Volume and angle structures on closed 3 manifolds
Download
1 / 39

Volume and Angle Structures on closed 3-manifolds - PowerPoint PPT Presentation


  • 172 Views
  • Uploaded on

Volume and Angle Structures on closed 3-manifolds. Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University. 1. H n , S n , E n n-dim hyperbolic, spherical and Euclidean spaces with curvature λ = -1,1,0. Conventions and Notations.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Volume and Angle Structures on closed 3-manifolds' - tassos


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Volume and angle structures on closed 3 manifolds l.jpg

Volume and Angle Structures on closed 3-manifolds

Feng Luo

Rutgers University

Oct. 28, 2006

Texas Geometry/Topology conference

Rice University


Slide2 l.jpg

1.Hn, Sn, En n-dim hyperbolic, spherical and

Euclidean spaces with curvature λ = -1,1,0.

Conventions and Notations

2.σnis an n-simplex, vertices labeled as 1,2,…,n, n+1.

3.indices i,j,k,l are pairwisedistinct.

4. Hn(or Sn) is the space of all hyperbolic (or spherical)

n-simplexes parameterized by the dihedral angles.

5.En = space of all Euclidean n-simplexes modulo similarity

parameterized by the dihedral angles.


Slide3 l.jpg

For instance, the space of allhyperbolic triangles,

H2 ={(a1, a2, a3) | ai >0 and a1 + a2 + a3 < π}.

The space ofEuclidean trianglesup to similarity,

E2 ={(a,b,c) | a,b,c >0, and a+b+c=π}.

Note.The corresponding spaces for 3-simplex, H3, E3, S3are not convex.


The space of all spherical triangles s 2 a 1 a 2 a 3 a 1 a 2 a 3 a i a j a k l.jpg
The space of allspherical triangles, S2 ={(a1, a2, a3) | a1 + a2 + a3 > π, ai + aj < ak + π}.


Slide5 l.jpg

let V =V(x) be the volume where x=(x12,x13,x14,x23,x24,x34).

d(V) = /2  lij dxij

The Schlaefli formula

Given σ3 in H3, S3with edge lengths lij and dihedral angles xij,


V x ij l ij 2 l.jpg
∂V/∂xij = (λlij )/2

Define the volume of a Euclidean simplex to be 0.

Corollary 1.The volume function

V: H3U E3U S3  R

is C1-smooth.

Schlaefli formula suggests:

natural length = (curvature) X length.


Slide7 l.jpg

Schlaefli formula suggests: a way to find geometric structures on triangulated closed 3-manifold (M, T).

Following Murakami, an H-structure on (M, T):

1.Realize each σ3 in T by a hyperbolic 3-simplex.

2. The sum of dihedral angles at each edge in T is 2π.

The volume V of an H-structure = the sum of the

volume of its simplexes


Slide8 l.jpg

H(M,T) structures on triangulated closed 3-manifold (M, T).= the space of all H-structures, a smooth manifold.

V: H(M,T)–> R is the volume.

Prop. 1.(Murakami, Bonahon, Casson, Rivin,…)

If V: H(M,T) R has a critical point p,

then the manifold M is hyperbolic.

Here is a proof using Schlaelfi:


Slide9 l.jpg

Suppose p=(p structures on triangulated closed 3-manifold (M, T).1,p 2 ,p3 ,…, pn) is a critical point.

ThendV/dt(p1-t, p2+t, p3,…,pn)=0 at t=0.

By Schlaefli, it is:

le(A)/2 -le(B)/2 =0


Slide10 l.jpg

4.Milnor’s conj.:V: Hn (or Sn)  R can be extended

continuously to the compact closure of Hn (or Sn )in

Rn(n+1)/2 .


Classical geometric tetrahedra l.jpg
Classical geometric tetrahedra structures on triangulated closed 3-manifold (M, T).

EuclideanHyperbolic Spherical

From dihedral angle point of view,

vertex trianglesare spherical triangles.


Angle structure l.jpg
Angle Structure structures on triangulated closed 3-manifold (M, T).

An angle structure (AS) on a 3-simplex:

assigns each edge a dihedral angle in (0, π)

so that each vertex triangle is aspherical triangle.

Eg. Classical geometric tetrahedra are AS.


Angle structure on 3 mfd l.jpg
Angle structure on 3-mfd structures on triangulated closed 3-manifold (M, T).

An angle structure (AS) on (M, T):

realize each 3-simplex in T by an AS

so that the sum of dihedral angles at each edge is 2π.

Note: The conditions are linear equations and linear inequalities


Slide14 l.jpg

There is a structures on triangulated closed 3-manifold (M, T).natural notion of volume of AS on 3-simplex (to be defined below using Schlaefli).

AS(M,T) = space of all AS’s on (M,T).

AS(M,T) is a convex bounded polytope.

Let V: AS(M, T)  R be the volume map.


Slide15 l.jpg

Theorem 1 structures on triangulated closed 3-manifold (M, T)..If T is a triangulation of a closed 3-manifold M

and volume V has a local maximum point in AS(M,T),

then,

  • M has a constant curvature metric, or

  • there is a normal 2-sphere intersecting each edge in at most one point.

    In particular, if T has only one vertex, M is reducible.

    Furthermore, V can be extended continuously to the compact closure of AS(M,T).

    Note.The maximum point of V always exists in the closure.


Slide16 l.jpg

Theorem 2. structures on triangulated closed 3-manifold (M, T).(Kitaev, L) For any closed 3-manifold M,

there is a triangulation T of M supporting an angle structure.

In fact, all 3-simplexes are hyperbolic or spherical tetrahedra.


Questions l.jpg
Questions structures on triangulated closed 3-manifold (M, T).

  • How to definethe volume of an angle structure?

  • How does an angle structure look like?


Slide18 l.jpg
Classical volume structures on triangulated closed 3-manifold (M, T).V can be defined on H3U E3U S3 by integrating the Schlaefli 1-form ω =/2  lij dxij .

  • ω depends on the length lij

  • lij depends on the face angles ybca by the cosine law.

    3. ybca depends on dihedral angles xrs by the cosine law.

    4. Thus ω can be constructed from xrs by the cosine law.

  • d ω =0.

    Claim:all above can be carried out for angle structures.


Angle structure19 l.jpg
Angle Structure structures on triangulated closed 3-manifold (M, T).

Face angle is well defined by the cosine law, i.e.,

face angle = edge length of the vertex triangle.


The cosine law l.jpg
The structures on triangulated closed 3-manifold (M, T).Cosine Law

For a hyperbolic, spherical or Euclidean triangle of inner angles

and edge lengths ,

(S)

(H)

(E)


The cosine law21 l.jpg
The Cosine Law structures on triangulated closed 3-manifold (M, T).

There is only one formula

The right-hand side makes sense for all x1, x2, x3 in (0, π).

Define the M-length Lijof the ij-th edge in AS using the above formula.

Lij = λ geometric length lij


Let as 3 all angle structures on a 3 simplex l.jpg
Let AS(3) = all angle structures on a 3-simplex. structures on triangulated closed 3-manifold (M, T).

Prop. 2.(a) The M-length of the ij-th edge is independent of the choice of triangles ijk, ijl.

(b) The differential 1-form on AS(3)

ω =1/2  lij dxij .

is closed, lij is the M-length.

  • For classical geometric 3-simplex

    lij = λX (classical geometric length)


Slide23 l.jpg

Theorem 3 structures on triangulated closed 3-manifold (M, T)..There is a smooth function V: AS(3) –> R s.t.,

(a) V(x) = λ2 (classical volume)

if x is a classical geometric tetrahedron,

(b) (Schlaefli formula) let lij be the M-length of the ij-th edge,

(c) V can be extended continuously to the compact closure of AS(3) in .

We call V the volume of AS.

Remark.(c ) implies an affirmative solution of a conjecture of Milnor in 3-D. We have established Milnor conjecture in all dimension. Rivin has a new proof of it now.


Main ideas of the proof theorem 1 l.jpg
Main ideas of the proof theorem 1. structures on triangulated closed 3-manifold (M, T).

Step 1. Classify AS on 3-simplex into:

Euclidean, hyperbolic,sphericaltypes.

First, let us see that,

AS(3) ≠ classical geometric tetrahedra


The i th flip map l.jpg
The i-th Flip Map structures on triangulated closed 3-manifold (M, T).


Slide26 l.jpg

The structures on triangulated closed 3-manifold (M, T).i-th flip map Fi : AS(3) AS(3)

sends a point (xab) to (yab) where


Angles change under flips l.jpg
angles change under flips structures on triangulated closed 3-manifold (M, T).


Lengths change under flips l.jpg
Lengths change under flips structures on triangulated closed 3-manifold (M, T).


Slide29 l.jpg

Prop. 3 structures on triangulated closed 3-manifold (M, T).. For any AS x on a 3-simplex,

exactly one of the following holds,

  • x is in E3, H3 or S3, a classical geometric tetrahedron,

    2. there is an index i so that Fi (x) is in E3 or H3,

    3. there are two distinct indices i, j so that

    Fi Fj (x) is in E3 or H3.

    The type of AS = the type of its flips.


Slide30 l.jpg

Flips generate a Z structures on triangulated closed 3-manifold (M, T).2 + Z2 + Z2 action on AS(3).

Step 2.Type is determined by the length of one edge.


Slide31 l.jpg

Classification of types structures on triangulated closed 3-manifold (M, T).

Prop. 4.Let l be the M-length of one edge in an AS.

Then,

(a) It is spherical type iff 0 < l < π.

(b) It is of Euclidean type iff l is in {0,π}.

(c) It is of hyperbolic type iff l is less than 0 or larger than π.

An AS is non classical iff one edge length is at least π.


Slide32 l.jpg

Step 3. structures on triangulated closed 3-manifold (M, T).At the critical point p of volume V on AS(M, T),

Schlaefli formula shows the edge length is well defined, i.e.,

independent of the choice of the 3-simplexes adjacent to it.

(same argument as in the proof of prop. 1).

Step 4.Steps 1,2,3 show at the critical point,

all simplexes have the same type.


Slide33 l.jpg

Step 5. structures on triangulated closed 3-manifold (M, T).If all AS on the simplexes in p come from classical hyperbolic (or spherical) simplexes,

we have a constant curvature metric.

(the same proof as prop. 1)

Step 6.Show that at the local maximum point,

not all simplexes are classical Euclidean.


Slide34 l.jpg

Step 7. ( structures on triangulated closed 3-manifold (M, T).Main Part)

If there is a 3-simplex in p which is not a classical geometric tetrahedron,

then the triangulation T contains a normal surface X of positive Euler characteristic

which intersects each 3-simplex in at most one normal disk.


Slide35 l.jpg

L structures on triangulated closed 3-manifold (M, T).et Y be all edges of lengths at least π.

The intersection of Y with each 3-simplex consists of,

  • three edges from one vertex (single flip), or

  • four edges forming a pair of opposite edges (double-flip), or,

  • empty set.

    This produces a normal surface X in T.

    Claim. the Euler characteristic of X is positive.


Slide36 l.jpg

X is a union of triangles and quadrilaterals. structures on triangulated closed 3-manifold (M, T).

  • Each triangle is a spherical triangle (def. AS).

  • Each quadrilateral Q is in a 3-simplex obtained from double flips of a Euclidean or hyperbolic tetrahedron (def. Y).

  • Thus four inner angles of Q, -a,-b, -c, -d satisfy that a,b,c,d, are angles at two pairs of opposite sides of Euclidean or hyperbolic tetrahedron. (def. flips)


Slide37 l.jpg

The Key Fact structures on triangulated closed 3-manifold (M, T).

Prop. 5.If a,b,c,d are dihedral angles at two pairs of

opposite edges of aEuclidean or hyperbolic tetrahedron,

Then


Slide38 l.jpg

Summary: for the normal surface X

1. Sum of inner angles of a quadrilateral > 2π.

2. Sum of the inner angles of a triangle > π.

3. Sum of the inner angles at each vertex = 2π.

Thus the Euler characteristic of X is positive.

Thank you



ad