Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University May, 18, 2006 Georgia Topology Conference

1. Hn, Sn, En n-dim hyperbolic, spherical and 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 Corollary 1. The volume function 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.

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

V: H(M,T) –> R is the volume. H(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:

Then dV/dt(p1-t, p2+t, p3,…,pn)=0 at t=0. By Schlaefli, it is: 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

It is difficult to determine if H(M,T) is non-empty. 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. Milnor’s conj.: V: Hn  R can be extended continuously to the compact closure of Hn in Rn(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 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.

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 . ω depends on the length lij 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. d ω =0. Claim: all above can be carried out for angle structures.

Angle Structure face angle = edge length of the vertex triangle. 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 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.

(b) The differential 1-form on AS(3) ω= 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. For classical geometric 3-simplex lij = λX (classical geometric length)

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.

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.

Step 2. Type is determined by the length of one edge. 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 6. Show that at the local maximum point, 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, three edges from one vertex, or, (single flip) 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.

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)

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

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.