Hyperbolic Discrete Ricci Curvature Flows

Hyperbolic Discrete Ricci Curvature Flows 1 1 Mathematics Science Center Tsinghua University Tsinghua University 010

Unified Framework of Discrete Curvature Flow Unified framework for both Discrete Ricci flow and Yamabe flow Curvature flow Energy E(u)= Hessian of E denoted as Δ, Newton s method. du dt = K K, ( K i K i )du i, i dk=δdu.

Hyperbolic Ricci Flow Computational results for genus and genus 3 surfaces.

Hyperbolic Ricci flow

Hyperbolic Ricci flow Poincaré Model Klein Model

Hyperbolic Functions sinhx = ex e x,cosh x = ex + e x tanhx = sinhx coshx,cothx = coshx sinhx sinh = cosh,cosh = sinh, tanh = 1 cosh,coth = 1 sinh cosh sinh = 1.

Cosine law v i l j θ i v k θ k l k l i θ j v j cosθ i sinθ i sinhl i = coshl j coshl k coshl i sinhl j sinhl k = sinθ j sinhl j = sinθ k sinhl k

Cosine law v k θ k l l i j θ i θ j l k v i v j A=sinhl j sinhl k sinθ i cosθ i sinhl i θ i l i = coshl j coshl k coshl i sinhl j sinhl k = sinhl j sinhl k sin θ i θ i l i = sinhl i A

Cosine law v k cosθ i θ i l j = coshl j coshl k coshl i sinhl j sinhl k = S jc k (S j S k ) ( C i + C j C k )S k C j sinθ i (S j S k ) l j θ k l i = (S j C j )C k S k + C i C j S k sinθ i S j S k v i θ i l k θ j v j = C k+ C i C j AS j A=sinhl j sinhl k sinθ i = S is j cosθ k AS j = S i A cosθ k

Cosine law dθ 1 dθ dθ 3 = 1 A S 1 0 0 0 S 0 0 0 S 3 1 cosθ 3 cosθ cosθ 3 1 cosθ 1 cosθ cosθ 1 1 dl 1 dl dl 3

Hyperbolic cosine law coshl i = coshr j coshr k + sinhr j sinhr k I jk sinhl i dl i dr j = sinhr j cosh r k + I jk coshr j sinhr k dl i dr j I jk = sinhr j coshr k + coshr j sinhr k I jk sinhl i = coshl i coshr j coshr k sinhr j sinhr k

Hyperbolic cosine law dl i dr j In Euclidean case dual = sinhr coshl j coshr k + coshr j sinhr i coshr j coshr k k sinhr j sinhr k sinhl i = sinh r j coshr k + coshr j coshl i cosh r j coshr k sinhl i sinhr j = (sinh r j cosh r j )cosh r k + coshr j coshl i sinhl i sinhr j = coshr j coshl i coshr k sinhl i sinhr j dl i = r j + li rk dr j l i r j l i sinhl i,cosh r j r j.

hyperbolic cosine law u = logtanh r du dr = = = 1 1 1 tanh r cosh r 1 sinh 1 cosh r 1 sinhr

Hyperbolic cosine law M= dl 1 dl dl 3 =MD du 1 du du 3 coshr 0 3 +coshl 1 coshr coshr +coshl 1 coshr 3 sinhl 1 sinhr sinhl 1 sinhr 3 coshr 3 +coshl coshr 1 coshr sinhl sinhr 1 0 1 +coshl coshr 3 sinhl sinhr 3 coshr +coshl 3 coshr 1 coshr 1 +coshl 3 coshr sinhl 3 sinhr 1 sinhl 3 sinhr 0 D = sinhr 1 0 0 0 sinhr 0 0 0 sinhr 3

Compute Teichmüller coordinates Step 1. Compute the hyperbolic uniformization metric. Step. Compute the Fuchsian group generators.

Teichmüller space Theorem (Teichmüller space) The Teichmüller space of a genus g closed surface is of 6g 6 dimension. Proof. Suppose {a 1,b 1,,a g,b g } are the generators of π 1 (M), then the corresponding deck transformations are {α 1,β 1,,α g,β g }. Under the hyperbolic uniformization metric, each α k and β k are Möbius transformations, therefore require 3 parameters, and α 1 β 1 α 1 1 β 1 1 α β α 1 β 1 α g β g αg 1 β 1 g = id, therefore β g can be determined from other generators. Therefore, there are total 3(g 1) parameters. On the other hand, the dimension of hyperbolic isometries is 3, so the dimension of Teichmüller space of a genus g surface is 6g 6.

Discrete Yamabe Flow Discrete conformal metric deformation: u 1 l 3 l y3 θ 1 y u l 1 θ 3 θ y 1 u 3 conformal factor y k = e u i l k e u j R sinh y k = e u i sinh l k e u j H sin y k = e u i sin l k e u j S Properties: K i u j = K j u i and dk=δdu.

Hyperbolic Yamabe Flow sinh y k 1 cosh y k y k u i y k u i = e u i sinh l k eu j = u i e u i sinh l k eu j = u i sinh y k = u i tanh y k

Hyperbolic Yamabe Flow dy 1 dy dy 3 = tanh y 1 0 0 0 tanh y 0 0 0 tanh y 3 u 1 0 0 0 u 0 0 0 u 3 du 1 du du 3

Maximal hyperbolic Ricci flow Discrete Riemann mapping Let Ω be a simply connected domain contained in the unit disk. Compute a triangulation and construct a mesh. Run hyperbolic ricci flow, such that v i M,u i. Then the resulting mapping is the discrete Riemann mapping. Theorem Discrete Riemann mappingconformal mapping from a simply connected planar domain to the unit disk exists, and unique up to a rigid motion.

Maximal Hyperbolic Ricci flow