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Volume and Angle Structures on triangulated closed 3-manifolds

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

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Volume and Angle Structures on triangulated closed 3-manifolds

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  1. Volume and Angle Structures on triangulated closed 3-manifolds Feng Luo Rutgers University May, 18, 2006 Georgia Top. Conference

  2. 1.Hn, Sn, En n-dim hyperbolic, spherical and Euclidean spaces with curvature λ = -1,1,0. Conventions and Notations 2. σn is an n-simplex, vertices labeled as 1,2,…,n, n+1. 3. indices i,j,k,l are pairwisedistinct. 4.Hn(or Sn or En) is the space of all hyperbolic (spherical, or Euclidean) n-simplexes parameterized by the dihedral angles.

  3. 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 triangle 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, S3are not convex.

  4. let V =V(x) be the volume where x=(x12,x13,x14,x23,x24,x34). (Schlaefli) d(V) = /2  lij dxij The Schlaefli formula Given σ3 in H3, S3with edge length lij and dihedral angle xij,

  5. Define the volume of a Euclidean simplex to be 0. Corollary 1. The volume function V: H3U E3U S3  R is C1-smooth. Schlaefli formula also suggests: the length should be (curvature) length.

  6. Schlaefli formula suggests a way to find geometric structures on triangulated closed 3-manifold (M, T) where T is a triangulation. 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 is defined to be the sum of the volume of its simplexes

  7. The space of all H-structures, parameterized by dihedral angles, is a smooth manifold H(M,T). The volume V: H(M,T) –> R is a smooth function. Prop. 2.(Murakami, Casson, …) If V: H(M,T)  R has a critical point p, then the manifold M is hyperbolic. The main ingredient of the proof: Schlaefli formula.

  8. The difficulties in carrying out the above approach: • It is difficult to determine if H(M,T) is non-empty. • H3 and S3 are known to be non-convex. • It is not even known if H(M,T) is connected. 4. Milnor’s question: V: Hn R can be extended continuously to the compact closure of Hn in Rn(n+1)/2 .

  9. Classical geometric Euclidean Hyperbolic Spherical From dihedral angle point of view, vertex triangles are spherical triangles.

  10. 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. 2. An angle structure on (M, T): An 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.

  11. There is a natural notion of volume of AS on 3-simplex (to be defined below using Schlaefli). Let AS(M,T) be the space of all AS’s on a closed triangulated 3-manifold (M,T). It is a convex bounded polytope in a Euclidean space. Let V: AS(M, T)  R be the volume map.

  12. Theorem 1. Suppose T is a triangulation of a closed 3-manifold M and the 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, the 2-sphere is non-separating. 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.

  13. Theorem 2. (Kitaev, L) For any closed 3-manifold M, there is a triangulation T of M supporting an angle structureso that all 3-simplexes are classical geometric tetrahedra.

  14. Questions • How to definethe volume of an angle structure? • How does angle structure look like?

  15. Volume V can be defined on H3U E3U S3 by integrating the Schlaefli 1-form ω =/2  lij dxij . • ω depends on the length lij which is obtained from the face angles ybca by the cosine law. The face angle ybca is obtained from dihedral angle xrs by the cosine law. Thus, the length is obtained from the dihedral angles by using the cosine law twice. 2. d ω =0. We claim both steps can be carried out for angle structures.

  16. 3-Dim Angle Structure Each face angle is well defined by the cosine law.

  17. The Cosine Law For a hyperbolic, spherical or Euclidean triangle of inner angles and edge lengths , (S) (H) (E)

  18. The Cosine Law There is only one formula The right-hand side makes sense for all x1, x2, x3 in (0, π). We define the M-length Lij of ij-the edge in an AS using the above formula. Let AS(3) be the space of all angle structures, parameterized by the dihedral angles, on a 3-simplex.

  19. Edge Length of AS Prop. 2. (a) For an AS, the M-length of the ij-th edge is well defined, i.e., independent of the choice of the two triangles △ ijk, △ ijl. • The differential 1-form on AS(3) ω= is a closed continuous 1-form, lij is the M-length. (c) For classical geometry 3-simplex Lij = λ lij

  20. Theorem 3. There is a smooth function V: AS(3) –> R so that the following holds, (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.

  21. Main ideas of the proof theorem 1. Step 1. Classify AS on 3-simplex into three types: Euclidean, hyperbolic, spherical types.

  22. The i-th Flip Map

  23. Flip map on angles

  24. Lengths change under flips

  25. AS(3) = space of all AS’s on , open convex polytope in . The i-th flip map : AS(3) AS(3) sends a point ( ) to ( ) where

  26. 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 simplex, 2. There is an index i so that (x) is in E3 or H3, 3. There are two distinct indices i, j so that (x) is in E3 or H3. The type of AS: the type of its flips.

  27. Step 2. Classify AS into three types: hyperbolic, Euclidean and spherical. Its type is determined by the length of one edge.

  28. Classification of types Prop. 4. Let l be the M-length of an edge in an AS on a tetrahedron. Then, (a) It is spherical type iff l is in (0, 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. In particular, an AS is non classical iff one edge length is at least π.

  29. Step 3. Show that at the critical point p of the volume V on AS(M, T), the Schlaefli formula implies that the edge length is well defined. Namely, independent of the choice of the 3-simplexes adjacent to it. Step 4. Using the steps 1,2,3 to show at the critical point, all simplexes have the same type.

  30. Step 5. Show that if all AS on the simplexes come from classical hyperbolic (or spherical) simplexes, we have a constant curvature metric. Step 6. Show that at the local maximum point, not all simplexes are classical Euclidean.

  31. 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.

  32. In this case, consider the set Y of all edge of lengths at least π. By definition, 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. It turns out the Euler characteristic of the surface is positive.

  33. The normal surface X is a union of triangles and quadrilaterals by the construction. • Each triangle is a spherical triangle (def. of AS). • Each quadrilateral Q comes from double flips of a Euclidean or hyperbolic tetrahedron (def.of Y). • Thus Q has four dihedral angles -a,-b, -c, -d so that a,b,c,d, are the dihedral angles at two pairs of opposite sides of Euclidean or hyperbolic tetrahedron. (def. of flips)

  34. Prop. 5.If a,b,c,d are dihedral angles at two pairs of opposite edges of a Euclidean or hyperbolic tetrahedron, then

  35. 1. The sum of inner angles of the normal quadrilateral is greater than 2π. 2. The sum of the inner angles of the normal triangle is greater than π. 3. The sum of the inner angles at each vertex is 2π. Thus the Euler characteristic is positive by the Gauss-Bonnet formula.

  36. Thank you.

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