Linear and nonlinear theories of. Discrete analytic functions. Integrable structure

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1 Linear and nonlinear theories of discrete analytic functions. Integrable structure Technical University Berlin Painlevé Equations and Monodromy Problems, Cambridge, September 18, 2006 DFG Research Unit 565 Polyhedral Surfaces

2 Discrete Differential Geometry Aim: Development of discrete equivalents of the geometric notions and methods of differential geometry. The latter appears then as a limit of refinements of the discretization. Intelligent discretizations lead to: interesting geometric objects in discrete geometry new methods (difference equations) deep understanding of smooth theory (unifies surfaces and their transformations) solution of problems in differential geometry (Weil s problem: convex surfaces from convex metrics; Alexandrov s solution with polyhedra) represent smooth shape by a discrete shape with just few elements; best approximation (Applications)

3 This talk Based on a joint work with Ch. Mercat and Yu. Suris Discrete complex analysis, discrete holomorphic Integrability (geometric definition) Isomonodromic Green s function Linear and nonlinear theories (circle patterns) Discrete Painlevé equations

4 Harmonic and holomorphic on the square lattice [Ferrand 44, Duffin 56] conjugate harmonic Cauchy-Riemann u x = v y u y = v x holomorphic w = u + iv w y = i w x discrete Cauchy-Riemann u r u l = v u v d u u u d = v u v d u w = w 1 iv w 4 w 2 = i(w 3 w 1 ) w 4 w 2 w 3

5 Quad-graph f harmonic on a graph G = (V, E), f = 0 f (x 0 ) = ν(x 0, x k )(f (x k ) f (x 0 )), ν : E R + x k x 0 G G cell decomposition of C

6 Quad-graph f harmonic on a graph G = (V, E), f = 0 f (x 0 ) = x k x 0 ν(x 0, x k )(f (x k ) f (x 0 )), ν : E R + G G G cell decomposition of C G dual

7 Quad-graph f harmonic on a graph G = (V, E), f = 0 f (x 0 ) = x k x 0 ν(x 0, x k )(f (x k ) f (x 0 )), ν : E R + G G D G cell decomposition of C G dual D double - quad-graph

8 Harmonic and holomorphic on a graph [Mercat 01] x 1 f : V (D) = V (G) V (G ) C discrete holomorphic if it satisfies discrete Cauchy-Riemann equations y 1 e e y 0 f (y 1 ) f (y 0 ) f (x 1 ) f (x 0 ) = iν(x 1 0, x 1 ) = iν(y 0, y 1 ) ν(e) = 1/ν(e ) f : V (D) C discrete holomorphic f V (G), f V (G ) discrete harmonic f : V (G) C discrete harmonic there exists unique (up to additive constant) extension to discrete holomorphic f : V (D) C Applications in computer graphics [Gu, Yau 05] x 0

9 Rhombic quad-graphs Quad-graph = quadrilateral cell decomposition Rhombic quad-graph = there exists a rhombic representation in R 2 combinatorial characterization [Kenyon, Schlenker 04] no strip crosses itself or periodic strips cross at most once Quasicristallic embedding of a rhombic quad-graph = finite number of slopes α 1, α 2,..., α d C

10 Rhombic quad-graphs Quad-graph = quadrilateral cell decomposition Rhombic quad-graph = there exists a rhombic representation in R 2 combinatorial characterization [Kenyon, Schlenker 04] no strip crosses itself or periodic strips cross at most once Quasicristallic embedding of a rhombic quad-graph = finite number of slopes α 1, α 2,..., α d C

11 Rhombic quad-graphs Quad-graph = quadrilateral cell decomposition Rhombic quad-graph = there exists a rhombic representation in R 2 combinatorial characterization [Kenyon, Schlenker 04] no strip crosses itself or periodic strips cross at most once α 2 α 1 Quasicristallic embedding of a α 3 α α 5 4 rhombic quad-graph = finite number of slopes α 1, α 2,..., α d C

12 Green s function D quasicristallic embedding of a rhombic quad-graph labelling α : E(D) C, α = 1 weights ν(e) = tan φ 2, ν(e ) = cot φ 2 f (y 1 ) f (y 0 ) f (x 1 ) f (x 0 ) = i tan φ 2 α 2 y 1 α 1 e e α 2 x 1 Green s function x 0 α 1 y 0 g x0 (x) = δ xx0, g x0 (x) log x x 0, x Problem - compute explicitly

13 Method. Results lift rhombic quad-graph D to Ω D Z d, d= number of different slopes P : V (D) Z d extend holomorphic functions from Ω D to Z d integrable Laplacians = Laplacians on rhombic quad-graphs discrete flat connection (Lax representation) isomonodromic solutions Green s function comes from linearization of a nonlinear integrable theory (circle patterns)

14 Surfaces and transformations Classical theory of (special classes of) surfaces (constant curvature, isothermic, etc.) General and special Quad-surfaces special transformations (Bianchi, B"acklund, Darboux) discrete symmetric

15 Basic idea Do not distinguish discrete surfaces and their transformations. Discrete master theory. Example - planar quadrilaterals as discrete conjugate systems. Multidimensional Q-nets [Doliwa, Santini 97].

16 Basic idea Do not distinguish discrete surfaces and their transformations. Discrete master theory. Example - planar quadrilaterals as discrete conjugate systems. Multidimensional Q-nets [Doliwa, Santini 97].

17 Integrability as Consistency Equation Consistency c d a b f (a, b, c, d) = 0

18 Integrability as Consistency Equation Consistency c d a b f (a, b, c, d) = 0

19 Integrability as Consistency Equation Consistency c d a b f (a, b, c, d) = 0

20 Integrability as Consistency Equation Consistency c d a b f (a, b, c, d) = 0

21 Circular nets Martin, de Pont, Sharrock [ 86], Nutbourne [ 86], B. [ 96], Cieslinski, Doliwa, Santini [ 97], Konopelchenko, Schief [ 98], Akhmetishin, Krichever, Volvovski [ 99],... three coordinate nets of a discrete orthogonal coordinate system elementary cube Miquel theorem

22 Rhombic = Integrable Discrete Cauchy-Riemann equations (dcr) on a quad-graph integrable (3D consistent) iff the weights come from parallelogram immersions of the quad-graph weight iν = ratio of diagonals ν R iff rhombi Rhombic dcr x 12 x = α 2 + α 1 x 1 x 2 α 1 α 2 Lax representation ( (affine transformation ) x 3 x 13, λ = α 3 ) λ + α 2α(x + x1 ) L(x 1, x, α; λ) = 0 λ α

23 Extension to Z d Discrete holomorphic f : Z d C: f (n + e i + e k ) f (n) f (n + e i ) f (n + e k ) = α i + α k α i α k on each square specified by its values on the axes ) m ( λ + αk discrete exponential e(me k ; λ) = λ α k discrete logarithm f (me k ) = { 2( ) m even 3 5 m 1 log α k m odd

24 Discrete Log as Green s function Discrete Green s function on a quasicristallic quad-graph D is the real part (i.e. restriction to G) of the discrete logarithm function g : Z d C Integral representation for discrete logarithm g : Z d C (Integral representation for discrete Green s function on D [Kenyon 02]) g(n) = 1 2π γ γ loops around α 1,..., α d C Isomonodromic log λ e(n, λ)dλ 2λ

25 Isomonodromic discrete Log ψ(n + e k, λ) = L k (n, λ)ψ(n, λ), L k (n, λ) Lax matrices dψ(n, λ) A(n, λ) = ψ 1 (n, λ) dλ isomonodromic A(n, λ) meromorphic in λ, poles (position and order) independent of n Z A(n, λ) = A(n) λ + ( d B k ) (n) k=1 + Ck (n) λ + α k λ α k Constraint [Nijhoff, Ramani, Grammaticos, Ohta 01] for Z 2 d k=1 n k(f (n + e k ) f (n e k )) = 1 ( 1) n 1+...n g discrete logarithm f (me k ) = { 2( log α k m 1 ) m even m odd

26 Nonlinear Theory. Circle Patterns Circle packings - discrete analogs of conformal maps [Thurston 85] Conformal maps can be discretized as orthogonal circle patterns [Schramm 97] f : C C is a conformal map if f x f y and f x = f y

27 Nonlinear Theory. Equations z 2 Cross-ratio equation (z 1 z)(z 2 z 12 ) (z 12 z 1 )(z z 2 ) = α2 1 α 2 2 z α 2 α 1 α 1 α 2 z 12 z 1 equivalent z 1 z = α 1 ww 1 to Hirota equation α 1 (ww 1 w 2 w 12 ) = α 2 (ww 2 w 1 w 12 ) cross-ratio and Hirota equations are 3D-consistent Lax representation ( (Möbius transformation ) w 3 w 13 ) 1 αw L(w 1, w, α; λ) = 1 λα/w w 1 /w

28 Integrable Circle Patterns Circle patterns: combinatorial data G and intersection angles Circle patterns from z and w: z - centers and intersection points of circles w( ) R + - radii, w( ) S 1 - rotation angles α = 1 rhombic realization w 1 Combinatorial data and intersection angles belong to an integrable circle pattern iff they admit an isoradial realization. rhombic immersion of the double D.

29 Z a circle pattern

30 Nontrivial. Unstable

31 Z a circle pattern. Construction ψ(n + e k, λ) = L k (n, λ)ψ(n, λ), A(n, λ) = A(n, λ) = A(n) λ L k (n, λ) Lax matrices dψ(n, λ) ψ 1 (n, λ), isomonodromic dλ + d k=1 B k (n) λ α 2 k Constraint [Nijhoff, Ramani, Grammaticos, Ohta 01] for Z 2 d k=1 n w(n + e k ) w(n e k ) k w(n + e k ) + w(n e k ) = a 1 2 (1 ( 1)n 1+...n g ) discrete z a : w(0) = 1, w(e k ) = α a 1 k z(0) = 0, z(e k ) = αk a Asymptotics on the coordinate axes: n w(2ne k ) = n a 1 (1 + O(1/n)), z(ne k ) = (nα k ) a (1 + O(1/n))

32 Z a circle pattern. Analysis Coordinate planes in Z d are immersed (neighboring quads do not overlap) [Agafonov, B. 00] Equation for rotation angles x n = w 2n 1 on the diagonal of the ij-coordinate plane q = α j /α i S 1 ( ) (n+1)(xn 2 xn+1 + x n /q 1) n(1 x 2 ( ) n q + x n x n+1 q 2 ) xn 1 + x n q q + x n x n 1 = ax n q 2 1 2q 2 immersed iff unitary solution with 0 < arg x n < arg q uniqueness of the solution discrete Painlevé II equation [Nijhoff, Ramani, Grammaticos, Ohta 01]

33 Z a circle pattern. Theorems embedded [Agafonov 03] uniqueness for a = 4/n follows from [He 99] uniqueness for arbitrary a [Bücking 06]

34 Non-orthogonal circle patterns Hexagonal [B., Hoffmann 03] Quasicristallic circle patterns from circle patterns with Z N combinatorics

35 Linearization isoradial pattern holomorphic functions on rhombic quad graphs circle patterns one-parameter family ɛ of circle patterns, ɛ = 0 isoradial, w ɛ solution of the Hirota equations. f = wɛ 1 dw ɛ dɛ ɛ=0 satisfies dcr equations Discrete Green s function (discrete log of the linear theory) is the d da -derivative of the circle pattern za at a = 1. f (n) = d da w a(n) a=1