Volume and Angle Structures on closed 3-manifolds

Feng Luo

Rutgers University

May, 18, 2006

Georgia Topology Conference

Conventions and Notations

1. Hn, Sn, En n-dim hyperbolic, spherical and Euclidean spaces with curvature λ = -1,1,0.

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

3. indices i,j,k,l are pairwise distinct.

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.

For instance, the space of all hyperbolic triangles, H2 ={(a1, a2, a3) | ai >0 and a1 + a2 + a3 < π}.

The space of all spherical triangles,

S2 ={(a1, a2, a3) | a1 + a2 + a3 > π, ai + aj < ak + π}.

The space of Euclidean triangles up to similarity,

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

Note. The corresponding spaces for 3-simplex, H3, E3, S3 are not convex.

The Schlaefli formula

Given σ3 in H3, S3 with edge lengths lij and dihedral angles xij,

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

d(V) = /2 lij dxij

∂V/∂xij = (λlij )/2

Define the volume of a Euclidean simplex to be 0.

Corollary 1. The volume function

V: H3 U E3 U S3 R is C1-smooth.

Schlaefli formula suggests: natural length = (curvature) X length.

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

Prop. 1.(Murakami, Bonahon, Casson, Rivin,…) If V: H(M,T) R has a critical point p, then the manifold M is hyperbolic.

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

V: H(M,T) –> R is the volume.

Here is a proof using Schlaelfi:

Suppose p=(p1,p 2 ,p3 ,…, pn) is a critical point.

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

By Schlaefli, it is:

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

The difficulties in carrying out the above approach:

1. It is difficult to determine if H(M,T) is non-empty.

2. H3 and S3 are known to be non-convex.

3. It is not even known if H(M,T) is connected.

4. Milnor’s conj.: V: Hn R can be extendedcontinuously to the compact closure of Hn inRn(n+1)/2 .

Classical geometric tetrahedra

Euclidean Hyperbolic Spherical

From dihedral angle point of view,

vertex triangles are spherical triangles.

Angle Structure

1. An angle structure (AS) on a 3-simplex: assigns each edge a dihedral angle in (0, π) so that each vertex triangle is a spherical triangle.

Eg. Classical geometric tetrahedra are AS.

2. An angle structure 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

There is a 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.

Theorem 1. If T is a triangulation of a closed 3-manifold Mand volume V has a local maximum point in AS(M,T),

then,

1. M has a constant curvature metric, or

2. 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.

Theorem 2. (Kitaev, L) For any closed 3-manifold M,

there is a triangulation T of M supporting an angle structure

so that all 3-simplexes are hyperbolic or spherical tetrahedra.

Questions

• How to define the volume of an angle structure?

• How does an angle structure look like?

Volume V can be defined on H3 U E3 U S3 by integrating the

Schlaefli 1-form ω =/2 lij dxij . 1. ω depends on the length lij

2. lij depends on the face angles ybc a 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.

5. d ω =0.

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

Angle Structure

Face angle is well defined by the cosine law, i.e., face angle = edge length of the vertex triangle.

The Cosine Law For a hyperbolic, spherical or Euclidean triangle of inner angles

and edge lengths , (S)

(H)

(E)

1 2 3, ,x x x

1 2 3, ,y y y

cos( ) (cos cos cos ) /(sin sin )i i j k j ky x x x x x cosh( ) (cos cos cos ) /(sin sin )i i j k j ky x x x x x

1 (cos cos cos ) /(sin sin )i j k j kx x x x x

The Cosine Law

There is only one formula

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

Define the M-length Lij in R of 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.

cos( ) (cos cos cos ) /(sin sin )i i j k j ky x x x x x

Edge Length of AS

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=

is a closed, lij is the M-length.

(c) For classical geometric 3-simplex

lij = λX (classical geometric length)

1/ 2 ij ijij

l dx

Theorem 3. There is a smooth function

V: AS(3) –> R so that,

(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 also established Milnor conjecture in all dimension. Rivin has a new proof of it now.

1C

( ) 1/ 2( )ij ijij

d V l dx

6[0, ]

Main ideas of the proof theorem 1.

Step 1. Classify AS on 3-simplex into:

Euclidean, hyperbolic, spherical types.

First, let us see that,

AS(3) ≠ classical geometric tetrahedra

The i-th Flip Map

The i-th flip map Fi : AS(3) AS(3)

sends a point (xab) to (yab) where

,ij ij

jk jk

y x

y x

angles change under flips

Lengths change under flips

Prop. 3. For any AS x on a 3-simplex, exactly one of the following holds,

1. 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.

Flips generate a Z2 + Z2 + Z2 action on AS(3).

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

Classification of types

Prop. 4. Let l be the M-length of an 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.

Step 3. 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.

Step 5. 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.

Step 7. (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.

Let Y be all edges of lengths at least π. The intersection of Y with each 3-simplex

consists of,(a) three edges from one vertex, or, (single flip)

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

(c) empty set.

This produces a normal surface X in T.

Claim. the Euler characteristic of X is positive.

X is a union of triangles and quadrilaterals.

• 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)

The punchline

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

opposite edges of a Euclidean or hyperbolic tetrahedron,

Then

Euclidean or hyperbolic tetrahedron

2a b c d

( ) ( ) ( ) ( ) 2a b c d

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.

Thank you.