# Volume and Angle Structures on closed 3-manifolds

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Volume and Angle Structures on closed 3-manifolds

Feng LuoRutgers University May, 18, 2006Georgia Topology Conference

Conventions and Notations

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

2. n 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 < }.

The space of all spherical triangles,

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

The space of Euclidean triangles up to similarity,

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

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

The Schlaefli formula Given 3 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 = (lij )/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 3 in T by a hyperbolic 3-simplex. 2. The sum of dihedral angles at each edge in T is 2.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:

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. Milnors 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 Structure1. An angle structure (AS) on a 3-simplex: assigns each edge a dihedral angle in (0, ) 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 2.

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 ASs 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,

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.

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 =/2 lij dxij . depends on the length lijlij depends on the face angles ybc a by the cosine law. 3. ybca depends on dihedral angles xrs by the cosine law.4. Thus can be constructed from xrs by the cosine law.

d =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)

The Cosine Law There is only one formula

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

Define the M-length Lij in R of the ij-th edge in AS using the above formula. Lij = geometric length lij

Let AS(3) = all angle structures on a 3-simplex.

Edge Length of ASProp. 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)

= is a closed, lij is the M-length.For classical geometric 3-simplex

lij = X (classical geometric length)

Theorem 3. There is a smooth function V: AS(3) > R so that,

(a) V(x) = 2 (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.

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

angles change under flips

Lengths change under flips

Prop. 3. 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.

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

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

Classification of typesProp. 4. Let l be the M-length of an edge in an AS. Then, (a) It is spherical type iff 0 < l < .

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

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

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

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 . The intersection of Y with each 3-simplex consists of,three edges from one vertex, or, (single flip)four edges forming a pa