formulasF �n+1;m � F �n;m = Fn+1;m � Fn;mkFn;mFn+1;mk2 ; F �n;m+1 � F �n;m = �Fn;m+1 � Fn;mkFn;mFn;m+1k2 (36)is isothermic.Proof. It is convenient to use the complex language (30) (elementary quadri-laterals are planar). Thus we assume a; b; a0; b0 2 C.For the elementary quadrilateral (Fn;m; Fn+1;m; Fn+1;m+1; Fn;m+1) we havea+ b0 = b+ a0; aa0 = �bb0: (37)Formulas (36) implya� = 1�a; a0 � = 1�a0 ; b� = �1�b ; b0 � = � 1�b0 (38)12

a

a’

b b’

a’*

a*

b* b’*Figure 2: An elementary quadrilateral and its dualfor the edges of the dual quadrilateral. Combining (37) with (38) one getsa� + b0 � = b� + a0 �; a�a0 � = �b�b0 �;the dual quadrilateral closes up and has a cross-ratio �1.Remark. For a discrete isothermic surface in the sense of de�nition (35)the dual surface is de�ned byF �n+1;m � F �n;m = 1�2n Fn+1;m � Fn;mkFn;mFn+1;mk2 ;F �n;m+1 � F �n;m+1 = � 1�2m Fn;m+1 � Fn;mkFn;mFn;m+1k2 : (39)The cross-ratios of the corresponding quadrilaterals of F and F � coincideQ(F �n;m; F �n+1;m; F �n+1;m+1; F �n;m+1) = ��m�n I (40)5 Lax representation for discrete isothermicsurfacesDe�nition 6 of discrete isothermic surfaces is geometrically motivated andlooks natural. In this section we explain how it was found. Namely, wediscretize the Lax representation (24, 25) in a natural way preserving all itssymmetries and show how De�nition 6 appears in this approach.An integrable discretization of the system (24, 25) looks as follows. Toeach point (n;m) of the Z2-lattice one associates a matrix n;m. These13

matrices at two neighbouring vertices are related byn+1;m = Un;mn;m; n;m+1 = Vn;mn;m; (41)where the matrices Un;m and Vn;m are associated to the edges connecting thepoints (n+ 1;m); (n;m) and (n;m+ 1); (n;m) respectively. Having in mindthe continuum limit (� is a characteristic size of edges)U = I + �U + :::; V = I + �V + :::with U; V of the form (25), and preserving the group structure and the de-pendency of � in the continuous case, it is natural to setUn;m = An;m �p0n;mi�p00n;mi An;m ! ; An;m = an;m1+ bn;mk+ cn;mj (42)Vn;m = Bn;m �q0n;mj��q00n;mj Bn;m ! ; Bn;m = dn;m1+ en;mk+ fn;mj;where the �elds p; q; a; b; c; d; e; f live on the corresponding edges.These matrices can be simpli�edUn;m ! gn+1;mgn;m Un;m; Vn;m ! gn;m+1gn;m Vn;mby a �-independent gauge transformationn;m ! gn;mn;mwith gn;m 2 R at vertices. By such a transformation, which preserves thestructure (42) of U and V, one can achieve the normalization p0n;m = 1=p00n;m �pn;m; q0n;m = 1=q00n;m � qn;m for all n;m. Then U and V are of the formUn;m = An;m �pn;mi�p�1n;mi An;m ! ;Vn;m = Bn;m �qn;mj��q�1n;mj Bn;m ! : (43)Let us take four adjacent vertices (n;m); (n+1;m); (n+1;m+1); (n;m+1)and consider the compatibility conditionV 0U = U 0V; (44)14

n,m n+1,m

n,m+1 n+1,m+1

U

U’

V V’Figure 3: Compatibility conditionchanging notations for the matrices Un;m;Un;m+1;Vn;m;Vn+1;m and their co-e�cients as it is shown in Fig.3.Then for the coe�cients of U ;U 0;V;V 0 one getspp0 = qq0; (45)B0A = A0B; (46)pB0i+ q0jA = qA0j+ p0iB; (47)1pB0i � 1q0 jA = �1qA0j+ 1p 0iB (48)With the notationun;m(�) = detUn;m; vn;m(�) = detVn;m(49) implies vn+1;m(�)un;m(�) = un;m+1(�)vn;m(�): (49)Now note that the zeros of both sides of (49) (considered as functions of �)should coincide. This implies(�2 + detBn+1;m)2(�2 + detAn;m)2 = (�2 + detAn;m+1)2(�2 + detBn;m)2:We suppose that the zeros of un;m+1(�) and un;m(�) coincide. This is equi-valent to detAn;m = �n; detBn;m = �m; (50)where we have incorporated into the notation that �n does not depend on mand �m not on n. 15

Now let us de�ne discrete surfaces Fn;m and F �n;m in R3 by the Symformula (28) �1n;m@n;m@� j�=0 � F̂n;m = 0 Fn;mF �n;m 0 ! ; (51)where n;m(�) is the solution of the initial value problem (41, 43) with0;0(�) = I.Lemma 4 The discrete surface Fn;m de�ned by (51) is isothermic in thesense of the generalized de�nition (35):Q(Fn;m; Fn+1;m; Fn+1;m+1; Fn;m+1) = ��m�n I:The surface F �n;m in (51) is its dual (39).Proof. Formula (51) implies for the edges 0 Fn+1;m � Fn;mF �n+1;m � F �n;m 0 ! = �1n;mU�1n;m @Un;m@� n;m == 0 pn;m��1n;mA�1n;mi�n;mp�1n;m��1n;mA�1n;mi�n;m 0 ! ;and a similar formula for the edges [Fn;m; Fn;m+1]; [F �n;m; F �n;m+1]: Here as inthe smooth case �n;m denotes the components ofn;m(� = 0) = �n;m 00 �n;m ! :Considering the local geometry of the quadrilaterals (Fn;m; Fn+1;m; Fn+1;m+1,Fn;m+1) and (F �n;m; F �n+1;m; F �n+1;m+1; F �n;m+1) we can neglect a general rotation��1n;m:::�n;m. Then the edges of these quadrilaterals in notations of Fig.3 aregiven by the vectors in Fig.4.The cross-ratio of the quadrilateral (Fn;m; Fn+1;m, Fn+1;m+1; Fn;m+1) isequal toQn;m = (�pA�1)(�q0A�1B0 �1jA)�1(p0B�1A0 �1iB)(qB�1j)�1 == �A�1iA�1kBjB = ��m�n I;16

qB −1 j q’A −1 B’ −1 jA

p’B −1 A’ −1 i B

pA −1iFn,m Fn+1,m

Fn,m+1 Fn+1,m+1

−(q’) −1A −1B’−1 jA

p−1A −1 i

(p’)−1B −1 A’ −1 i B

−q−1B−1 j

F*n,m+1 F*n+1,m+1

F*n,m F*n+1,mFigure 4: Quadrilaterals (Fn;m; Fn+1;m; Fn+1;m+1; Fn;m+1) and(F �n;m; F �n+1;m; F �n+1;m+1; F �n;m+1) obtained from the Sym formulawhere we have used equations (45, 46), the equalities�niA�1 = Ai; jB = �mB�1jand the notations (50). One can easily check that the edges of the surfaceF � : Z2 ! R3 are related to the edges of F : Z2 ! R3 by (39).The special case detAn;m = detBn;m = 1;leads to discrete isothermic surfaces as de�ned in De�nition 6.Now let us show that any discrete isothermic surface generates the Laxrepresentation (43). Consider an elementary quadrilateral (Fn;m; Fn+1;m,Fn+1;m+1; Fn;m+1). In a �xed frame its edges (see Fig 5) are represented bythe matrices X;Y;X 0; Y 0, which satisfy the following identities:X + Y 0 = X 0 + Y (52)XY 0�1X 0Y 0�1 = �1: (53)17

n,m n+1,m

n,m+1 n+1,m+1

X

X’

Y Y’Figure 5: An elementary quadrilateral in R3The dual quadrilateral is also closed:X 0 � Y 0�1 = X 0�1 � Y �1: (54)To compare these identities with the system (45-48) let us introduce amoving frame, de�ned at each vertex of the surface. This frame is describedby unitary quaternions �n;m. We denote by ~X; ~Y the coordinate matricesof the vectors X and Y in the frame �n;m taken at the basic vertex of thevector X = ��1n;m ~X�n;m; X = ��1n;m ~Y �n;m:Then for the vectors ~X 0; ~Y 0 one hasX 0 = ��1n;m+1 ~X 0�n;m+1 = ��1n;mB�1n;m ~X 0Bn;m�n;m; (55)Y 0 = ��1n+1;m ~Y 0�n+1;m = ��1n;mA�1n;m ~Y 0An;m�n;m;where An;m = �n+1;m��1n;m; Bn;m = �n;m+1��1n;mstill have to be de�ned.Theorem 7 Let F : Z2! R3 be a discrete isothermic surface,Xn;m = Fn+1;m � Fn;m; Yn;m = Fn;m+1 � Fn;mthe vectors of its edges andpn;m = kXn;mk; qn;m = kYn;mk (56)18

the lengths of these edges. Let �n;m be a unitary quaternion describing theframe at the vertex (n;m) and~Xn;m = �n;mXn;m��1n;m; ~Yn;m = �n;mYn;m��1n;m (57)be a representation of the vectors Xn;m; Yn;m in this frame. Suppose that theframes at negbouring points are related by�n+1;m = An;m�n;m; �n;m+1 = Bn;m�n;m; (58)where An;m = �i 1pn;m ~Xn;m; Bn;m = �j 1qn;m ~Yn;m: (59)ThenA = An;m; A0 = An;m+1; B = Bn;m; B0 = Bn+1;m;p = pn;m; p0 = pn;m+1; q = qn;m; q0 = qn+1;msatisfy the system (45-48), which is equivalent to the Lax representation(43,44) for the discrete isothermic surface F .Proof. Identity (45) follows immediately from (53, 56). Let us show �rst thatthe frames �n;m are well-de�ned, i.e. the compatibility condition (46) for thesystem (58,59) is satis�ed. Taking into account (55) we see, that B0A = A0Bis equivalent to �j 1q0A�Y 0 = �j 1p0B�X 0: (60)Formulas (57) implyA� = �i1p�X; B� = �j1q�Y:Substituting these identities into (60) and using (45) and the obvious relati-ons Y = �q2Y �1; X = �p2X�1;we see that (46) follows from (53). In the same way, usingiB�1 = Bi; iA�1 = Ai;one shows that (52, 54) imply (47, 48).19

6 Construction of discrete isothermic surfa-cesThe discrete isothermic surfaces can be constructed in the same way as thediscrete K-surfaces we discussed in our paper [3].Using the de�nition of discrete isothermic surfaces and starting with someinitial stairway loopFn;m; Fn+1;m; Fn+1;m+1; Fn+2;m; :::; Fn+N;m+N = Fn;m;one can recursively determine the coordinates Fn;m of all the points of thediscrete isothermic surface in a unique way (see Fig.6).n,m

n+1,m+1

n,m+1n,m+1

b)a)Figure 6: Discrete isothermic surfaces as evolutions of initial stairways:a) with Qn;m = �1, b) with Qn;m = ��m=�nIndeed, given the points Fn;m, Fn+1;m, Fn+1;m+1 the cross-ratio (34) uni-quely determines the point Fn;m+1.To construct the discrete isothermic surface in the sense of the generalizedDe�nition 1 one should start with an initial stairway strip as shown in Fig.6b.The factorization property (35) of the cross-ratio allows us to calculate all�n; �m up to a common factor and, as a corollary, the cross-ratios of all thequadrilaterals of the discrete isothermic surface F : Z2! R3. After that onecan proceed as before in Fig.6a reconstructing the surface step by step.As in the case of a discrete K-surface, one can construct by this elementarymethod various cylinders and some other interesting surfaces. We shouldmention here also, that the evolution of the stairway loops described above20

is unstable. A small variation of the initial loop yields dramatic changes ofthe surface. This simple geometrical method does not allow us to controlthe global behaviour of the surface (for example to control the periodicity ofthe evolution of the initial stairway loop) and to construct compact discreteisothermic surfaces.To construct them one should apply methods from the theory of integrableequations, which are based on an analytic solution of the problem (and aredescribed for the case of the discrete K-surface in [3]). But this could be thesubject of a future paper.7 Discrete isothermic minimal surfaces.De�nition 7 A discrete isothermic surface F : X2 ! R3 is called minimalif for any vertex F : n;m of this surface there is a plane Pn;m such that thedistances of the points Fn�1;m; Fn+1;m;Fn;m�1; Fn;m+1 to this plane are equaland < Fn+1;m � Fn;m; Nn;m >=< Fn�1;m � Fn:m; Nn;m >== � < Fn;m+1 � Fn;m; Nn;m >= � < Fn;m�1 � Fn;m; Nn;m >= �n;m;where Nn;m is a normal vector to Pn;m (see Fig. 7).Remark. A mean curvature H for discrete isothermic surfaces has been de�-ned in [4]. De�nition 7 corresponds to the special case H = 0 of the de�nitionof the discrete constant mean curvature surfaces.Lemma 5 For all points of a discrete isothermic minimal surface�n;m = �is constant on Z2.Proof. Since the formulas (36) for the n� and m� edges of a dual surfacedi�er by a sign, we introduce 4 vectorsH = Fn+1;m � Fn;m; H 0 = Fn�1;m � Fn;m;V = Fn;m � Fn;m+1; V 0 = Fn;m � Fn;m�1with the basic point at the vertex Fn;m. Then all the endpoints of these21

Fn,mFn,m−1

Fn,m+1

Fn+1,m

Fn−1,m

Nn,m

Pn,m∆

∆

∆

∆Figure 7: De�nition of discrete minimal isothermic surfacesvectors lie in a plane P�n;m and �n;m is the distance between these twoplanes (Fig. 8). If we identify Fn;m = F �n;m, formulas (36) show thatall the points F �n�1;m; F �n+1;m, F �n;m�1, F �n;m+1 of the dual surface lie on asphere Sn;m of radius 12�n;m , which is the image of P�n;m under inversionof R3 with respect to the unit sphere with a center at Fn;m. The pointsF �n�1;m�1; F �n+1;m�1; F �n+1;m+1; F �n�1;m+1 also lie in Sn;m. Indeed, since F � isdiscrete isothermic, F �n+1;m+1 lies in a circle determined by F �n;m; F �n+1;m,F �n;m+1, which lies in Sn;m. Six points F �n;m+1, F �n+1;m+1, F �n;m, F �n+1;m, F �n;m�1,F �n+1;m�1 are common for Sn;m and Sn+1;m, therefore these spheres coincideSn;m = Sn+1;m; �n;m = �n+1;m:Actually we proved already that the dual surface lies on a sphere. Withoutloss of generality one can assume the normalization� = 1=2: (61)Then the dual surface lies on the unit sphere. Moreover the Gauss map Nof F coincides with F �.Theorem 8 The following statements are equivalent:22

Fn,m

Fn+1,mFn−1,m

Fn,m+1

Fn,m−1

F*n,m+1 F*n−1,m

F*n,m−1F*n+1,m

Pn,m

P ∆n,m

Nn,m

HH’V

V’Figure 8: Dual discrete minimal isothermic surface(i) F : Z2 ! R3 is a discrete isothermic minimal surface normalized by(61).(ii) The dual surface F � : Z2 ! R3 lies on a sphere and without loss ofgenerality one can assumeF � = N : Z2 ! S2:This theorem allows us to parametrize discrete minimal surfaces by "holo-morphic" data N .De�nition 8 A map g : Z2 ! R2 = C is called discrete holomorphic if thecross-ratio of all its elementary quadrilaterals are equal to -1q(gn;m; gn+1;m; gn+1;m+1; gn;m+1) =(gn+1;m � gn;m)(gn;m+1 � gn+1;m+1)(gn+1;m+1 � gn+1;m)(gn;m � gn;m+1) = �1;gn;m are complex number here. 23

A discrete isothermic surface N : Z2 ! S2 in S2 can be obtained froma discrete holomorphic function g : Z2 ! C by stereographic projectionC! S2 (N1 + iN2; N3) = ( 2g1 + jgj2 ; jgj2 � 1jgj2 + 1):Combining this formula with (36) one gets an analogue of the Weierstrassrepresentation in the discrete case.Theorem 9 Let g : Z2 ! C be discrete holomorphic. Then the formulasFn+1;m � Fn;m == 12Re� 1gn+1;m � gn;m (1� gn+1;mgn;m; i(1 + gn+1;mgn;m); gn+1;m + gn;m)�Fn;m+1 � Fn;m == �12Re� 1gn;m+1 � gn;m (1� gn;m+1gn;m; i(1 + gn;m+1gn;m); gn;m+1 + gn;m)�describe a discrete minimal isothermic surface. All discrete minimal isother-mic surfaces are described in this way.Figure 9: Discrete Enneper surface24

The identical map gn;m = n+ im(which is obviously discrete holomorphic) generates the discrete Ennepersurface (Fig. 9). The discrete exponential mapgn;m = exp(�n + i�m); � = 2�=N;where � = 2 arcsinh(sin �2 );generates the discrete catenoid. (Fig. 10)Figure 10: Discrete catenoidReferences[1] W. Blaschke: Vorlesungen �uber Di�erentialgeometrie III, Berlin (1929)[2] A. Bobenko: Surfaces in terms of 2 by 2 matrices. Old and new integrablecases, In: Fordy A., Wood J. (eds) "Harmonic Maps and IntegrableSystems", Vieweg (1994) 25

[3] A. Bobenko, U. Pinkall: Discrete Surfaces with Constant Negative Gaus-sian Curvature and the Hirota Equation, Sfb 288 Preprint No.127 (1994)(to be published in Journ. of Di�. Geom.)[4] A. Bobenko, U. Pinkall: Discrete Surfaces with Constant Curvature andIntegrable Systems, (in preparation)[5] F. Burstall, U. Hertrich-Jeromin, F. Pedit, U. Pinkall: Curved Flats andIsothermic Surfaces, Sfb 288 Preprint No. 132 (1994)[6] P. Calapso: Sulla super�cie a linee di curvature isotherme, Rend. Circ.Mat. Palermo 17, 275-286 (1992)[7] J. Cie�sli�nski, P. Goldstein, A. Sym: Isothermic Surfaces in E3 as SolitonSurfaces, Short Report (1994)[8] D. Ferus, F. Pedit, U. Pinkall, I. Sterling: Minimal Tori in S4, J. ReineAngew. Math. 429, 1-47 (1992)[9] F. Pedit, H. Wu: Discretizing constant curvature surfaces via loop groupfactorizations: the discrete sine- and sinh-Gordon equations, PreprintGANG (1994)26