Riemann Surface
David Gu1
1Department of Computer ScienceUniversity of New York at Stony Brook
SMI 2012 Course
David Gu Conformal Geometry
Thanks
Thanks for the invitation.
David Gu Conformal Geometry
Collaborators
The work is collaborated with Shing-Tung Yau, Feng Luo, TonyChan, Paul Thompson, Yalin Wang, Ronald Lok Ming Lui, HongQin, Dimitris Samaras, Jie Gao, Arie Kaufman, and many othermathematicians, computer scientists and medical doctors.
David Gu Conformal Geometry
Klein’s Program
Klein’s Erlangen Program
Different geometries study the invariants under differenttransformation groups.
Geometries
Topology - homeomorphisms
Conformal Geometry - Conformal Transformations
Riemannian Geometry - Isometries
Differential Geometry - Rigid Motion
David Gu Conformal Geometry
Motivation
Scientific View
Conformal structure is a natural geometric structure; manyphysics phenomena are governed by conformal geometry, suchas electro-magnetic fields, brownian motion, phase transition,percolation ... Conformal geometry is part of nature.
David Gu Conformal Geometry
Motivation
Practical View
Conformal geometry offers the theoretic frameworks for
Shape Space
Mapping Space
David Gu Conformal Geometry
Shape Space
A shape embedded in E3 has the following level of information
1 Topological structure (genus, boundaries)2 Conformal structure, (conformal module )3 Riemannian metric, (conformal factor)4 Embedding structure, (mean curvature )
David Gu Conformal Geometry
Shape Space
Surfaces can be classified according to their geometricstructures
Each topological equivalent class has infinite manyconformal equivalent classes, which form a finitedimensional Riemannian manifold.
Each conformal equivalent class has infinite manyisometric classes, which form an infinite dimensionalspace.
David Gu Conformal Geometry
Mapping Space
Given two metric surfaces, fixing the homotopy type, there areinfinite many of diffeomorphisms between them, which form themapping space.
Each mapping uniquely determines a differential - Beltramidifferential.
Each Beltrami differential essentially uniquely determines adiffeomorphism.
The mapping space is converted to a convex functionalspace.
Variational calculus can be carried out in the functionalspace.
David Gu Conformal Geometry
Motivation
Conformal geometry lays down the theoretic foundation for
Surface mapping
Shape classification
Shape analysis
Applied in computer graphics, computer vision, geometricmodeling, wireless sensor networking and medical imaging,and many other engineering, medical fields.
David Gu Conformal Geometry
Sample of Applications
1 Computational Topology: Shortest word problem2 Geometric Modeling: Manifold Spline3 Graph Theory: Graph embedding4 Meshing: Smooth surface meshing that ensures curvature
convergence.5 Shape Analysis: A discrete Heat kernel determines the
discrete metric.
David Gu Conformal Geometry
History
History
In pure mathematics, conformal geometry is theintersection of complex analysis, algebraic topology,Riemann surface theory, algebraic curves, differentialgeometry, partial differential equation.
In applied mathematics, computational complex functiontheory has been developed, which focuses on theconformal mapping between planar domains.
Recently, computational conformal geometry has beendeveloped, which focuses on the conformal mappingbetween surfaces.
David Gu Conformal Geometry
History
History
Conventional conformal geometric method can only handle themappings among planar domains.
Applied in thin plate deformation (biharmonic equation)
Membrane vibration
Electro-magnetic field design (Laplace equation)
Fluid dynamics
Aerospace design
David Gu Conformal Geometry
Reasons for Booming
Data Acquisition
3D scanning technology becomes mature, it is easier to obtainsurface data.
David Gu Conformal Geometry
System Layout
David Gu Conformal Geometry
3D Scanning Results
David Gu Conformal Geometry
3D Scanning Results
David Gu Conformal Geometry
System Layout
David Gu Conformal Geometry
Reasons for Booming
Our group has developed high speed 3D scanner, which cancapture dynamic surfaces 180 frames per second.
Computational Power
Computational power has been increased tremendously. Withthe incentive in graphics, GPU becomes mature, which makesnumerical methods for solving PDE’s much easier.
David Gu Conformal Geometry
Fundamental Problems
1 Given a Riemannian metric on a surface with an arbitrarytopology, determine the corresponding conformal structure.
2 Compute the complete conformal invariants (conformalmodules), which are the coordinates of the surface in theTeichmuller shape space.
3 Fix the conformal structure, find the simplest Riemannianmetric among all possible Riemannian metrics
4 Given desired Gaussian curvature, compute thecorresponding Riemannian metric.
5 Given the distortion between two conformal structures,compute the quasi-conformal mapping.
6 Compute the extremal quasi-conformal maps.7 Conformal welding, glue surfaces with various conformal
modules, compute the conformal module of the gluedsurface.
David Gu Conformal Geometry
Complete Tools
Computational Conformal Geometry Library
1 Compute conformal mappings for surfaces with arbitrarytopologies
2 Compute conformal modules for surfaces with arbitrarytopologies
3 Compute Riemannian metrics with prescribed curvatures4 Compute quasi-conformal mappings by solving Beltrami
equation
David Gu Conformal Geometry
Books
The theory, algorithms and sample code can be found in thefollowing books.
You can find them in the book store.
David Gu Conformal Geometry
Source Code Library
Please email me [email protected] for updated code libraryon computational conformal geometry.
David Gu Conformal Geometry
Conformal Mapping
David Gu Conformal Geometry
Holomorphic Function
Definition (Holomorphic Function)
Suppose f : ℂ→ ℂ is a complex function,f : x + iy → u(x ,y)+ iv(x ,y), if f satisfies Riemann-Cauchyequation
∂u∂x
=∂v∂y
,∂u∂y
=−∂v∂x
,
then f is a holomorphic function.
Denote
dz = dx+ idy ,dz = dx− idy ,∂
∂z=
12(
∂∂x
− i∂
∂y),
∂∂ z
=12(
∂∂x
+ i∂
∂y)
then if ∂ f∂ z = 0, then f is holomorphic.
David Gu Conformal Geometry
biholomorphic Function
Definition (biholomorphic Function)
Suppose f : ℂ→ ℂ is invertible, both f and f−1 are holomorphic,then then f is a biholomorphic function.
γ0
γ1
γ2
D0
D1
David Gu Conformal Geometry
Conformal Map
f
z wτβ ◦ f ◦ φ−1
α
Uα Vβ
φα τβ
S1 ⊂ {(Uα, φα)} S2 ⊂ {(Vβ , τβ)}
The restriction of the mapping on each local chart isbiholomorphic, then the mapping is conformal.
David Gu Conformal Geometry
Conformal Mapping
David Gu Conformal Geometry
Conformal Geometry
Definition (Conformal Map)
Let φ : (S1,g1)→ (S2,g2) is a homeomorphism, φ is conformalif and only if
φ∗g2 = e2ug1.
Conformal Mapping preserves angles.
θ
θ
David Gu Conformal Geometry
Conformal Mapping
Conformal maps Properties
Map a circle field on the surface to a circle field on the plane.
David Gu Conformal Geometry
Quasi-Conformal Map
Diffeomorphisms: maps ellipse field to circle field.
David Gu Conformal Geometry
Conformal Structure
David Gu Conformal Geometry
Riemann Surface
Intuition1 A conformal structure is a equipment on a surface, such
that the angles can be measured, but the lengths, areascan not.
2 A Riemannian metric naturally induces a conformalstructure.
3 Different Riemannian metrics can induce a sameconformal structure.
David Gu Conformal Geometry
Manifold
Definition (Manifold)
M is a topological space, {Uα} α ∈ I is an open covering of M,M ⊂∪αUα . For each Uα , φα : Uα → ℝ
n is a homeomorphism.The pair (Uα ,φα) is a chart. Suppose Uα ∩Uβ ∕= /0, thetransition function φαβ : φα(Uα ∩Uβ )→ φβ (Uα ∩Uβ ) is smooth
φαβ = φβ ∘φ−1α
then M is called a smooth manifold, {(Uα ,φα)} is called anatlas.
David Gu Conformal Geometry
Manifold
φαφβ
Uα Uβ
S
φαβ
φα (Uα) φβ (Uβ)
David Gu Conformal Geometry
Conformal Atlas
Definition (Conformal Atlas)
Suppose S is a topological surface, (2 dimensional manifold), Ais an atlas, such that all the chart transition functionsφαβ : ℂ→ ℂ are bi-holomorphic, then A is called a conformalatlas.
Definition (Compatible Conformal Atlas)
Suppose S is a topological surface, (2 dimensional manifold),A1 and A2 are two conformal atlases. If their union A1 ∪A2 isstill a conformal atlas, we say A1 and A2 are compatible.
David Gu Conformal Geometry
Conformal Structure
The compatible relation among conformal atlases is anequivalence relation.
Definition (Conformal Structure)
Suppose S is a topological surface, consider all the conformalatlases on M, classified by the compatible relation
{all conformal atlas}/∼
each equivalence class is called a conformal structure.
In other words, each maximal conformal atlas is a conformalstructure.
David Gu Conformal Geometry
Conformal Structure
Definition (Conformal equivalent metrics)
Suppose g1,g2 are two Riemannian metrics on a manifold M, if
g1 = e2ug2,u : M → ℝ
then g1 and g2 are conformal equivalent.
Definition (Conformal Structure)
Consider all Riemannian metrics on a topological surface S,which are classified by the conformal equivalence relation,
{Riemannian metrics on S}/∼,
each equivalence class is called a conformal structure.
David Gu Conformal Geometry
Relation between Riemannian metric and ConformalStructure
Definition (Isothermal coordinates)
Suppose (S,g) is a metric surface, (Uα ,φα) is a coordinatechart, (x ,y) are local parameters, if
g = e2u(dx2 +dy2),
then we say (x ,y) are isothermal coordinates.
Theorem
Suppose S is a compact metric surface, for each point p, thereexits a local coordinate chart (U,φ), such that p ∈ U, and thelocal coordinates are isothermal.
David Gu Conformal Geometry
Riemannian metric and Conformal Structure
Corollary
For any compact metric surface, there exists a naturalconformal structure.
Definition (Riemann surface)
A topological surface with a conformal structure is called aRiemann surface.
David Gu Conformal Geometry
Riemannian metric and Conformal Structure
Theorem
All compact orientable metric surfaces are Riemann surfaces.
David Gu Conformal Geometry
Uniformization
David Gu Conformal Geometry
Conformal Canonical Representations
Theorem (Poincar e Uniformization Theorem)
Let (Σ,g) be a compact 2-dimensional Riemannian manifold.Then there is a metric g = e2λ g conformal to g which hasconstant Gauss curvature.
Spherical Euclidean HyperbolicDavid Gu Conformal Geometry
Uniformization of Open Surfaces
Definition (Circle Domain)
A domain in the Riemann sphere ℂ is called a circle domain ifevery connected component of its boundary is either a circle ora point.
Theorem
Any domain Ω in ℂ, whose boundary ∂Ω has at most countablymany components, is conformally homeomorphic to a circledomain Ω∗ in ℂ. Moreover Ω∗ is unique upto Mobiustransformations, and every conformal automorphism of Ω∗ isthe restriction of a Mobius transformation.
David Gu Conformal Geometry
Uniformization of Open Surfaces
Spherical Euclidean Hyperbolic
David Gu Conformal Geometry
Conformal Canonical Representation
Simply Connected Domains
David Gu Conformal Geometry
Conformal Canonical Forms
Topological Quadrilateral
p1 p2
p3p4
p1 p2
p3p4
David Gu Conformal Geometry
Conformal Canonical Forms
Multiply Connected Domains
David Gu Conformal Geometry
Conformal Canonical Forms
Multiply Connected Domains
David Gu Conformal Geometry
Conformal Canonical Forms
Multiply Connected Domains
David Gu Conformal Geometry
Conformal Canonical Forms
Multiply Connected Domains
David Gu Conformal Geometry
Conformal Canonical Representations
Definition (Circle Domain in a Riemann Surface)
A circle domain in a Riemann surface is a domain, whosecomplement’s connected components are all closed geometricdisks and points. Here a geometric disk means a topologicaldisk, whose lifts in the universal cover or the Riemann surface(which is ℍ
2, ℝ2 or S2 are round.
Theorem
Let Ω be an open Riemann surface with finite genus and atmost countably many ends. Then there is a closed Riemannsurface R∗ such that Ω is conformally homeomorphic to a circledomain Ω∗ in R∗. More over, the pair (R∗,Ω∗) is unique up toconformal homeomorphism.
David Gu Conformal Geometry
Conformal Canonical Form
Tori with holes
David Gu Conformal Geometry
Conformal Canonical Form
High Genus Surface with holes
David Gu Conformal Geometry
Teichmuller Space
David Gu Conformal Geometry
Teichmuller Space
Shape Space - Intuition
1 All metric surfaces are classified by conformal equivalencerelation.
2 All conformal equivalences form a finite dimensionalmanifold.
3 The manifold has a natural Riemannian metric, whichmeasures the distortions between two conformalstructures.
David Gu Conformal Geometry
Teichmuller Theory: Conformal Mapping
Definition (Conformal Mapping)
Suppose (S1,g1) and (S2,g2) are two metric surfaces,φ : S1 → S2 is conformal, if on S1
g1 = e2λ φ∗g2,
where φ∗g2 is the pull-back metric induced by φ .
Definition (Conformal Equivalence)
Suppose two surfaces S1,S2 with marked homotopy groupgenerators, {ai ,bi} and {αi ,βi}. If there exists a conformal mapφ : S1 → S2, such that
φ∗[ai ] = [αi ],φ∗[bi ] = [βi ],
then we say two marked surfaces are conformal equivalent.
David Gu Conformal Geometry
Teichmuller Space
Definition (Teichmuller Space)
Fix the topology of a marked surface S, all conformalequivalence classes sharing the same topology of S, form amanifold, which is called the Teichmuller space of S. Denotedas TS .
Each point represents a class of surfaces.
A path represents a deformation process from one shapeto the other.
The Riemannian metric of Teichmuller space is welldefined.
David Gu Conformal Geometry
Teichmuller Space
Topological Quadrilateral
p1 p2
p3p4
Conformal module: hw . The Teichmuller space is 1 dimensional.
David Gu Conformal Geometry
Teichmuller Space
Multiply Connected Domains
Conformal Module : centers and radii, with Mobius ambiguity.The Teichmuller space is 3n−3 dimensional, n is the number ofholes.
David Gu Conformal Geometry
Topological Pants
Genus 0 surface with 3 boundaries is conformally mapped tothe hyperbolic plane, such that all boundaries becomegeodesics.
David Gu Conformal Geometry
Teichmuller Space
Topological Pants
γ0
γ1
γ0
γ2
γ0
2
γ1
2
γ2
2γ2
2
γ1
2
γ0
2
τ2
τ0τ0
τ1 τ1
γ0
2
γ0
2
γ1
2
γ1
2
τ0
τ2
τ2
τ1
τ1
τ2
2
τ2
2
The lengths of 3 boundaries are the conformal module. TheDavid Gu Conformal Geometry
Teichmuller Space
Topological Pants Decomposition - 2g−2 pairs of Pants
γ
P1
P2
p1
p2
θ
David Gu Conformal Geometry
Compute Teichmuller coordinates
Step 1. Compute the hyperbolic uniformization metric.
Step 2. Compute the Fuchsian group generators.
David Gu Conformal Geometry
Compute Teichmuller Coordinates
Step 3. Pants decomposition using geodesics and compute thetwisting angle.
David Gu Conformal Geometry
Teichmuller Coordinates
Compute Teichmuller coordinates
Compute the pants decomposition using geodesics andcompute the twisting angle.
David Gu Conformal Geometry
Quasi-Conformal Maps
David Gu Conformal Geometry
Quasi-Conformal Mappings
Intuition1 All diffeomorphisms between two compact Riemann
surfaces are quasi-conformal.2 Each quasi-conformal mapping corresponds to a unique
Beltrami differential.3 The space of diffeomorphisms equals to the space of all
Beltrami differentials.4 Variational calculus can be carried out on the space of
diffeomorphisms.
David Gu Conformal Geometry
Quasi-Conformal Map
Most homeomorphisms are quasi-conformal, which mapsinfinitesimal circles to ellipses.
David Gu Conformal Geometry
Beltrami-Equation
Beltrami Coefficient
Let φ : S1 → S2 be the map, z,w are isothermal coordinates ofS1, S2, Beltrami equation is defined as ∥µ∥∞ < 1
∂φ∂ z
= µ(z)∂φ∂z
David Gu Conformal Geometry
Diffeomorphism Space
Theorem
Given two genus zero metric surface with a single boundary,
{Diffeomorphisms} ∼={Beltrami Coefficient}
{Mobius}.
David Gu Conformal Geometry
Solving Beltrami Equation
The problem of computing Quasi-conformal map is convertedto compute a conformal map.
Solveing Beltrami Equation
Given metric surfaces (S1,g1) and (S2,g2), let z,w beisothermal coordinates of S1,S2,w = φ(z).
g1 = e2u1dzdz (1)
g2 = e2u2dwdw, (2)
Then
φ : (S1,g1)→ (S2,g2), quasi-conformal with Beltramicoefficient µ .
φ : (S1,φ∗g2)→ (S2,g2) is isometric
φ∗g2 = eu2∣dw ∣2 = eu2∣dz +µddz∣2.
φ : (S1, ∣dz +µddz∣2)→ (S2,g2) is conformal.
David Gu Conformal Geometry
Quasi-Conformal Map Examples
D0
D1
D2D3
pq p q
D0
D1
D2D3
pq
David Gu Conformal Geometry
Quasi-Conformal Map Examples
David Gu Conformal Geometry
Variational Calculus on Diffeomorphism Space
Let {µ(t)} be a family of Beltrami coefficients depends on aparameter t ,
µ(t)(z) = µ(z)+ tν(z)+ tε(t)(z)
for z ∈ ℂ, with suitable µ , ν ,ε(t) ∈ L∞(ℂ),
f µ(t)(w) = f µ(w)+ tV (f µ ,ν)(w)+o(∣t ∣),
where as t → 0,
V (f µ ,ν)(w)=−f µ(w)(f µ(w)−1)
π
∫ℂ
ν(z)((f µ )z(z))2
f µ(z)(f µ (z)−1)(f µ(z)− f µ(w))dxdy
David Gu Conformal Geometry
Solving Beltrami Equation
David Gu Conformal Geometry
Summary
Conformal structure is more flexible than Riemannianmetric
Conformal structure is more rigid than topology
Conformal geometry can be used for a broad range ofengineering applications.
David Gu Conformal Geometry
Thanks
For more information, please email to [email protected].
Thank you!
David Gu Conformal Geometry