Discrete analytic functions. Integrable structure Alexander Bobenko - - PowerPoint PPT Presentation

Discrete analytic functions. Integrable structure

Alexander Bobenko

Technical University Berlin

Geometry Conference in honour of Nigel Hitchin, Madrid, September 4-8, 2006 DFG Research Unit 565 “Polyhedral Surfaces”

Alexander Bobenko Discrete analytic functions

Integrable Systems in Surface Theory

Hitchin, Harmonic maps from a 2-torus to the 3-sphere, [’91] First applications of integrable systems in the global surface theory

Alexander Bobenko Discrete analytic functions

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)

Alexander Bobenko Discrete analytic functions

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)

Alexander Bobenko Discrete analytic functions

Harmonic and holomorphic on the square lattice

[Ferrand ’44, Duffin ’56] conjugate harmonic Cauchy-Riemann ∂u ∂x = ∂v ∂y ∂u ∂y = −∂v ∂x discrete Cauchy-Riemann ur − ul = vu − vd uu−ud = vu−vd holomorphic w = u + iv ∂w ∂y = i ∂w ∂x iv u w = w1 w4 w3 w2 w4 − w2 = i(w3 − w1)

Alexander Bobenko Discrete analytic functions

Quad-graph

f harmonic on a graph G = (V, E), ∆f = 0 ∆f(x0) =

• xk∼x0

ν(x0, xk)(f(xk) − f(x0)), ν : E → R+ G G cell decomposition of C

Alexander Bobenko Discrete analytic functions

Quad-graph

f harmonic on a graph G = (V, E), ∆f = 0 ∆f(x0) =

• xk∼x0

ν(x0, xk)(f(xk) − f(x0)), ν : E → R+ G G∗ G cell decomposition of C G∗ dual

Alexander Bobenko Discrete analytic functions

Quad-graph

f harmonic on a graph G = (V, E), ∆f = 0 ∆f(x0) =

• xk∼x0

ν(x0, xk)(f(xk) − f(x0)), ν : E → R+ G G∗ D G cell decomposition of C G∗ dual D double - quad-graph

Alexander Bobenko Discrete analytic functions

Harmonic and holomorphic on a graph

[Mercat ’01] f : V(D) = V(G) ∪ V(G∗) → C discrete holomorphic if it satisfies discrete Cauchy-Riemann equations f(y1) − f(y0) f(x1) − f(x0) = iν(x0, x1) = − 1 iν(y0, y1) ν(e) = 1/ν(e∗) x0 x1 e∗ e y0 y1

◮ 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]

Alexander Bobenko Discrete analytic functions

Rhombic quad-graphs

◮ Quad-graph = quadrilateral cell decomposition ◮ Rhombic quad-graph = there exists a rhombic

representation in R2

◮ combinatorial characterization [Kenyon, Schlenker ’04]

◮ no strip crosses itself or periodic ◮ strip cross at most once

Quasicristallic embedding of a rhombic quad-graph = finite number of slopes α1, α2, . . . , αd ∈ C

Alexander Bobenko Discrete analytic functions

Rhombic quad-graphs

◮ Quad-graph = quadrilateral cell decomposition ◮ Rhombic quad-graph = there exists a rhombic

representation in R2

◮ combinatorial characterization [Kenyon, Schlenker ’04]

◮ no strip crosses itself or periodic ◮ strip cross at most once

Quasicristallic embedding of a rhombic quad-graph = finite number of slopes α1, α2, . . . , αd ∈ C

Alexander Bobenko Discrete analytic functions

Rhombic quad-graphs

◮ Quad-graph = quadrilateral cell decomposition ◮ Rhombic quad-graph = there exists a rhombic

representation in R2

◮ combinatorial characterization [Kenyon, Schlenker ’04]

◮ no strip crosses itself or periodic ◮ strip cross at most once

α1 α2 α3 α4 α5 Quasicristallic embedding of a rhombic quad-graph = finite number of slopes α1, α2, . . . , αd ∈ C

Alexander Bobenko Discrete analytic functions

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(y1) − f(y0) f(x1) − f(x0) = i tan φ 2 α1 α2 x0 y1 y0 x1 α2 α1 e∗ e Green’s function ∆gx0(x) = δxx0, gx0(x) → log |x − x0|, x → ∞ Problem - compute explicitly

Alexander Bobenko Discrete analytic functions

• Method. Results

◮ lift rhombic quad-graph D

to ΩD ⊂ Zd, d= number of different slopes P : V(D) → Zd

◮ extend holomorphic

functions from ΩD to Zd

◮ integrable Laplacians = Laplacians on rhombic

quad-graphs

◮ Zero curvature (Lax) representation ◮ isomonodromic solutions ⇒ Green’s function ◮ comes from linearization of a nonlinear integrable theory

(circle patterns)

Alexander Bobenko Discrete analytic functions

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

Alexander Bobenko Discrete analytic functions

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].

Alexander Bobenko Discrete analytic functions

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].

Alexander Bobenko Discrete analytic functions

Integrability as Consistency

◮ Equation ◮ Consistency

• ✁

a b c d f(a, b, c, d) = 0

Alexander Bobenko Discrete analytic functions

Integrability as Consistency

◮ Equation ◮ Consistency

• ✁

a b c d f(a, b, c, d) = 0

Alexander Bobenko Discrete analytic functions

Integrability as Consistency

◮ Equation ◮ Consistency

• ✁

a b c d f(a, b, c, d) = 0

Alexander Bobenko Discrete analytic functions

Integrability as Consistency

◮ Equation ◮ Consistency

• ✁

a b c d f(a, b, c, d) = 0

Alexander Bobenko Discrete analytic functions

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

Alexander Bobenko Discrete analytic functions

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

x12 − x x1 − x2 = α2 + α1 α1 − α2

◮ Lax representation (affine transformation x3 → x13, λ = α3)

L(x1, x, α; λ) = λ + α −2α(x + x1) λ − α

• Alexander Bobenko

Discrete analytic functions

Extension to Zd

Discrete holomorphic f : Zd → C:

◮ f(n + ei + ek) − f(n)

f(n + ei) − f(n + ek) = αi + αk αi − αk

• n each square

◮ specified by its values on the axes ◮ discrete exponential e(mek; λ) =

λ + αk λ − αk m

◮ discrete logarithm

f(mek) = { 2(1 + 1

3 + 1 5 + . . . 1 m−1)

m even log αk m odd

Alexander Bobenko Discrete analytic functions

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 : Zd → C

◮ Integral representation for discrete logarithm g : Zd → C

(Integral representation for discrete Green’s function on D [Kenyon’s ’02]) g(n) = 1 2π

• γ

log λ 2λ e(n, λ)dλ γ loops around α1, . . . , αd ∈ C

◮ Isomonodromic

Alexander Bobenko Discrete analytic functions

Isomonodromic discrete Log

ψ(n + ek, λ) = Lk(n, λ)ψ(n, λ), Lk(n, λ) Lax matrices

◮ A(n, λ) = dψ(n, λ)

dλ ψ−1(n, λ)

◮ isomonodromic ⇔ A(n, λ) meromorphic in λ, poles

(position and order) independent of n ∈ Z

◮ A(n, λ) = A(n)

λ + d

k=1

Bk(n) λ + αk + Ck(n) λ − αk

• ◮ Constraint [Nijhoff, Ramani, Grammaticos, Ohta ’01] for Z2

d

k=1 nk(f(n + ek) − f(n − ek)) = 1 − (−1)n1+...ng ◮ discrete logarithm

f(mek) = { 2(1 + 1

3 + 1 5 + . . . 1 m−1)

m even log αk m odd

Alexander Bobenko Discrete analytic functions

Nonlinear Theory. Circle Patterns

Circle packings - discrete analogs of conformal maps [Thurston ’85] Conformal maps can be discretized F : Z2 → C as orthogonal circle patterns [Schramm ’97] f : C → C is a conformal map if fx ⊥ fy and |fx| = |fy|

Alexander Bobenko Discrete analytic functions

Nonlinear Theory. Equations

Cross-ratio equation (z1 − z)(z2 − z12) (z12 − z1)(z − z2) = α2

1

α2

2

z2 z z12 α2 z1 α1 α1 α2

◮ equivalent z1 − z = α1ww1

to Hirota equation α1(ww1 − w2w12) = α2(ww2 − w1w12)

◮ cross-ratio and Hirota equations are 3D-consistent ◮ Lax representation (Möbius transformation w3 → w13)

L(w1, w, α; λ) =

• 1

−αw1 −λα/w w1/w

• Alexander Bobenko

Discrete analytic functions

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(•) ∈ S1 - 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.

Alexander Bobenko Discrete analytic functions

Z a circle pattern

Alexander Bobenko Discrete analytic functions

Z a circle pattern. Construction

ψ(n + ek, λ) = Lk(n, λ)ψ(n, λ), Lk(n, λ) Lax matrices

◮ A(n, λ) = dψ(n, λ)

dλ ψ−1(n, λ), isomonodromic

◮ A(n, λ) = A(n)

λ + d

k=1

Bk(n) λ − α−2

k ◮ Constraint [Nijhoff, Ramani, Grammaticos, Ohta ’01] for Z2

d

k=1 nk

w(n + ek) − w(n − ek) w(n + ek) + w(n − ek) = a − 1 2 (1 − (−1)n1+...ng)

◮ discrete za:

w(0) = 1, w(ek) = αa−1

k

⇔ z(0) = 0, z(ek) = αa

k

Asymptotics on the coordinate axes: n → ∞ w(2nek) = na−1(1 + O(1/n)), z(nek) = (nαk)a(1 + O(1/n))

Alexander Bobenko Discrete analytic functions

Z a circle pattern. Analysis

Coordinate planes in Zd are immersed (neighboring quads do not overlap) [Agafonov, B. ’00] Equation for rotation angles xn = w2n−1 on the diagonal of the ij-coordinate plane q = αj/αi ∈ S1 (n+1)(x2

n−1)

xn+1 + xn/q q + xnxn+1

• −n(1−x2

n

q2 ) xn−1 + xnq q + xnxn−1

• = axn

q2 − 1 2q2

◮ immersed iff unitary solution with 0 < arg xn < arg q ◮ uniqueness of the solution ◮ discrete Painlevé II equation [Nijhoff, Ramani,

Grammaticos, Ohta ’01]

Alexander Bobenko Discrete analytic functions

Z a circle pattern. Theorems

◮ embedded [Agafonov ’03] ◮ uniqueness for a = 4/n follows from [He ’99] ◮ uniqueness for arbitrary a [Bücking ’06]

Alexander Bobenko Discrete analytic functions

Non-orthogonal circle patterns

◮ Hexagonal [B., Hoffmann ’03] ◮ Quasicristallic circle patterns from circle patterns with ZN

combinatorics

Alexander Bobenko Discrete analytic functions

Linearization

isoradial pattern circle patterns holomorphic functions

• n rhombic quad graphs

◮ one-parameter family ǫ of circle patterns, ǫ = 0 isoradial,

wǫ solution of the Hirota equations. f = w−1

ǫ

dwǫ dǫ |ǫ=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 dawa(n)|a=1

Alexander Bobenko Discrete analytic functions

Discrete integrable systems 2D

◮ Equation ◮ Consistency ⇒

Integrability

• ✁

x u v y

◮ Lax representation,

Darboux transformation [B., Suris ’02] Q(x, y, u, v) = 0, Q affine with respect to all variables, square symmetry

Alexander Bobenko Discrete analytic functions

Discrete integrable systems 2D. Classification

[Adler, B., Suris ’03] (Q1) α(x − v)(u − y) − β(x − u)(v − y) + δ2αβ(α − β) = 0, (Q2) α(x − v)(u − y) − β(x − u)(v − y) + αβ(α − β)(x + y + u + v) −αβ(α − β)(α2 − αβ + β2) = 0, (Q3) sin(α)(xu + vy) − sin(β)(xv + uy) − sin(α − β)(xy + uv) +δ2 sin(α − β) sin(α) sin(β) = 0, (Q4) sn(α)(xu + vy) − sn(β)(xv + uy) − sn(α − β)(xy + uv) +sn(α − β)sn(α)sn(β)(1 + k2xyuv) = 0, (H1) (x − y)(u − v) + β − α = 0, (H2) (x − y)(u − v) + (β − α)(x + y + u + v) + β2 − α2 = 0, (H3) α(xu + vy) − β(xv + uy) + δ(α2 − β2) = 0

Alexander Bobenko Discrete analytic functions

Discrete integrable systems 3D

Q(f, . . . , f123) = 0, Q affine with respect to all variables, with cube symmetry (system on a cubical complex) or less symmetry (system on Z4).

Alexander Bobenko Discrete analytic functions

Problems

◮ 3D: Classify 3D equations which are 4D consistent.

Example with cubical symmetry - discrete BKP equation (f1 − f3)(f2 − f123) (f3 − f2)(f123 − f1) = (f − f13)(f12 − f23) (f13 − f12)(f23 − f)

◮ 4D: Are there 4D equations which are 5D consistent? ◮ Describe smooth limits

Alexander Bobenko Discrete analytic functions