Volume and Angle Structures on closed 3-manifolds

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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

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• Conventions and Notations

1. Hn, Sn, En n-dim hyperbolic,
spherical and Euclidean spaces
with curvature ? -1,1,0.
2. sn is an n-simplex, vertices labeled
as 1,2,,n, n1.
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.
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For instance,
the space of all hyperbolic triangles, H2
(a1, a2, a3) ai gt0 and a1 a2 a3 lt
p.

The space of Euclidean triangles up to
similarity, E2 (a,b,c) a,b,c gt0,
and abcp. Note. The corresponding spaces
for 3-simplex, H3, E3, S3 are not convex.
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The space of all spherical triangles, S2
(a1, a2, a3) a1 a2 a3 gt p, ai aj
lt ak p.
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• The Schlaefli formula
• Given s3 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




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?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.

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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 s3 in T by a hyperbolic
3-simplex.
2. The sum of dihedral angles at each edge in T
is 2p.
The volume V of an H-structure the sum of the
volume of its simplexes
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H(M,T) the space of all H-structures,
a smooth manifold. V
H(M,T) gt 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

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• Suppose p(p1,p 2 ,p3 ,, pn) is a critical
  point.
• Then dV/dt(p1-t, p2t, p3,,pn)0 at t0.
• By Schlaefli, it is
• le(A)/2 -le(B)/2 0

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• 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 (or Sn) ? R can be
extended continuously to the compact closure of
Hn (or Sn )in Rn(n1)/2 .

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Classical geometric tetrahedra

• Euclidean Hyperbolic
  Spherical

From dihedral angle point of view,
vertex triangles are spherical triangles.
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Angle Structure

• An angle structure (AS) on a 3-simplex
  
• assigns each edge a dihedral angle in (0, p)
• so that each vertex triangle is a spherical
  triangle.
• Eg. Classical geometric tetrahedra are AS.

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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 2p.
• Note The conditions are linear equations and
  linear inequalities

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• 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.

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• 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.

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• 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.

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Questions

• How to define the volume of an angle structure?
• How does an angle structure look like?

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Classical volume V can be defined on H3 U E3 U
S3 by integrating the Schlaefli 1-form ? ?/2
? lij dxij .

• ? 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 ? can be constructed from xrs by the
  cosine law.
• d ? 0.
• Claim all above can be carried out for angle
  structures.

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Angle Structure

• Face angle is well defined by the cosine law,
  i.e.,
• face angle edge length of the vertex
  triangle.

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The Cosine Law

• For a hyperbolic, spherical or Euclidean
  triangle of inner angles
• and edge lengths ,
• (S)
• (H)
• (E)

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The Cosine Law

• There is only one formula
• The right-hand side makes sense for all x1,
  x2, x3 in (0, p).
• Define the M-length Lij of the ij-th edge in
  AS using the above formula.
• Lij ?
  geometric length lij

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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)
• ? 1/2 ? lij dxij .
• is closed, lij is the M-length.
• For classical geometric 3-simplex
• lij ?X (classical
  geometric length)

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• Theorem 3. There is a smooth function V
  AS(3) gt R s.t.,
• (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
  established Milnor conjecture in all dimension.
  Rivin has a new proof of it now.

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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

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The i-th Flip Map

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• The i-th flip map Fi AS(3) ?AS(3)
• sends a point (xab) to (yab) where

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angles change under flips
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Lengths change under flips
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• 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.

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Flips generate a Z2 Z2 Z2 action
on AS(3). Step 2. Type is determined by the
length of one edge.
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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 lt
l lt p. (b) It is of Euclidean type iff l is in
0,p. (c) It is of hyperbolic type iff l is
less than 0 or larger than p. An AS is non
classical iff one edge length is at least p.
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• 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.

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• 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.

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• 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.

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• Let Y be all edges of lengths at least p.
• The intersection of Y with each 3-simplex
  consists of,
• three edges from one vertex (single flip), or
• 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.

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• 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)

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• 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

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• Summary for the normal surface X
  
  
  
• 1. Sum of inner angles of a quadrilateral gt 2p.
• 2. Sum of the inner angles of a triangle gt p.
• 3. Sum of the inner angles at each vertex 2p.
• Thus the Euler characteristic of X is
  positive.
• Thank you

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• Thank you.