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andAngleStructuresonclosed3-manifolds

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structures manifolds and Angle Structures on closed 3
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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. 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 0, and a+b+c=π}. Note. The corresponding spaces for 3-simplex, H3, E3, S3 are not convex. The space of all spherical triangles, S2 ={(a1, a2, a3) | a1 + a2 + a3 π, ai + aj 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. 1. 2. H3 and S3 are known to be non-convex. 2. 3. It is not even known if H(M,T) is connected. 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 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. Eg. 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, • 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. 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 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 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, π). Define the M-length Lij 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. 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) Theorem 3. 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. 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 types Prop. 4. Let l be the M-length of one edge in an AS. Then, (a) It is spherical type iff 0 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.
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