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 discrete Cauchy-Riemann ∂ u ∂ x = ∂ v u r − u l = v u − v d ∂ y ∂ u ∂ y = − ∂ v u u − u d = v u − v d ∂ x w 4 holomorphic u w = u + iv w = w 1 w 3 iv ∂ w ∂ y = i ∂ w w 2 ∂ x w 4 − w 2 = i ( w 3 − w 1 ) Alexander Bobenko Discrete analytic functions
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 Alexander Bobenko Discrete analytic functions
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 ∗ 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 ( x 0 ) = ν ( x 0 , x k )( f ( x k ) − f ( x 0 )) , ν : E → R + x k ∼ x 0 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 x 1 [Mercat ’01] f : V ( D ) = V ( G ) ∪ V ( G ∗ ) → C discrete holomorphic if it satisfies e e ∗ discrete Cauchy-Riemann equations y 1 y 0 f ( y 1 ) − f ( y 0 ) 1 f ( x 1 ) − f ( x 0 ) = i ν ( x 0 , x 1 ) = − i ν ( y 0 , y 1 ) ν ( e ) = 1 /ν ( e ∗ ) x 0 ◮ 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 R 2 ◮ 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 α 1 , α 2 , . . . , α d ∈ C Alexander Bobenko Discrete analytic functions
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 ◮ strip cross at most once Quasicristallic embedding of a rhombic quad-graph = finite number of slopes α 1 , α 2 , . . . , α d ∈ C Alexander Bobenko Discrete analytic functions
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 ◮ strip cross at most once α 2 α 1 Quasicristallic embedding of a α 3 rhombic quad-graph = finite α 5 α 4 number of slopes α 1 , α 2 , . . . , α d ∈ 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 x 1 α 1 y 1 e f ( x 1 ) − f ( x 0 ) = i tan φ f ( y 1 ) − f ( y 0 ) α 2 e ∗ α 2 2 y 0 α 1 x 0 Green’s function ∆ g x 0 ( x ) = δ xx 0 , g x 0 ( x ) → log | x − x 0 | , x → ∞ Problem - compute explicitly Alexander Bobenko Discrete analytic functions
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 ◮ 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 General and special classes of) surfaces (constant Quad-surfaces curvature, isothermic, etc.) special transformations discrete → symmetric (Bianchi, B"acklund, Darboux) 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
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☎ ✂✄ ✝ ✆ ☎✆ ✝✞ ☎ ☎ ✄ ✂ ✝ ✂ ✂ ✁ �✁ � � � ✞ ✝ Integrability as Consistency ◮ Equation ◮ Consistency c d a b f ( a , b , c , d ) = 0 Alexander Bobenko Discrete analytic functions
☎ ✂✄ ✝ ✆ ☎✆ ✝✞ ☎ ☎ ✄ ✂ ✝ ✂ ✂ ✁ �✁ � � � ✞ ✝ Integrability as Consistency ◮ Equation ◮ Consistency c d a b f ( a , b , c , d ) = 0 Alexander Bobenko Discrete analytic functions
☎ ✂✄ ✝ ✆ ☎✆ ✝✞ ☎ ☎ ✄ ✂ ✝ ✂ ✂ ✁ �✁ � � � ✞ ✝ Integrability as Consistency ◮ Equation ◮ Consistency c d a b f ( a , b , c , d ) = 0 Alexander Bobenko Discrete analytic functions
☎ ✂✄ ✝ ✆ ☎✆ ✝✞ ☎ ☎ ✄ ✂ ✝ ✂ ✂ ✁ �✁ � � � ✞ ✝ Integrability as Consistency ◮ Equation ◮ Consistency c d a b 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 elementary cube discrete orthogonal coordinate → Miquel theorem system Alexander Bobenko Discrete analytic functions
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