Surfaces of finite types Chihchy Fwu Department of Mathematics Soochow University November 2001

How to visualize abstractly defined surfaces?

What is the configuration space of the following linkage [W. Thurston 1982]

It is a surface of genus 3 Vertice = 8  6 / 4 = 12 Edges = 8  6 / 2 = 24 Faces = 8  = 12 – = 2 ( 1 – g )

Outline Review classical surfaces Type number Computation application

Circle Group SO(2) X =  cos t sin t   -sin t cos t  X ” + X = 0

Orthogonal group SO(n) X = ( x ij )  R n  n Killing form ds 2 = - tr( dX dX t ) Orthonormal basis of Lie algebra E ij = e i e j t – e j e i t Casimir operator  E ij 2 = - I Maurer Cartan form  = dX X t Structure equation d  =    Laplacian  X + X = 0

Spectrum of a Manifold Eigenfunction  with eigenvalue   +  = 0 Eigenvalues 0 < 1 < 2 < 3 < …. E( k ) space of eigenfunctions is a finite dimensional vector space

Immersion of homogenous space M = G / H Riemannian homogeneous space G compact Lie group H acts on T [H] M irreducibly. {  1,  2, …,  N } an orthonormal basis of E( k ) The evaluation map  : M  R N  (x) = (  1 (x),  2 (x), …,  N (x) ) is an isometric immersion and   = k .

Some examples Sphere S n = SO(n+1) / SO(n) S 2n+1 = SU(n+1) / SU(n) S 4n+3 = Sp(n+1) / Sp(n) Projective space RP n = SO(n+1) / S(O(n)  O(1)) CP n = SU(n+1) / S(U(n)  U(1)) QP n = Sp(n+1) / Sp(n)  Sp(1)

spherical harmonics E( k ) consists of harmonic polynomials homogeneous of degree k in x, y, z dim E( k ) = 2 k +1 k = k ( k + 1)

E( 1 ) = { x, y, z } canonical immersion

E( 2 ) = { x 2 - y 2, y 2 - z 2, z 2 - x 2, 2xy, 2yz, 2zx } Veronese surface as an immersion of the real projective plane RP 2  R 5  S 5 = S 4  R 4

Roman Surface ( 2xy, 2yz, 2zx )

Crosscap ( 2yz, 2zx, z 2 -x 2 )

E( 3 ) Immersion of the 3rd degree

E( 4 ) Boy surface immersion of degree 4

Boy surface viewed from bottom

Boy surface viewed from top

3 different views of Boy surface

Harmonic immersions of Flat torus T 2 (a,b)   S 1 ( a ) × S 1 ( b )  S 3  R 4, a 2 + b 2 = 1  (u,v) = ( a cos u, a sin u, b cos v, b sin v ) The harmonic functions on a flat torus are ordinary trig polynomials

Anchor ring = Flat torus under stereographic projection

Flat torus under spherical harmonic immersion of degree 2

Another spherical harmonic immersion of Flat torus

Type number X : M n  R N is a Riemannian manifold. E( k ) is the space of kth eigenfunctions of the Laplacian. The position vector has a Fourier expansion X =  X k Type number = # { k | X k  0 } All previous examples have type number either 1 : homogeneous spaces, or 2 : flat torus

QUESTION Is type number necessarily finite ? What is the type number of a surface ?

Spectral problem Eigenfunction  with eigenvalue   +  = 0 Rayleigh quotient Q(  ) = |  | 2 / |  | 2

Rayleigh-Ritz Approximation parametrize the surface in a single patch  = I  I with boundary properly identified. subdivide  into cells {e k } by a grid {x k } define a finite element  k at each vertex x k approximate the function  by  u k  k with coefficients vector u=(u k ) k ≦ K set K = 1600

Generalized eigenvalue problem Replace the Rayleigh quotient with the discrete form  Au, u  /  Bu, u  SVD gives A U = B U D U = ( u 1 … u K ) the matrix of eigenvectors, and D = diag( 1 … K ) the matrix of eigenvalues A u = B u

Determine the type number discretized position vector X = U Y Y = ( Y 1,Y 2, Y 3 ) Fourier coefficients y = Y 1 2 +Y Y 3 2 = ( y k ) Type number = # { k | y k > residue} Here residue = 10 -6

Surfaces to be investigated sphere flat torus a class of anchor rings a class of that under inversion a class of knotted tori a surface of genus two

Reconstructed sphere

Reconstructed Roman surface

Reconstructed flat torus

Reconstructed flat torus with higher degree eigenfunctions

Class of anchor rings

Generators are the geodesic circles at (1,0) of H 2

Type numbers of anchor rings

Class of inversions T 2 (a,b)  S 3  S 3   R 3  R 3 O(1,4) is the conformal groups of both R 3 and S 3 Inversion : x, u  R 4, | x | = 1  ( x ) = u – ( |u| 2 –1) / | x- u| 2 (x - u )

Inversion of an anchor ring

Type numbers of inversions

Class of knotted tori tube around a torus knot

Type numbers of knotted tori

Surface of genus 2 type number > 1525

Surface of genus 3 type number = ?

Surface of genus 5 type number = ?

All surfaces are of finite type, but some are more finite than the others [rephase George Orwell]

Surface of higher genus Uniformization : a surface M g of genus g > 1 is covered by the unit disk  M g =  \ SL(2, R) / SO(2)  =  1 ( M g ) fundamental group =  a 1, b 1,… a g, b g | [a 1,b 1 ][a 2,b 2 ]…[a g,b g ] = 1  M g  CP 3  R N

Fundamental domain ( g = 2)

Riemann surface of genus 2

View from bottom

View from top

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