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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**1.Hn, Sn, En n-dim hyperbolic, spherical and**Euclidean spaces with curvature λ = -1,1,0. Conventions and Notations 2.σnis an n-simplex, vertices labeled as 1,2,…,n, n+1. 3.indices i,j,k,l are pairwisedistinct. 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 allhyperbolic triangles,**H2 ={(a1, a2, a3) | ai >0 and a1 + a2 + a3 < π}. The space ofEuclidean trianglesup 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.**The space of allspherical triangles, S2 ={(a1, a2, a3) |**a1 + a2 + a3 > π, ai + aj < ak + π}.**let V =V(x) be the volume where x=(x12,x13,x14,x23,x24,x34).**d(V) = /2 lij dxij The Schlaefli formula Given σ3 in H3, S3with edge lengths lij and dihedral angles xij,**∂V/∂xij = (λlij )/2**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 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 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**H(M,T)= the space of all H-structures, a smooth manifold.**V: H(M,T)–> 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:**Suppose p=(p1,p 2 ,p3 ,…, pn) is a critical point.**ThendV/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.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**EuclideanHyperbolic Spherical From dihedral angle point of view, vertex trianglesare 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 aspherical 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 definethe volume of an angle structure? • How does an angle structure look like?**Classical volumeV can be defined on H3U E3U S3 by**integrating the Schlaefli 1-form ω =/2 lij dxij . • ω depends on the length lij • lij depends on the face angles ybca 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 Lijof 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,sphericaltypes. First, let us see that, AS(3) ≠ classical geometric tetrahedra**The i-th flip map Fi : AS(3) AS(3)**sends a point (xab) to (yab) where**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 < 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.**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 π.**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.**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)**The Key Fact**Prop. 5.If a,b,c,d are dihedral angles at two pairs of opposite edges of aEuclidean or hyperbolic tetrahedron, Then**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