Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas...
-
date post
21-Dec-2015 -
Category
Documents
-
view
218 -
download
1
Transcript of Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas...
![Page 1: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/1.jpg)
Volume and Angle Structures on closed 3-manifolds
Feng Luo
Rutgers University
Oct. 28, 2006
Texas Geometry/Topology conference
Rice University
![Page 2: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/2.jpg)
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.
![Page 3: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/3.jpg)
For instance, the space of all hyperbolic triangles, H2 ={(a1, a2, a3) | ai >0 and a1 + a2 + a3 < 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.
![Page 4: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/4.jpg)
The space of all spherical triangles,
S2 ={(a1, a2, a3) | a1 + a2 + a3 > π, ai + aj < ak + π}.
![Page 5: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/5.jpg)
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
![Page 6: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/6.jpg)
∂V/∂xij = (λlij )/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.
![Page 7: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/7.jpg)
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
![Page 8: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/8.jpg)
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:
![Page 9: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/9.jpg)
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
![Page 10: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/10.jpg)
The difficulties in carrying out the above approach:
1. 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 (or Sn) R can be extendedcontinuously to the compact closure of Hn (or Sn )inRn(n+1)/2 .
![Page 11: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/11.jpg)
Classical geometric tetrahedra
Euclidean Hyperbolic Spherical
From dihedral angle point of view,
vertex triangles are spherical triangles.
![Page 12: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/12.jpg)
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.
![Page 13: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/13.jpg)
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
![Page 14: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/14.jpg)
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.
![Page 15: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/15.jpg)
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,
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.
![Page 16: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/16.jpg)
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.
![Page 17: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/17.jpg)
Questions
• How to define the volume of an angle structure?
• How does an angle structure look like?
![Page 18: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/18.jpg)
Classical volume V can be defined on H3 U E3 U S3 by integrating the
Schlaefli 1-form ω =/2 lij dxij .
1. ω depends on the length lij
2. 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.
5. d ω =0.
Claim: all above can be carried out for angle structures.
![Page 19: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/19.jpg)
Angle Structure
Face angle is well defined by the cosine law, i.e., face angle = edge length of the vertex triangle.
![Page 20: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/20.jpg)
The Cosine Law For a hyperbolic, spherical or Euclidean triangle of inner angles
and edge lengths , (S)
(H)
(E)
1 2 3, ,x x x
1 2 3, ,y y y
cos( ) (cos cos cos ) /(sin sin )i i j k j ky x x x x x cosh( ) (cos cos cos ) /(sin sin )i i j k j ky x x x x x
1 (cos cos cos ) /(sin sin )i j k j kx x x x x
![Page 21: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/21.jpg)
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
cos( ) (cos cos cos ) /(sin sin )i i j k j ky x x x x x
![Page 22: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/22.jpg)
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.
(c) For classical geometric 3-simplex
lij = λX (classical geometric length)
![Page 23: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/23.jpg)
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.
1C
( ) 1/ 2( )ij ijij
d V l dx
6[0, ]
![Page 24: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/24.jpg)
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
![Page 25: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/25.jpg)
The i-th Flip Map
![Page 26: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/26.jpg)
The i-th flip map Fi : AS(3) AS(3)
sends a point (xab) to (yab) where
,ij ij
jk jk
y x
y x
![Page 27: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/27.jpg)
angles change under flips
![Page 28: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/28.jpg)
Lengths change under flips
![Page 29: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/29.jpg)
Prop. 3. For any AS x on a 3-simplex, exactly one of the following holds,
1. 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.
![Page 30: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/30.jpg)
Flips generate a Z2 + Z2 + Z2 action on AS(3).
Step 2. Type is determined by the length of one edge.
![Page 31: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/31.jpg)
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.
![Page 32: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/32.jpg)
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.
![Page 33: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/33.jpg)
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.
![Page 34: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/34.jpg)
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.
![Page 35: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/35.jpg)
Let 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.
![Page 36: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/36.jpg)
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)
![Page 37: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/37.jpg)
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
2a b c d
( ) ( ) ( ) ( ) 2a b c d
![Page 38: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/38.jpg)
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
![Page 39: Volume and Angle Structures on closed 3-manifolds Feng Luo Rutgers University Oct. 28, 2006 Texas Geometry/Topology conference Rice University.](https://reader035.fdocument.pub/reader035/viewer/2022062407/56649d6b5503460f94a4a6c4/html5/thumbnails/39.jpg)
Thank you.