Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University

Conventions and Notations 1. H n, S n, E n n-dim hyperbolic, spherical and Euclidean spaces with curvature λ = -1,1,0. 2. σ n is an n-simplex, vertices labeled as 1,2,…,n, n indices i,j,k,l are pairwise distinct. 4. H n (or S n ) is the space of all hyperbolic (or spherical) n-simplexes parameterized by the dihedral angles. 5. E n = space of all Euclidean n-simplexes modulo similarity parameterized by the dihedral angles.

For instance, the space of all hyperbolic triangles, H 2 ={(a 1, a 2, a 3 ) | a i >0 and a 1 + a 2 + a 3 < π}. The space of Euclidean triangles up to similarity, E 2 ={(a,b,c) | a,b,c >0, and a+b+c=π}. Note. The corresponding spaces for 3-simplex, H 3, E 3, S 3 are not convex.

The space of all spherical triangles, S 2 ={(a 1, a 2, a 3 ) | a 1 + a 2 + a 3 > π, a i + a j < a k + π}.

The Schlaefli formula Given σ 3 in H 3, S 3 with edge lengths l ij and dihedral angles x ij, let V =V(x) be the volume where x=(x 12,x 13,x 14,x 23,x 24,x 34 ). d(V) = /2  l ij dx ij

∂V/∂x ij = (λl ij )/2 Define the volume of a Euclidean simplex to be 0. Corollary 1. The volume function V: H 3 U E 3 U S 3  R is C 1 -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 σ 3 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=(p 1,p 2,p 3,…, p n ) is a critical point. Then dV/dt(p 1 -t, p 2 +t, p 3,…,p n )=0 at t=0. By Schlaefli, it is: l e (A)/2 -l e (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. H 3 and S 3 are known to be non-convex. 3. It is not even known if H(M,T) is connected. 4. Milnor’s conj.: V: H n (or S n )  R can be extended continuously to the compact closure of H n (or S n )in R n(n+1)/2.

Classical geometric tetrahedra Euclidean Hyperbolic Spherical From dihedral angle point of view, vertex triangles are spherical triangles.

Angle Structure 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. E g. Classical geometric tetrahedra are AS.

Angle structure on 3-mfd 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

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 M and 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. In fact, 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?

Classical volume V can be defined on H 3 U E 3 U S 3 by integrating the Schlaefli 1-form ω = /2  l ij dx ij. 1.ω depends on the length l ij 2.l ij depends on the face angles y bc a by the cosine law. 3. y bc a depends on dihedral angles x rs by the cosine law. 4. Thus ω can be constructed from x rs 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)

There is only one formula The right-hand side makes sense for all x 1, x 2, x 3 in (0, π). Define the M-length L ij of the ij-th edge in AS using the above formula. L ij = λ geometric length l ij

Let AS(3) = all angle structures on a 3-simplex. 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  l ij dx ij. is closed, l ij is the M-length. (c)For classical geometric 3-simplex l ij = λ X ( classical geometric length)

Theorem 3. There is a smooth function V: AS(3) –> R s.t., (a) V(x) = λ 2 (classical volume) if x is a classical geometric tetrahedron, (b) (Schlaefli formula) let l ij 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. 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 F i : AS(3)  AS(3) sends a point (x ab ) to (y ab ) 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, 1.x is in E 3, H 3 or S 3, a classical geometric tetrahedron, 2. there is an index i so that F i (x) is in E 3 or H 3, 3. there are two distinct indices i, j so that F i F j (x) is in E 3 or H 3. The type of AS = the type of its flips.

Flips generate a Z 2 + Z 2 + Z 2 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 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.

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.

L et 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 (single flip), or (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 Key Fact Prop. 5. If a,b,c,d are dihedral angles at two pairs of opposite edges of a Euclidean or hyperbolic tetrahedron, Then

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.