Volume and Angle Structures on closed 3-manifolds

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Volume and Angle Structures on closed 3-manifolds. Feng Luo Rutgers University May, 18, 2006 Georgia Topology Conference. 1. H n , S n , E n n-dim hyperbolic, spherical and Euclidean spaces with curvature λ = -1,1,0. Conventions and Notations. - PowerPoint PPT Presentation

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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