A relative version of the Turaev-Viro invariants and the volume of hyperbolic polyhedral 3-manifolds
AA relative version of the Turaev-Viro invariants and thevolume of hyperbolic polyhedral -manifolds Tian Yang
Abstract
We introduce a relative version of the Turaev-Viro invariants for an ideally triangulated compact -manifold with non-empty boundary and a coloring on the edges, and prove that they coincide withthe relative Reshetikhin-Turaev invariants [4, 17] of the double of the manifold with the link insideit consisting of the double of the edges of the ideal triangulation and with the coloring inheritedfrom the edges. When the coloring is zero, the relative Turaev-Viro invariants coincide with theTuraev-Viro invariants [30] of the manifold. We also propose the Volume Conjecture for the relativeTuraev-Viro invariants whose asymptotic behavior is related to the volume of the manifold in thehyperbolic polyhedral metric [19, 20] with singular locus the edges and cone angles determined bythe coloring, and prove the conjecture in the case that the cone angles are sufficiently small. Thissuggests an approach of solving the Volume Conjecture for the Turaev-Viro invariants proposed byChen-Yang [6] for hyperbolic -manifold with totally geodesic boundary. Let M be a compact -manifold with non-empty boundary, and let T be an ideal triangulation of M, that is, a finite collection T = { ∆ , . . . , ∆ | T | } of truncated Euclidean tetrahedra with faces identified inpairs by affine homeomorphisms. We also let E = { e , . . . , e | E | } be the set of edges of T . For a positiveinteger r (cid:62) , a coloring a of ( M, T ) assigns an integer a i in between and r − to the edge e i , andthe coloring a is r -admissible if for any { i, j, k } ⊂ { , . . . , | E |} such that e i , e j and e k are the edges ofa face of T , (1) a i + a j − a k (cid:62) , (2) a i + a j + a k (cid:54) r − , (3) a i + a j + a k is even. Definition 1.1.
Let r (cid:62) be an integer and let q be a r -th root of unity such that q is a primitive r -throot of unity. Then the r -th relative Turaev-Viro invariant of ( M, T ) with the coloring b = ( b , . . . , b | E | ) on the edges is defined by TV r ( M, E, b ) = (cid:88) a | E | (cid:89) i =1 H( a i , b i ) | T | (cid:89) s =1 (cid:12)(cid:12)(cid:12)(cid:12) a s a s a s a s a s a s (cid:12)(cid:12)(cid:12)(cid:12) , where the sum is over all the r -admissible colorings a = ( a , . . . , a | E | ) of ( M, T ) , H( a i , b i ) = ( − a i + b i q ( a i +1)( b i +1) − q − ( a i +1)( b i +1) q − q − , { a s , . . . , a s } are the colors of the edges of the tetrahedron ∆ s assigned by a and (cid:12)(cid:12)(cid:12)(cid:12) a s a s a s a s a s a s (cid:12)(cid:12)(cid:12)(cid:12) isthe quantum j -symbol of the -tuple ( a s , . . . , a s ) . (See Section 3.2.) a r X i v : . [ m a t h . G T ] S e p e note that if b = (0 , . . . , , then TV r ( M, E, b ) coincides with the Turaev-Viro invariant of M [30].Similar to the relationship between the Turaev-Viro invariants of M and the Reshetikhin-Turaev in-variants of its double [31, 27, 5], the relative Turaev-Viro invariants of ( M, T ) and the relative Reshetikhin-Turaev invariants [4, 17] of the double of M is related as follows. Theorem 1.2. At q = e πir , TV r ( M, E, b ) = (cid:18) πr √ r (cid:19) − χ ( M ) RT r ( D ( M ) , D ( E ) , b ); and at q = e πir , TV r ( M, E, b ) = 2 rankH ( M ; Z ) (cid:18) πr √ r (cid:19) − χ ( M ) RT r ( D ( M ) , D ( E ) , b ) , where χ ( M ) is the Euler characteristic of M, D ( M ) is the double of M and D ( E ) ⊂ D ( M ) is the linkconsisting of the union of the double of the edges. Theorem 1.2 can be proved following the same idea of Roberts [27]. See also [5] for the caseof manifolds with non-empty boundary and [9] for the case that q = e πir for odd r. For the readersconvenience, we include a sketch of the proof of Theorem 1.2 in Section 2.Various quantum invariants are expected to contain different geometric information of the manifold.See for example [15, 21, 6, 3, 32]. For the relative Turaev-Viro invariants, the corresponding geometricobject is the hyperbolic polyhedral metric. As defined in [19, 20], a hyperbolic polyhedral metric onan ideally triangulated -manifold ( M, T ) is obtained by replacing each tetrahedron in T by a truncatedhyperideal tetrahedron (see Secrtion 3.1) and replacing the gluing homeomorphisms between pairs of thefaces by isometries. The cone angle at an edge is the sum of the dihedral angles of the truncated hyper-ideal tetrahedra around the edge. If all the cone angles are equal to π, then the hyperbolic polyhedralmetric gives a hyperbolic metric on M with totally geodesic boundary. In [20, Theorem 1.2 (b)], Luoand the author proved that hyperbolic polyhedral metrics on ( M, T ) are rigid in the sense that they areup to isometry determined by their cone angles. Conjecture 1.3.
Let { b ( r ) } be a sequence of r -admissible colorings of ( M, T ) . For each i ∈ { , . . . , | E |} , let θ i = (cid:12)(cid:12)(cid:12) π − lim r →∞ πb ( r ) i r (cid:12)(cid:12)(cid:12) and let θ = ( θ , . . . , θ | E | ) . Then as r varies over all odd integers and at q = e πir , lim r →∞ πr log TV r ( M, E, b ( r ) ) = Vol( M E θ ) , where M E θ is M with the hyperbolic polyhedral metric on ( M, T ) with cone angles θ. We note that if b = (0 , . . . , , then Conjecture 1.3 recovers the Volume Conjecture for the Turaev-Viro invariants for hyperbolic -manifolds with totally geodesic boundary proposed by Chen and theauthor [6].The main result of this paper is the following Theorem 1.4.
Conjecture 1.3 is true for all ideally triangulated -manifold ( M, T ) with non-emptyboundary, and with sufficiently small cone angles θ.
2n [20, Proposition 6.14], Luo and the author proved that any hyperbolic polyhedral metric on ( M, T ) with all cone angles less than or equal to π can be smoothly deformed in the space of hyperbolic poly-hedral metrics to the hyperbolic polyhedral metric with all cone angles equal to . It is expected that thespace of all possible cone angles of hyperbolic polyhedral metrics on ( M, T ) is connected so that eachhyperbolic polyhedral metric can be smoothly deformed to the hyperbolic polyhedral metric with allcone angles equal to . In [16], Kojima proved that every hyperbolic -manifold M with totally geodesicboundary admits an ideal triangulation such that each tetrahedron is either isometric to a truncated hy-perideal tetrahedron or flat; and it is expected that every such M admits a geometric ideal triangulationthat each tetrahedron is truncated hyperideal. Therefore, for Kojima’s ideal triangulations, if one couldpush the cone angles in Theorem 1.4 from sufficiently small to π, then one solves Chen-Yang’s VolumeConjecture [6] for the Turaev-Viro invariants for hyperbolic -manifolds with totally geodesic boundary.As an immediate consequence of Theorems 1.2 and 1.4, we have Theorem 1.5.
The Volume Conjecture of the relative Reshetikhin-Turaev invariants [32, Conjecture 1.1]is true for all pairs ( D ( M ) , D ( E )) with sufficiently small cone angles. Outline of the proof of Theorem 1.4.
We follow the guideline of Ohtsuki’s method. In Proposition 4.1,we compute the relative Turaev-Viro invariant of ( M, T ) , writing them as a sum of values of a holomor-phic function f r at integer points. The function f r comes from Faddeev’s quantum dilogarithm function.Using Poisson Summation Formula, we in Proposition 4.3 write the invariants as a sum of the Fouriercoefficients of f r computed in Propositions 4.2. In Proposition 5.2 we show that the critical value of thefunctions in the leading Fourier coefficients has real part the volume of the deeply truncated tetrahedron.Then we estimate the leading Fourier coefficients in Sections 5.3 using the Saddle Point Method (Propo-sition 5.1). Finally, we estimate the non-leading Fourier coefficients and the error term respectively inSections 5.4 and 5.5 showing that they are neglectable, and prove Theorem 1.4 in Section 5.6. Acknowledgments.
The author would like to thank Giulio Belletti, Francis Bonahon, Xingshan Cui andFeng Luo for useful discussions. The author is partially supported by NSF Grant DMS-1812008.
We first recall the definition of the relative Reshetikhin-Turaev invariants following the skein theoreticalapproach [4, 17], and focus on the SO (3) -theory and the values at the root of unity q = e π √− r for oddintegers r (cid:62) . A framed link in an oriented -manifold M is a smooth embedding L of a disjoint union of finitelymany thickened circles S × [0 , (cid:15) ] , for some (cid:15) > , into M. The Kauffman bracket skein module K r ( M ) of M is the C -module generated by the isotopic classes of framed links in M modulo the follow tworelations:(1) Kauffman Bracket Skein Relation: = e π √− r + e − π √− r . (2) Framing Relation: L ∪ = ( − e π √− r − e − π √− r ) L. There is a canonical isomorphism (cid:104) (cid:105) : K r (S ) → C defined by sending the empty link to . The image (cid:104) L (cid:105) of the framed link L is called the Kauffmanbracket of L. K r ( A × [0 , be the Kauffman bracket skein module of the product of an annulus A with a closedinterval. For any link diagram D in R with k ordered components and S , . . . , S k ∈ K r ( A × [0 , , let (cid:104) S , . . . , S k (cid:105) D be the complex number obtained by cabling S , . . . , S k along the components of D considered as aelement of K r (S ) then taking the Kauffman bracket (cid:104) (cid:105) . On K r ( A × [0 , there is a commutative multiplication induced by the juxtaposition of annuli,making it a C -algebra; and as a C -algebra K r ( A × [0 , ∼ = C [ z ] , where z is the core curve of A. For aninteger a (cid:62) , let e a ( z ) be the a -th Chebyshev polynomial defined recursively by e ( z ) = 1 , e ( z ) = z and e a ( z ) = ze a − ( z ) − e a − ( z ) . Let I r = { , , . . . , r − } be the set of even integers in between and r − . Then the Kirby coloring Ω r ∈ K r ( A × [0 , isdefined by Ω r = µ r (cid:88) a ∈ I r [ a + 1] e a , where µ r = 2 sin πr √ r and [ a ] is the quantum integer defined by [ a ] = e aπ √− r − e − aπ √− r e π √− r − e − π √− r . Let M be a closed oriented -manifold and let L be a framed link in M with n components. Suppose M is obtained from S by doing a surgery along a framed link L (cid:48) , D ( L (cid:48) ) is a standard diagram of L (cid:48) (ie, the blackboard framing of D ( L (cid:48) ) coincides with the framing of L (cid:48) ). Then L adds extra componentsto D ( L (cid:48) ) forming a linking diagram D ( L ∪ L (cid:48) ) with D ( L ) and D ( L (cid:48) ) linking in possibly a complicatedway. Let U + be the diagram of the unknot with framing , σ ( L (cid:48) ) be the signature of the linking matrix of L (cid:48) and b = ( b , . . . , b n ) be a multi-elements of I r . Then the r -th relative Reshetikhin-Turaev invariantof M with L colored by b is defined as RT r ( M, L, b ) = µ r (cid:104) e b , . . . , e b n , Ω r , . . . , Ω r (cid:105) D ( L ∪ L (cid:48) ) (cid:104) Ω r (cid:105) − σ ( L (cid:48) ) U + . (2.1) Sketch of the proof of Theorem 1.2.
We focus on the case that q = e πir , and the case that q = e πir issimilar.Consider the handle decomposition of M dual to the ideal triangulation T , namely, the -handlescome from a tubular neighborhood of the edges, the -handles come from a tubular neighborhood ofthe farces and the -handles come from the complement of the - and -handles. Following the idea ofRoberts [27], we construct the following quantity CM r ( M, E, b ) . Let { (cid:15) , . . . , (cid:15) | E | } be the attachingcurves of the -handles and let { δ , . . . , δ | F | } be the meridians of the -handles. Thicken these curves tobands parallel to the surface of the -skeleton H and push each (cid:15) i slightly into H and circulate it by aframed trivial loop γ i . Embed H arbitrarily into S , cable each of the image of the (cid:15) - and δ -bands by theKirby coloring Ω r and cable the image of γ i by the b i -th Chebyshev polynomial. In this way, we get anelement S ( M,E, b ) in K r ( S ) , and we define CM r ( M, E, b ) = µ | T | r (cid:104) S ( M,E, b ) (cid:105) .
4n the one hand, since each face of T has three edges, each δ -band encloses exactly three (cid:15) -bands(see [27, Figure 11]). Writing Ω r = µ r (cid:88) a ∈ I r [ a + 1] e a and applying [9, Lemma 3.3] to each δ -band, we have CM r ( M, E, b ) = µ | T |−| E | + | F | r (cid:88) a | E | (cid:89) i =1 H( a i , b i ) | T | (cid:89) s =1 (cid:12)(cid:12)(cid:12)(cid:12) a s a s a s a s a s a s (cid:12)(cid:12)(cid:12)(cid:12) , =2 − rankH( M ; Z ) µ χ ( M ) r TV r ( M, E, b ) , (2.2)where c runs over all the r -admissible colorings of ( M, T ) with even integers and the last equality comesfrom [9, Lemma A.4, Theorem 2.9 and its proof].On the other hand, the image of the union of the (cid:15) - and δ -bands form a surgery diagram of the -manifold D ( M ) S × S ) | T |− and the signature of the linking matrix equals zero as argued in [27,Proof of Theorem 3.4]. Note that the solid tori attached along the (cid:15) -bands are the double of the -handles,and each γ i is isotopic to the meridian of a tubular neighborhood of (cid:15) i , hence is isotopic to the core ofthe solid torus attached to it, which is the double of the edge e i of T . Therefore, CM r ( M, E, b ) = µ | T |− r RT r ( D ( M ) S × S ) | T |− , D ( E ) , b )=RT r ( D ( M ) , D ( E ) , b ) , (2.3)where the last equality comes from the fact that RT r ( M M , L ∪ L , ( b , b )) = µ − r RT r ( M , L , b ) · RT r ( M , L , b ) and that RT r ( S × S ) = 1 . From (2.2) and (2.3), the result follows.
We first recall some of the basic results on hyper-ideal tetrahedra. Following [1] and [12], a truncated hy-perideal tetrahedron ∆ in H is a compact convex polyhedron that is diffeomorphic to a truncated tetra-hedron in E with four hexagonal faces { H , H , H , H } isometric to right-angled hyperbolic hexagonsand four triangular faces { T , T , T , T } isometric to hyperbolic triangles (see Figure 1). An edge ina hyper-ideal tetrahedron is the intersection of two hexagonal faces and the dihedral angle at an edgeis the angle between the two hexagonal faces adjacent to it. The angle between a hexagonal face and atriangular face is always π . Let e ij be the edge connecting the triangular faces T i and T j , and let θ ij and l ij respectively be the dihedral angle at and edge length of e ij . Then by the Cosine Law of hyperbolictriangles and right-angled hyperbolic hexagons, we have cosh l ij = c kl + c ik c jk + c il c jl + ( c ik c jl + c il c jk ) c ij − c kl c ij (cid:113) − c ij + c ik + c il + 2 c ij c ik c il (cid:113) − c ij + c jk + c jl + 2 c ij c jk c jl , (3.1)where { i, j, k, l } = { , , , } and c ij = cos θ ij ; and cos θ kl = ch ij + ch ik ch il + ch jk ch jl + ( ch ik ch jl + ch il ch jk ) ch ij − ch kl ch ij (cid:113) − ch ij + ch ik + ch jk + 2 ch ij ch ik ch jk (cid:113) − ch ij + ch il + ch jl + 2 ch ij ch il ch jl , (3.2)5 Figure 1where ch ij = cosh l ij . By [1], a truncated hyperideal tetrahedron is up to isometry determined by its six dihedral angles { θ , . . . , θ } , and by (3.2) is determined by its six edge lengths { l , . . . , l } . Definition 3.1 ([19, 20]) . Let
Vol and { θ , . . . , θ } respectively be the volume and the dihedral anglesof a hyperideal tetrahedron ∆ as functions of the edge lengths { l , . . . , l } . The co-volume function
Cov is defined by
Cov( l , . . . , l ) = Vol + 12 (cid:88) i =1 θ i · l i . The key property of the co-volume function is the following
Lemma 3.2.
For i ∈ { , . . . , } , ∂ Cov ∂l i = θ i . Proof.
By the Schl¨afli formula, we have ∂ Vol ∂θ i = − l i . Then by the chain rule and the product rue, we have ∂ Cov ∂l i = (cid:88) k =1 ∂ Vol ∂θ k · ∂θ k ∂l i + 12 (cid:88) k =1 ∂∂l i (cid:16) θ k · l k (cid:17) = − (cid:88) k =1 l k · ∂θ k ∂l i + 12 (cid:88) k =1 · ∂θ k ∂l i · l k + θ i θ i . Let log :
C(cid:114) ( −∞ , → C be the standard logarithm function defined by log z = log | z | + √− z − π < arg z < π. The dilogarithm function Li : C(cid:114) (1 , ∞ ) → C is defined by Li ( z ) = − (cid:90) z log(1 − u ) u du where the integral is along any path in C(cid:114) (1 , ∞ ) connecting and z, which is holomorphic in C(cid:114) [1 , ∞ ) and continuous in C(cid:114) (1 , ∞ ) . The dilogarithm function satisfies the follow properties (see eg. Zagier [34]).(1) Li (cid:16) z (cid:17) = − Li ( z ) − π − (cid:0) log( − z ) (cid:1) . (3.3)(2) In the unit disk (cid:8) z ∈ C (cid:12)(cid:12) | z | < (cid:9) , Li ( z ) = ∞ (cid:88) n =1 z n n . (3.4)(3) On the unit circle (cid:8) z = e √− θ (cid:12)(cid:12) (cid:54) θ (cid:54) π (cid:9) , Li ( e √− θ ) = π θ ( θ − π ) + 2 √− θ ) . (3.5)Here Λ : R → R is the Lobachevsky function defined by Λ( θ ) = − (cid:90) θ log | t | dt, which is an odd function of period π. See eg. Thurston’s notes [31, Chapter 7].The following variant of Faddeev’s quantum dilogarithm functions [10, 11] will play a key role in theproof of the main result. Let r (cid:62) be an odd integer. Then the following contour integral ϕ r ( z ) = 4 π √− r (cid:90) Ω e (2 z − π ) x x sinh( πx ) sinh( πxr ) dx (3.6)defines a holomorphic function on the domain (cid:110) z ∈ C (cid:12)(cid:12)(cid:12) − πr < Re z < π + πr (cid:111) , where the contour is Ω = (cid:0) − ∞ , − (cid:15) (cid:3) ∪ (cid:8) z ∈ C (cid:12)(cid:12) | z | = (cid:15), Im z > (cid:9) ∪ (cid:2) (cid:15), ∞ (cid:1) , for some (cid:15) ∈ (0 , . Note that the integrand has poles at n √− , n ∈ Z , and the choice of Ω is to avoidthe pole at . The function ϕ r ( z ) satisfies the following fundamental properties, whose proof can be found in [ ? ,Section 2.3]. Lemma 3.3. (1) For z ∈ C with < Re z < π, − e √− z = e r π √− (cid:16) ϕ r (cid:0) z − πr (cid:1) − ϕ r (cid:0) z + πr (cid:1)(cid:17) . (3.7)7
2) For z ∈ C with − πr < Re z < πr , e r √− z = e r π √− (cid:16) ϕ r ( z ) − ϕ r (cid:0) z + π (cid:1)(cid:17) . (3.8)Using (3.7) and (3.8), for z ∈ C with π + n − πr < Re z < π + nπr , we can define ϕ r ( z ) inductivelyby the relation n (cid:89) k =1 (cid:16) − e √− (cid:0) z − (2 k − πr (cid:1)(cid:17) = e r π √− (cid:16) ϕ r (cid:0) z − nπr (cid:1) − ϕ r ( z ) (cid:17) , (3.9)extending ϕ r ( z ) to a meromorphic function on C . The poles of ϕ r ( z ) have the form ( a + 1) π + bπr or − aπ − bπr for all nonnegative integer a and positive odd integer b. Let q = e π √− r , and let ( q ) n = n (cid:89) k =1 (1 − q k ) . Lemma 3.4. (1) For (cid:54) n (cid:54) r − , ( q ) n = e r π √− (cid:16) ϕ r (cid:0) πr (cid:1) − ϕ r (cid:0) πnr + πr (cid:1)(cid:17) . (3.10) (2) For r − (cid:54) n (cid:54) r − , ( q ) n = 2 e r π √− (cid:16) ϕ r (cid:0) πr (cid:1) − ϕ r (cid:0) πnr + πr − π (cid:1)(cid:17) . (3.11)We consider (3.11) because there are poles in ( π, π ) , and to avoid the poles we move the variablesto (0 , π ) by subtracting π. Let { n } = q n − q − n and { n } ! = (cid:81) nk =1 { k } . Then { n } ! = ( − n q − n ( n +1)2 ( q ) n , and as a consequence of Lemma 3.4, we have Lemma 3.5. (1) For (cid:54) n (cid:54) r − , { n } ! = e r π √− (cid:16) − π (cid:0) πnr (cid:1) + (cid:0) πr (cid:1) ( n + n )+ ϕ r (cid:0) πr (cid:1) − ϕ r (cid:0) πnr + πr (cid:1)(cid:17) . (3.12) (2) For r − (cid:54) n (cid:54) r − , { n } ! = 2 e r π √− (cid:16) − π (cid:0) πnr (cid:1) + (cid:0) πr (cid:1) ( n + n )+ ϕ r (cid:0) πr (cid:1) − ϕ r (cid:0) πnr + πr − π (cid:1)(cid:17) . (3.13)The function ϕ r ( z ) and the dilogarithm function are closely related as follows. Lemma 3.6. (1) For every z with < Re z < π,ϕ r ( z ) = Li ( e √− z ) + 2 π e √− z − e √− z ) 1 r + O (cid:16) r (cid:17) . (3.14) (2) For every z with < Re z < π,ϕ (cid:48) r ( z ) = − √− − e √− z ) + O (cid:16) r (cid:17) . (3.15) (3) [24, Formula (8)(9)] ϕ r (cid:16) πr (cid:17) = Li (1) + 2 π √− r log (cid:16) r (cid:17) − π r + O (cid:16) r (cid:17) . .3 Quantum j -symbols and their underlying geometry As is customary we define [ a ]! = a (cid:89) k =1 [ k ] . Recall that a triple ( a i , a j , a k ) of integers in { , . . . , r − } is r -admissible if(1) a i + a j − a k (cid:62) , (2) a i + a j + a k (cid:54) r − , (3) a i + a j + a k is even.For an r -admissible triple ( a , a , a ) , define ∆( a , a , a ) = (cid:115) [ a + a − a ]![ a + a − a ]![ a + a − a ]![ a + a + a + 1]! with the convention that √ x = (cid:112) | x |√− when the real number x is negative.A 6-tuple ( a , . . . , a ) is r -admissible if the triples ( a , a , a ) , ( a , a , a ) , ( a , a , a ) and ( a , a , a ) are r -admissible Definition 3.7.
The quantum j -symbol of an r -admissible 6-tuple ( a , . . . , a ) is (cid:12)(cid:12)(cid:12)(cid:12) a a a a a a (cid:12)(cid:12)(cid:12)(cid:12) = √− − (cid:80) i =1 a i ∆( a , a , a )∆( a , a , a )∆( a , a , a )∆( a , a , a ) min { Q ,Q ,Q } (cid:88) k =max { T ,T ,T ,T } ( − k [ k + 1]![ k − T ]![ k − T ]![ k − T ]![ k − T ]![ Q − k ]![ Q − k ]![ Q − k ]! , where T = a + a + a , T = a + a + a , T = a + a + a and T = a + a + a , Q = a + a + a + a , Q = a + a + a + a and Q = a + a + a + a . Closely related, a triple ( α , α , α ) ∈ [0 , π ] is admissible if(1) α i + α j − α k (cid:62) for { i, j, k } = { , , } , (2) α i + α j + α k (cid:54) π. A -tuple ( α , . . . , α ) ∈ [0 , π ] is admissible if the triples { , , } , { , , } , { , , } and { , , } are admissible. Definition 3.8. An r -admissible -tuple ( a , . . . , a ) is of the hyperideal type if for { i, j, k } = { , , } , { , , } , { , , } and { , , } , (1) (cid:54) a i + a j − a k < r − , (2) r − < a i + a j + a k (cid:54) r − , (3) a i + a j + a k is even. As a consequence of Lemma 3.5 we have 9 roposition 3.9.
The quantum j -symbol at the root of unity q = e π √− r can be computed as (cid:12)(cid:12)(cid:12)(cid:12) a a a a a a (cid:12)(cid:12)(cid:12)(cid:12) = 12 { } min { Q ,Q ,Q ,r − } (cid:88) k =max { T ,T ,T ,T } e r π √− U r (cid:0) πa r ,..., πa r , πkr (cid:1) , where U r is defined as follows. If ( a , . . . , a ) is of the hyperideal type, then U r ( α , . . . , α , ξ ) = π − (cid:16) πr (cid:17) + 12 (cid:88) i =1 3 (cid:88) j =1 ( η j − τ i ) − (cid:88) i =1 (cid:16) τ i + 2 πr − π (cid:17) + (cid:16) ξ + 2 πr − π (cid:17) − (cid:88) i =1 ( ξ − τ i ) − (cid:88) j =1 ( η j − ξ ) − ϕ r (cid:16) πr (cid:17) − (cid:88) i =1 3 (cid:88) j =1 ϕ r (cid:16) η j − τ i + πr (cid:17) + 12 (cid:88) i =1 ϕ r (cid:16) τ i − π + 3 πr (cid:17) − ϕ r (cid:16) ξ − π + 3 πr (cid:17) + (cid:88) i =1 ϕ r (cid:16) ξ − τ i + πr (cid:17) + (cid:88) j =1 ϕ r (cid:16) η j − ξ + πr (cid:17) , (3.16) where α i = πa i r for i = 1 , . . . , and ξ = πkr , τ = α + α + α , τ = α + α + α , τ = α + α + α and τ = α + α + α , η = α + α + α + α , η = α + α + α + α and η = α + α + α + α . If ( a , . . . , a ) is notof the hyperideal type, then U r will be changed according to Lemma 3.5. Definition 3.10. A -tuple ( α , . . . , α ) ∈ [0 , π ] is of the hyperideal type if(1) (cid:54) α i + α j − α k (cid:54) π, (2) π (cid:54) α i + α j + α k (cid:54) π. We notice that the six numbers | π − α | , . . . , | π − α | are the dihedral angles of an ideal or a hyperidealtetrahedron if and only if ( α , . . . , α ) is of the hyperideal type.By Lemma 3.6, U r = U − π √− r log (cid:0) r (cid:1) + O ( r ) , where U is defined by U ( α , . . . , α , ξ ) = π + 12 (cid:88) i =1 3 (cid:88) j =1 ( η j − τ i ) − (cid:88) i =1 ( τ i − π ) + ( ξ − π ) − (cid:88) i =1 ( ξ − τ i ) − (cid:88) j =1 ( η j − ξ ) − Li (1) − (cid:88) i =1 3 (cid:88) j =1 Li (cid:0) e i ( η j − τ i ) (cid:1) + 12 (cid:88) i =1 Li (cid:0) e i ( τ i − π ) (cid:1) − Li (cid:0) e i ( ξ − π ) (cid:1) + (cid:88) i =1 Li (cid:0) e i ( ξ − τ i ) (cid:1) + (cid:88) j =1 Li (cid:0) e i ( η j − ξ ) (cid:1) (3.17)on the region B H , C consisting of ( α , . . . , α , ξ ) ∈ C such that (Re( α ) , . . . , Re( α )) is of the hyper-ideal type and max { Re( τ i ) } (cid:54) Re( ξ ) (cid:54) min { Re( η j ) , π } . Let B H = B H , C ∩ R . α = ( α , . . . , α ) ∈ C such that (Re( α ) , . . . , Re( α )) is of the hyperideal type, we let ξ ( α ) be such that ∂U ( α, ξ ) ∂ξ (cid:12)(cid:12)(cid:12) ξ = ξ ( α ) = 0 . (3.18)Following the idea of [22], see also [3], it is proved that e − √− ξ ( α ) satisfies a concrete quadratic equa-tion. Therefore, for each such α, there is at most one ξ ( α ) such that ( α, ξ ( α )) ∈ B H , C . At this point, wedo not know whether ( α, ξ ( α )) ∈ B H , C for all such α, but in the next section we will show that it is thecase if all Re( α ) , . . . , Re( α ) are sufficiently close to π. For α ∈ C so that ( α, ξ ( α )) ∈ B H , C , we define W ( α ) = U ( α, ξ ( α )) . (3.19)Then as a special case of [3, Theorem 3.5], we have Theorem 3.11. W (cid:0) π ± √− l , . . . , π ± √− l (cid:1) = 2 π + 2 √− · Cov (cid:0) l , . . . , l (cid:1) for all { l , . . . , l } that form the set of edge lengths of a truncated hyperideal tetrahedron, where Cov isthe co-volume function defined in Definition 3.1.
Proposition 4.1.
Let b be a coloring of ( M, T ) . Then the relative Turaev-Viro invariant T V r ( M, E, b ) of ( M, T ) at the root of unity q = e π √− r can be computed as TV r ( M, E, b ) = ( − | E | (cid:0) r +1 (cid:1) rankH( M ; Z ) −| T | { } | E | + | T | (cid:88) a , k (cid:16) (cid:88) (cid:15) g (cid:15)r ( a , k ) (cid:17) , where (cid:15) = ( (cid:15) , . . . , (cid:15) | E | ) ∈ { , − } E runs over all multi-signs, a = ( a , . . . , a | E | ) runs over all multi-even integers in { , , . . . , r − } so that for each s ∈ { , . . . , | T |} the -tuple ( a s , a s , a s , a s , a s , a s ) is r -admissible, and k = ( k , . . . , k | T | ) runs over all multi-integers with each k s lying in between max { T s i } and min { Q s j , r − } , with g (cid:15)r ( a , k ) = e (cid:80) | E | i =1 (cid:15) i π √− ai + bi +1) r + r π √− W (cid:15)r ( π a r , π k r ) where π a r = (cid:16) πa r , . . . , πa | E | r (cid:17) , π k r = (cid:16) πk r , . . . , πk | T | r (cid:17) , and W (cid:15)r ( α, ξ ) = − | E | (cid:88) i =1 (cid:15) i ( α i − π )( β i − π ) + | T | (cid:88) s =1 U r ( α s , . . . , α s , ξ s ) with α i = πa i r and β i = πb i r for i = 1 , . . . , | E | . Proof.
First, we observe that if we let the summation in the definition of T V r ( M, E, b ) be over allmulti-even integers a instead of multi-integers, then the resulting quantity differs from T V r ( M, E, b ) bya factor rankH( M ; Z ) by [9, Lemma A.4, Theorem 2.9 and its proof]. Next, we observe that ( − a + b q ( a +1)( b +1) = − ( − r q (cid:0) a − r (cid:1)(cid:0) b − r (cid:1) + a + b +1 , and ( − a + b q − ( a +1)( b +1) = ( − r q − (cid:0) a − r (cid:1)(cid:0) b − r (cid:1) − a − b − . Then the result follows from Proposition 3.9. 11e notice that the summation in Proposition 4.1 is finite, and to use the Poisson Summation Formula,we need an infinite sum over integral points. To this end, we consider the following regions and a bumpfunction over them.Let α i = πa i r and β i = πb i r for i = 1 , . . . , | E | , ξ s = πk s r for s = 1 , . . . , | T | , τ s i = πT si r for i = 1 , . . . , , and η s j = πQ sj r for j = 1 , , . Let D A = (cid:110) ( α, ξ ) ∈ R | E | + | T | (cid:12)(cid:12)(cid:12) ( α s , . . . , α s ) is admissible, max { τ s i } (cid:54) ξ s (cid:54) min { η s j , π } , s = 1 , . . . , | T | (cid:111) , and let D H = (cid:110) ( α, ξ ) ∈ D A (cid:12)(cid:12)(cid:12) ( α s , . . . , α s ) is of the hyperideal type , s = 1 , . . . , | T | (cid:111) . For a sufficiently small δ > , let D δ H = (cid:110) ( α, ξ ) ∈ D H (cid:12)(cid:12)(cid:12) d (( α, ξ ) , ∂ D H ) < δ (cid:111) , where d is the Euclidean distance on R n . We let ψ : R | E | + | T | → R be the C ∞ -smooth bump functionsupported on (D H , D δ H ) , ie, ψ ( α, ξ ) = 1 , ( α, ξ ) ∈ D δ H < ψ ( α, ξ ) < , ( α, ξ ) ∈ D H (cid:114) D δ H ψ ( α, ξ ) = 0 , ( α, ξ ) / ∈ D H , and let f (cid:15)r ( a , k ) = ψ (cid:16) π a r , π k r (cid:17) g (cid:15)r ( a , k ) . In Proposition 4.1, the coloring a runs over multi-even integers. On the other hand, to use the PoissonSummation Formula, we need a sum over all integers. For this purpose, we for each i let a i = 2 a (cid:48) i andlet a (cid:48) = ( a (cid:48) , . . . , a (cid:48)| E | ) . Then by Proposition 4.1, TV r ( M, E, b ) = ( − | E | (cid:0) r +1 (cid:1) rankH( M ; Z ) −| T | { } | E | + | T | (cid:88) ( a (cid:48) , k ) ∈ Z | E | + | T | (cid:16) (cid:88) (cid:15) ∈{ , − } E f (cid:15)r (cid:0) a (cid:48) , k (cid:1)(cid:17) + error term . Let f r = (cid:88) (cid:15) ∈{ , − } E f (cid:15)r . Then TV r ( M, E, b ) = ( − | E | (cid:0) r +1 (cid:1) rankH( M ; Z ) −| T | { } | E | + | T | (cid:88) ( a (cid:48) , k ) ∈ Z | E | + | T | f r (cid:0) a (cid:48) , k (cid:1) + error term . Since f r is C ∞ -smooth and equals zero out of D H , it is in the Schwartz space on R | E | + | T | . Then bythe Poisson Summation Formula (see e.g. [28, Theorem 3.1]), (cid:88) ( a (cid:48) , k ) ∈ Z | E | + | T | f r (cid:0) a (cid:48) , k (cid:1) = (cid:88) ( m , n ) ∈ Z | E | + | T | (cid:98) f r ( m , n ) , where m = ( m , . . . , m | E | ) ∈ Z E , n = ( n , . . . , n | T | ) ∈ Z T , and (cid:98) f r ( m , n ) is the ( m , n ) -th Fouriercoefficient of f r defined by (cid:98) f r ( m , n ) = (cid:90) R | E | + | T | f r (cid:0) a (cid:48) , k (cid:1) e (cid:80) | E | i =1 π √− m i a (cid:48) i + (cid:80) | T | s =1 π √− n s k s d a (cid:48) d k , d a (cid:48) = (cid:81) | E | i =1 da (cid:48) i and d k = (cid:81) | T | s =1 dk s . By the change of variable, and by changing a (cid:48) i back to a i , the Fourier coefficients can be computedas Proposition 4.2. (cid:98) f r ( m , n ) = (cid:88) (cid:15) ∈{ , − } E (cid:98) f (cid:15)r ( m , n ) with (cid:98) f (cid:15)r ( m , n ) = r | E | + | T | | E | + | T | · π | E | + | T | (cid:90) D H ψ ( α, ξ ) e (cid:80) | E | i =1 (cid:15) i √− α i + β i + πr ) · e r π √− (cid:0) W (cid:15)r ( α,ξ ) − (cid:80) | E | i =1 πm i α i − (cid:80) | E | s =1 πn s ξ s (cid:1) dαdξ, where dα = (cid:81) | E | i =1 dα i , dξ = (cid:81) | T | s =1 dξ s and W (cid:15)r ( α, ξ ) = − | E | (cid:88) i =1 (cid:15) i ( α i − π )( β i − π ) + | T | (cid:88) s =1 U r ( , α s , . . . , α s , ξ s ) . In particular, (cid:98) f (cid:15)r ( , ) = r | E | + | T | | E | + | T | · π | E | + | T | (cid:90) D H ψ ( α, ξ ) e (cid:80) | E | i =1 (cid:15) i √− α i + β i + πr )+ r π √− W (cid:15)r ( α,ξ ) dαdξ. Proposition 4.3. TV r ( M, E, b ) = ( − | E | (cid:0) r +1 (cid:1) rankH( M ; Z ) −| T | { } | E | + | T | (cid:88) ( m , n ) ∈ Z | E | + | T | (cid:98) f r ( m , n ) + error term . We will estimate the leading Fourier coefficients, the non-leading Fourier coefficients and the errorterm respectively in Sections 5.3, 5.4 and 5.5, and prove Theorem 1.4 in Section 5.6.
Proposition 5.1.
Let D z be a region in C n and let D a be a region in R k . Let f ( z , a ) and g ( z , a ) becomplex valued functions on D z × D a which are holomorphic in z and smooth in a . For each positiveinteger r, let f r ( z , a ) be a complex valued function on D z × D a holomorphic in z and smooth in a . For a fixed a ∈ D a , let f a , g a and f a r be the holomorphic functions on D z defined by f a ( z ) = f ( z , a ) ,g a ( z ) = g ( z , a ) and f a r ( z ) = f r ( z , a ) . Suppose { a r } is a convergent sequence in D a with lim r a r = a ,f a r r is of the form f a r r ( z ) = f a r ( z ) + υ r ( z , a r ) r , { S r } is a sequence of embedded real n -dimensional closed disks in D z sharing the same boundary, and c r is a point on S r such that { c r } is convergent in D z with lim r c r = c . If for each r (1) c r is a critical point of f a r in D z , (2) Re f a r ( c r ) > Re f a r ( z ) for all z ∈ S (cid:114) { c r } ,
3) the Hessian matrix
Hess( f a r ) of f a r at c r is non-singular,(4) | g a r ( c r ) | is bounded from below by a positive constant independent of r, (5) | υ r ( z , a r ) | is bounded from above by a constant independent of r on D z , and(6) the Hessian matrix Hess( f a ) of f a at c is non-singular,then (cid:90) S r g a r ( z ) e rf a rr ( z ) d z = (cid:16) πr (cid:17) n g a r ( c r ) (cid:112) − det Hess( f a r )( c r ) e rf a r ( c r ) (cid:16) O (cid:16) r (cid:17)(cid:17) . A proof can be found in [32, Appendix].For a fixed { β , . . . , β | E | } , let θ i = 2 | β i − π | each i ∈ { , . . . , | E |} . The function W (cid:15)r is approximatedby the following function W (cid:15) ( α, ξ ) = − | E | (cid:88) i =1 (cid:15) i ( α i − π )( β i − π ) + | T | (cid:88) s =1 U ( α s , . . . , α s , ξ s ) . The approximation will be specified in the proof of Proposition 5.5. Notice that W (cid:15) is continuous on D H , C = (cid:8) ( α, ξ ) ∈ C | E | + | T | (cid:12)(cid:12) (Re( α ) , Re( ξ )) ∈ D H (cid:9) and for any δ > is analytic on D δ H , C = (cid:8) ( α, ξ ) ∈ C | E | + | T | (cid:12)(cid:12) (Re( α ) , Re( ξ )) ∈ D δ H (cid:9) , where Re( α ) = (Re( α ) , . . . , Re( α | E | )) and Re( ξ ) = (Re( ξ ) , . . . , Re( ξ | T | )) . In the rest of this paper, we assume that θ , . . . , θ | E | are sufficiently close to , or equivalently, β , . . . , β | E | are sufficiently close to π. In the special case β i = · · · = β | E | = π, a direct com-putation shows that ξ ( π, . . . , π ) = π . For δ > , we denote by D δ, C the L δ -neighborhood of (cid:0) π, . . . , π, π , . . . , π (cid:1) in C | E | + | T | , that is D δ, C = (cid:110) ( α, ξ ) ∈ C | E | + | T | (cid:12)(cid:12)(cid:12) d L (cid:16) ( α, ξ ) , (cid:16) π, . . . , π, π , . . . , π (cid:17)(cid:17) < δ (cid:111) , where d L is the real L norm on C n defined by d L ( x , y ) = max i ∈{ ,...,n } {| Re( x i ) − Re( y i ) | , | Im( x i ) − Im( y i ) |} , where x = ( x , . . . , x n ) and y = ( y , . . . , y n ) . We will also consider the region D δ = D δ, C ∩ R | E | + | T | . W (cid:15) Suppose { β , . . . , β | E | } are sufficiently close to π. Let θ i = 2 | β i − π | for i ∈ { , . . . , | E |} , and let µ i = 1 if β i (cid:62) π and let µ i = − if β i (cid:54) π so that µ i θ i = 2( β i − π ) . roposition 5.2. For each i ∈ { , . . . , | E |} , let l i be the length of the edge e i in M E θ and let α ∗ i = π + (cid:15) i µ i √− l i . (5.1) For each s ∈ { , . . . , | T |} , let ξ ∗ s = ξ ( α ∗ s , . . . , α ∗ s ) . Then W (cid:15) has a critical point z (cid:15) = (cid:0) α ∗ , . . . , α ∗| E | , ξ ∗ , . . . , ξ ∗| T | (cid:1) in D δ, C with critical value | T | π + 2 √− M E θ ) . Proof.
For each s ∈ { , . . . , | T |} , let α s = ( α s , . . . , α s ) and let α ∗ s = ( α ∗ s , . . . , α ∗ s ) . By (3.1), if θ i ’s are sufficiently small, then l i ’s are sufficiently close to and α ∗ i ’s are sufficientlyclose to π. Then by the continuity of ξ ( α s ) for each s, z (cid:15) ∈ D δ, C . We first have ∂ W (cid:15) ∂ξ s (cid:12)(cid:12)(cid:12) z (cid:15) = ∂U ( α ∗ s , ξ s ) ∂ξ s (cid:12)(cid:12)(cid:12) ξ ∗ s = 0 . (5.2)Now let W ( α s ) = U ( α s , ξ ( α s )) be the function defined in (3.19). Then for i ∈ { s , . . . , s } ,∂W ( α s ) ∂α i (cid:12)(cid:12)(cid:12) α ∗ s = ∂U ( α s , ξ s ) ∂α i (cid:12)(cid:12)(cid:12) ( α ∗ s ,ξ ∗ s ) + ∂U ( α s , ξ s ) ∂ξ s (cid:12)(cid:12)(cid:12) ( α ∗ s ,ξ ∗ s ) · ∂ξ ( α s ) ∂α i (cid:12)(cid:12)(cid:12) α ∗ s = ∂U ( α s , ξ s ) ∂α i (cid:12)(cid:12)(cid:12) ( α ∗ s ,ξ ∗ s ) . For each s ∈ { , . . . , | T |} , let ( l s , . . . , l s ) be the edge lengths of ∆ s . Then by Theorem 3.11 andLemma 3.2, we have ∂U ( α s , ξ s ) ∂α i (cid:12)(cid:12)(cid:12) ( α ∗ s ,ξ ∗ s ) = ∂W ( α s ) ∂α i (cid:12)(cid:12)(cid:12) α ∗ s = − (cid:15) i µ i √− · ∂W∂l i (cid:12)(cid:12)(cid:12) ( l s ,...,l s ) = (cid:15) i µ i θ s,i , where θ s,i is the dihedral angle of ∆ s at the edge e i . Then for each i ∈ { , . . . , | E |} ,∂ W (cid:15) ∂α i (cid:12)(cid:12)(cid:12) z (cid:15) = − (cid:15) i ( β i − π ) + | T | (cid:88) s =1 ∂U ( α s , ξ s ) ∂α i (cid:12)(cid:12)(cid:12) ( α ∗ s ,ξ ∗ s ) = − (cid:15) i ( β i − π ) + (cid:15) i µ i | E | (cid:88) s =1 θ s,i = (cid:15) i (cid:0) − β i − π ) + µ i θ i (cid:1) = 0 . (5.3)By (5.2) and (5.3), z (cid:15) is a critical point of W (cid:15) . Finally, we compute the critical value. For each s ∈ { , . . . , | T |} , let ( l s , . . . , l s ) and ( θ s , . . . , θ s ) respectively be the edge lengths and the dihedral angles of ∆ s . Then by Theorem 3.11, we have W (cid:15) ( z (cid:15) ) = − | E | (cid:88) i =1 (cid:15) i ( √− (cid:15) i µ i l i )( β i − π ) + | T | (cid:88) s =1 (cid:16) π + 2 √− (cid:16) Vol(∆ s ) + 12 (cid:88) k =1 θ s k l s k (cid:17)(cid:17) =2 | T | π + 2 | T | (cid:88) s =1 √− s ) + | E | (cid:88) i =1 √− (cid:16) − µ i ( β i − π ) + | T | (cid:88) s =1 θ s,i (cid:17) l i =2 | T | π + 2 √− M E θ ) . .2 Convexity of W (cid:15) Proposition 5.3.
For a sufficiently small δ > , the function W (cid:15) ( α, ξ ) is strictly concave down in { Re( α i ) } | E | i =1 and { Re( ξ s ) } | T | s =1 , and is strictly concave up in { Im( α i ) } | E | i =1 and { Im( ξ s ) } | T | s =1 on D δ , C . Proof.
We first consider the special case { α i } | E | i =1 and { ξ s } | T | s =1 are real. In this case, Im W (cid:15) ( α, ξ ) = | T | (cid:88) s =1 V ( α s , . . . , α s , ξ s ) for V : B H → R defined by V ( α , . . . , α , ξ ) = δ ( α , α , α ) + δ ( α , α , α ) + δ ( α , α , α ) + δ ( α , α , α ) − Λ( ξ ) + (cid:88) i =1 Λ( ξ − τ i ) + (cid:88) j =1 Λ( η j − ξ ) , (5.4)where δ is defined by δ ( α, β, γ ) = −
12 Λ (cid:16) α + β − γ (cid:17) −
12 Λ (cid:16) β + γ − α (cid:17) −
12 Λ (cid:16) γ + α − β (cid:17) + 12 Λ (cid:16) α + β + β (cid:17) . At (cid:0) π, . . . , π, π (cid:1) , we have ∂ Im V∂α si = − for i ∈ { , . . . , } , ∂ V∂α si α sj = − for i (cid:54) = j in { , . . . , } , ∂ Im V∂α si ξ s = 2 for i ∈ { , . . . , } and ∂ Im V∂ξ s = − . Then a direct computation shows that, at (cid:0) π, . . . , π, π (cid:1) , the Hessian matrix of Im V in { Re( α s i ) } i ∈{ ,..., } and Re( ξ s ) is negative definite. As a consequence, theHessian matrix of Im W (cid:15) in { Re( α i ) } i ∈ I and { Re( ξ s ) } cs =1 is negative definite at (cid:0) π, . . . , π, π , . . . , π (cid:1) . Then by the continuity, there exists a sufficiently small δ > such that ( α, ξ ) ∈ D δ , C , the Hessianmatrix of Im W (cid:15) with respect to { Re( α i ) } | E | i =1 and { Re( ξ s ) } | T | s =1 is still negative definite, implying that Im W (cid:15) is strictly concave down in { Re( α i ) } | E | i =1 and { Re( ξ s ) } | T | s =1 on D δ , C . Since W (cid:15) is holomorphic, Im W (cid:15) is strictly concave up in { Im( α i ) } | E | i =1 and { Im( ξ s ) } | T | s =1 on D δ , C . Proposition 5.4.
The Hessian matrix
Hess W (cid:15) of W (cid:15) with respect to { α i } | E | i =1 and { ξ s } | T | s =1 is non-singular on D δ , C . Proof.
By Proposition 5.3, the real part of the
Hess W (cid:15) is negative definite. Then by [18, Lemma], it isnonsingular. Proposition 5.5.
Suppose { β , . . . , β | E | } are in { π − (cid:15), π + (cid:15) } for a sufficiently small (cid:15) > . For (cid:15) ∈ { , − } E , let z (cid:15) be the critical point of W (cid:15) described in Proposition 5.2. Then (cid:98) f (cid:15)r (0 , . . . ,
0) = C (cid:15) ( z (cid:15) ) (cid:112) − det Hess W (cid:15) ( z (cid:15) ) e r π Vol( M Eθ ) (cid:16) O (cid:16) r (cid:17)(cid:17) , where each C (cid:15) ( z (cid:15) ) depends continuously on { β , . . . , β | E | } and when β = · · · = β | E | = π,C (cid:15) ( z (cid:15) ) = ( − | T | r | E |−| T | | E | + | T | π | E | + | T | . Lemma 5.6.
For each (cid:15) ∈ { , − } E , max D H Im W (cid:15) (cid:54) Im W (cid:15) (cid:16) π, . . . , π, π , . . . , π (cid:17) = 2 | T | v where v is the volume of the regular ideal octahedron, and the equality holds if and only if α = · · · = α | E | = π and ξ = · · · = ξ | T | = π . Proof. On D H , we have Im W (cid:15) ( α, ξ ) = | T | (cid:88) s =1 V ( α s , . . . , α s , ξ s ) for V defined in (5.4). Then the result is a consequence of the result of Costantino [7] and the Murakami-Yano formula [22] (see Ushijima [ ? ] for the case of hyperideal tetrahedra). Indeed, by [7], for a fixed α = ( α , . . . , α ) of the hyperideal type, the function f ( ξ ) defined by f ( ξ ) = V ( α, ξ ) is strictly concavedown and the unique maximum point ξ ( α ) exists and lies in (max { τ i } , min { η j , π } ) , ie, ( α, ξ ( α )) ∈ B H . Then by [ ? ], V ( α, ξ ( α )) = Vol(∆ | π − α | ) , the volume of the hyperideal tetrahedron ∆ | π − α | withdihedral angles | π − α | , . . . , | π − α | . Since ξ ( π, . . . , π ) = π and the regular ideal octahedron ∆ (0 ,..., has the maximum volume among all the hyperideal tetrahedra, V (cid:0) π, . . . , π, π (cid:1) = Vol(∆ (0 ,..., ) (cid:62) Vol(∆ | π − α | ) = V ( α, ξ ( α )) (cid:62) V ( α, ξ ) for any ( α, ξ ) ∈ B H . For the equality part, suppose ( α , . . . , α | E | , ξ , . . . , ξ | T | ) (cid:54) = (cid:0) π, . . . , π, π , π (cid:1) . If ( α , . . . , α ) (cid:54) =( π, . . . , π ) , then Im W (cid:15) ( α, ξ ) (cid:54) (cid:80) | T | s =1 Vol(∆ s ) < | T | v , where ∆ s is the truncated hyperideal tetrahe-dron with dihedral angles | π − α s | , . . . , | π − α s | . If ( α s , . . . , α s ) = ( π, . . . , π ) for all s ∈ { , . . . , | T |} but, say, ξ (cid:54) = π , then the strict concavity of f ( ξ ) implies that Im W (cid:15) ( π, . . . , π, ξ , . . . , ξ | T | ) < Im W (cid:15) (cid:0) π, . . . , π, π , . . . , π (cid:1) . Proof of Proposition 5.5.
Let δ > be as in Proposition 5.3. By Lemma 5.3, Proposition 5.6 and thecompactness of D H (cid:114) D δ , | T | v > max D H (cid:114) D δ Im W (cid:15) . By Proposition 5.2 and continuity, if { β , . . . , β | E | } are sufficiently close to π, then the critical point z (cid:15) of W (cid:15) as in Proposition 5.2 lies in D δ , C , and Im W (cid:15) ( z (cid:15) ) = Vol( M E θ ) is sufficiently close to | T | v sothat Im W (cid:15) ( z (cid:15) ) > max D H (cid:114) D δ Im W (cid:15) . Therefore, we only need to estimate the integral on D δ . To do this, we consider as drawn in Figure2 the surface S (cid:15) = S (cid:15) top ∪ S (cid:15) side in D δ , C , where S (cid:15) top = { ( α, ξ ) ∈ D δ , C | ((Im( α )) , Im( ξ )) = Im( z (cid:15) ) } and S (cid:15) side = { ( α, ξ ) + t √− · Im( z (cid:15) ) | ( α, ξ ) ∈ ∂ D δ , t ∈ [0 , } . By analyticity, the integral remains the same if we deform the domain from D δ to S (cid:15) . By Proposition 5.3, Im W (cid:15) is concave down on S (cid:15) top . Since z (cid:15) is the critical points of Im W (cid:15) , it is theonly absolute maximum on S (cid:15) top . op S side S Re( α ) Re( ξ ) D H D δ ε z ε ε Figure 2: The deformed surface S (cid:15) On the side S (cid:15) side , for each ( α, ξ ) ∈ ∂ D δ , consider the function g (cid:15) ( α,ξ ) ( t ) = Im W (cid:15) (( α, ξ ) + t √− · Im( z (cid:15) )) on [0 , . By Lemma 5.3, g (cid:15) ( α,ξ ) ( t ) is concave up for any ( α, ξ ) ∈ ∂ D δ . As a consequence, g (cid:15) ( α,ξ ) ( t ) (cid:54) max { g (cid:15) ( α,ξ ) (0) , g (cid:15) ( α,ξ ) (1) } . Now by the previous two steps, since ( α, ξ ) ∈ ∂ D δ ,g (cid:15) ( α,ξ ) (0) = Im W (cid:15) ( α, ξ ) < Im W (cid:15) ( z (cid:15) ); and since ( α, ξ ) + √− · Im( z (cid:15) ) ∈ S (cid:15) top ,g (cid:15) ( α,ξ ) (0) = Im W (cid:15) (( α, ξ ) + √− · Im( z (cid:15) )) < Im W (cid:15) ( z (cid:15) ) . As a consequence, Im W (cid:15) ( z (cid:15) ) > max S (cid:15) side Im W (cid:15) . Therefore, we proved that z (cid:15) is the unique maximum point of Im W (cid:15) on S (cid:15) ∪ (cid:0) D H (cid:114) D δ (cid:1) , and W (cid:15) has critical value | T | π + 2 √− · Vol( M E θ ) at z (cid:15) . By Proposition 5.4, det Hess W (cid:15) ( z (cid:15) ) (cid:54) = 0 . Finally, we estimate the difference between W (cid:15)r and W (cid:15) . By Lemma 3.6, (3), we have ϕ r (cid:16) πr (cid:17) = Li (1) + 2 π √− r log (cid:16) r (cid:17) − π r + O (cid:16) r (cid:17) ; and for z with < Rez < π have ϕ r (cid:16) z + kπr (cid:17) = ϕ r ( z ) + ϕ (cid:48) r ( z ) · kπr + O (cid:16) r (cid:17) . Then by Lemma 3.6, in (cid:8) ( α, ξ ) ∈ D δ H , C (cid:12)(cid:12) | Im( α i ) | < L for i = { , . . . , | E |} , | Im( ξ s ) | < L for s = { , . . . , | T |}} for some L > , W (cid:15)r ( α, ξ ) = W (cid:15) ( α, ξ ) − | T | π √− r log (cid:16) r (cid:17) + 4 π √− · κ ( α, ξ ) r + ν r ( α, ξ ) r , κ ( α, ξ )= | T | (cid:88) s =1 (cid:16) (cid:88) i =1 √− τ s i − √− ξ s − √− π − √− π
2+ 14 (cid:88) i =1 3 (cid:88) j =1 log (cid:0) − e √− η sj − τ si ) (cid:1) − (cid:88) i =1 log (cid:0) − e √− τ si − π ) (cid:1) + 32 log (cid:0) − e √− ξ s − π ) (cid:1) − (cid:88) i =1 log (cid:0) − e √− ξ s − τ si ) (cid:1) − (cid:88) j =1 log (cid:0) − e √− η sj − ξ s ) (cid:1)(cid:17) and | ν r ( α, ξ ) | bounded from above by a constant independent of r. Then e (cid:80) | E | i =1 (cid:15) i √− (cid:0) α i + β i + πr (cid:1) + r π √− W (cid:15)r ( α,ξ ) = (cid:16) r (cid:17) −| T | e (cid:80) | E | i =1 (cid:15) i √− α i + β i )+ κ ( α,ξ ) · e r π √− (cid:16) W (cid:15) ( α,ξ )+ νr ( α,ξ ) − (cid:80) | E | i =1 (cid:15)i π r (cid:17) . Now let D z = (cid:8) ( α, ξ ) ∈ D δ H , C (cid:12)(cid:12) | Im( α i ) | < L for i = { , . . . , | E |} , | Im( ξ s ) | < L for s = { , . . . , | T |}} for some L > . Let a r = (( β , . . . , β | E | ) (recall that β i = πb ( r ) i r depends on r ), f a r ( α, ξ ) = W (cid:15) ( α, ξ ) , g a r ( α, ξ ) = ψ ( α, ξ ) e (cid:80) | E | i =1 (cid:15) i √− α i + β i )+ κ ( α,ξ ) , f a r r ( α, ξ ) = W (cid:15)r ( α, ξ )+ | T | π √− r log (cid:0) r (cid:1) ,υ r ( α, ξ ) = ν r ( α, ξ ) − (cid:80) | E | i =1 (cid:15) i π , S r = S (cid:15) ∪ (cid:0) D H (cid:114) D δ (cid:1) and z (cid:15) is the critical point of f in D z . Thenall the conditions of Proposition 5.1 are satisfied and the result follows.When β = · · · = β | E | = π, a direct computation shows that C (cid:15) ( z (cid:15) ) = r | E | + | T | | E | + | T | π | E | + | T | (cid:16) πr (cid:17) | E | + | T | (cid:16) r (cid:17) −| T | g (cid:16) π, . . . , π, π , . . . , π (cid:17) = ( − | T | r | E |−| T | | E | + | T | π | E | + | T | . Corollary 5.7. If (cid:15) > is sufficiently small and all { β , . . . , β | E | } are in { π − (cid:15), π + (cid:15) } , then (cid:88) (cid:15) ∈{ , − } E C (cid:15) ( z (cid:15) ) (cid:112) − det Hess W (cid:15) ( z (cid:15) ) (cid:54) = 0 . Proof. If β i = · · · = β | E | = π, then all z (cid:15) = (cid:0) π, . . . , π, π , . . . , π (cid:1) and all W (cid:15) are the same functions.As a consequence, all the C (cid:15) ( z (cid:15) ) ’s and all Hessian determinants det Hess W (cid:15) ( z (cid:15) ) ’s are the same at thispoint, imply that the sum is not equal to zero. Then by continuity, if (cid:15) is small enough, then the sumremains none zero. Remark . We suspect that all C (cid:15) ( z (cid:15) ) ’s and all det Hess W (cid:15) ( z (cid:15) ) ’s are always the same for any given { β , . . . , β | E | } . Proposition 5.9.
Suppose { β , . . . , β | E | } are in { π − (cid:15), π + (cid:15) } for a sufficiently small (cid:15) > . If ( m , n ) (cid:54) =(0 , . . . , , then (cid:12)(cid:12)(cid:12) (cid:98) f (cid:15)r ( m , n ) (cid:12)(cid:12)(cid:12) < O (cid:16) e r π (cid:0) Vol( M Eθ ) − (cid:15) (cid:48) (cid:1)(cid:17) for some (cid:15) (cid:48) > . roof. Recall that if β = · · · = β | E | = π, then the total derivative D W (cid:15) (cid:16) π, . . . , π, π , . . . , π (cid:17) = (0 , . . . , . Hence there exists a δ > and an (cid:15) > such that if { β , . . . , β | E | } are in { π − (cid:15), π + (cid:15) } , then forall ( α, ξ ) ∈ D δ , C and for any unit vector u = ( u , . . . , u | E | , w , . . . , w | T | ) ∈ R | E | + | T | , the directionalderivatives | D u Im W (cid:15) ( α, ξ ) | = (cid:12)(cid:12)(cid:12)(cid:12) | E | (cid:88) i =1 u i ∂ Im W (cid:15) ∂ Im( α i ) + | T | (cid:88) s =1 w s ∂ Im W (cid:15) ∂ Im( ξ s ) (cid:12)(cid:12)(cid:12)(cid:12) < π − (cid:15) (cid:48)(cid:48) (cid:112) | E | + 2 | T | for some (cid:15) (cid:48)(cid:48) > . On D H , we have Im (cid:16) W (cid:15) ( α, ξ ) − | E | (cid:88) i =1 πm i α i − | T | (cid:88) s =1 πn s ξ s (cid:17) = Im W (cid:15) ( α, ξ ) . Then by Lemma 5.3, Proposition 5.6 and the compactness of D H (cid:114) D δ , | T | v > max D H (cid:114) D δ Im (cid:16) W (cid:15) ( α, ξ ) − | E | (cid:88) i =1 πm i α i − | T | (cid:88) s =1 πn s ξ s (cid:17) + (cid:15) (cid:48)(cid:48)(cid:48) for some (cid:15) (cid:48)(cid:48)(cid:48) > . By Proposition 5.2 and continuity, if { β , . . . , β | E | } are sufficiently close to π, then thecritical point z (cid:15) of W (cid:15) as in Proposition 5.2 lies in D δ , C , and Im W (cid:15) ( z (cid:15) ) = 2Vol( M E θ ) is sufficientlyclose to | T | v so that Im W (cid:15) ( z (cid:15) ) > max D H (cid:114) D δ Im (cid:16) W (cid:15) ( α, ξ ) − | E | (cid:88) i =1 πm i α i − | T | (cid:88) s =1 πn s ξ s (cid:17) + (cid:15) (cid:48)(cid:48)(cid:48) . (5.5)Therefore, we only need to estimate the integral on D δ . If ( m , n ) (cid:54) = (0 , . . . , , then there is at least one element of { m , . . . , m | E | } or of { n , . . . , n | T | } that is nonzero. Without loss of generality, assume that m (cid:54) = 0 . If m > , then consider the surface S + = S + top ∪ S + side in D δ , C where S + top = { ( α, ξ ) ∈ D δ , C | (Im( α ) , Im( ξ )) = ( δ , , . . . , } and S + side = { ( α, ξ ) + ( t √− δ , , . . . , | ( α, ξ ) ∈ ∂ D δ , t ∈ [0 , } . On the top, for any ( α, ξ ) ∈ S + top , by the Mean Value Theorem, (cid:12)(cid:12) Im W (cid:15) ( z (cid:15) ) − Im W (cid:15) ( α, ξ ) (cid:12)(cid:12) = (cid:12)(cid:12) D u Im W (cid:15) ( z ) (cid:12)(cid:12) · (cid:13)(cid:13) z (cid:15) − ( α, ξ ) (cid:13)(cid:13) < π − (cid:15) (cid:48)(cid:48) (cid:112) | E | + 2 | T | · (cid:112) | E | + 2 | T | δ =2 πδ − (cid:15) (cid:48)(cid:48) δ , z is some point on the line segment connecting z (cid:15) and ( α, ξ ) , u = z (cid:15) − ( α,ξ ) (cid:107) z (cid:15) − ( α,ξ ) (cid:107) and (cid:112) | E | + 2 | T | δ is the diameter of D δ , C . Then Im (cid:16) W (cid:15) ( α, ξ ) − | E | (cid:88) i =1 πm i α i − | T | (cid:88) s =1 πn s ξ s (cid:17) =Im W (cid:15) ( α, ξ ) − πm δ < Im W (cid:15) ( z (cid:15) ) + 2 πδ − (cid:15) (cid:48)(cid:48) δ − πδ =Im W (cid:15) ( z (cid:15) ) − (cid:15) (cid:48)(cid:48) δ . On the side, for any point ( α, ξ ) + ( t √− δ , , . . . , ∈ S + side , by the Mean Value Theorem again,we have (cid:12)(cid:12) Im W (cid:15) (cid:0) ( α, ξ ) + ( t √− δ , , . . . , (cid:1) − Im W (cid:15) ( α, ξ ) (cid:12)(cid:12) < π − (cid:15) (cid:48)(cid:48) (cid:112) | E | + 2 | T | tδ . Then Im W (cid:15) (cid:0) ( α, ξ ) + ( t √− δ , , . . . , (cid:1) − πm tδ < Im W (cid:15) ( α, ξ ) + 2 π − (cid:15) (cid:48)(cid:48) (cid:112) | E | + 2 | T | tδ − πtδ < Im W (cid:15) ( α, ξ ) < Im W (cid:15) ( z (cid:15) ) − (cid:15) (cid:48)(cid:48)(cid:48) , where the last inequality comes from that ( α, ξ ) ∈ ∂ D δ ⊂ D H (cid:114) D δ and (5.5).Now let (cid:15) (cid:48) = min { (cid:15) (cid:48)(cid:48) δ , (cid:15) (cid:48)(cid:48)(cid:48) } , then on S + ∪ (cid:0) D H (cid:114) D δ (cid:1) , Im (cid:16) W (cid:15) ( α, ξ ) − | E | (cid:88) i =1 πm i α i − | T | (cid:88) s =1 πn s ξ s (cid:17) < Im W (cid:15) ( z (cid:15) ) − (cid:15) (cid:48) , and the result follows.If m < , then we consider the surface S − = S − top ∪ S − side in D δ , C where S − top = { ( α, ξ ) ∈ D δ , C | (Im( α ) , Im( ξ )) = ( − δ , , . . . , } and S − side = { ( α, ξ ) − ( t √− δ , , . . . , | ( α, ξ ) ∈ ∂ D δ , t ∈ [0 , } . Then the same estimate as in the previous case proves that on S − ∪ (cid:0) D H (cid:114) D δ (cid:1) , Im (cid:16) W (cid:15) ( α, ξ ) − | E | (cid:88) i =1 πm i α i − | T | (cid:88) s =1 πn s ξ s (cid:17) < Im W (cid:15) ( z (cid:15) ) − (cid:15) (cid:48) , from which the result follows. The goal of this section is to estimate the error term in Proposition 4.3.
Proposition 5.10.
The error term in Proposition 4.3 is less than O (cid:0) e r π (Vol( M Eθ ) − (cid:15) (cid:48) ) (cid:1) for some (cid:15) (cid:48) > . For the proof we need the following estimate, which first appeared in [13, Proposition 8.2] for q = e π √− r , and for the root q = e π √− r in [8, Proposition 4.1].21 emma 5.11. For any integer < n < r and at q = e π √− r , log |{ n } ! | = − r π Λ (cid:18) nπr (cid:19) + O (log( r )) . Proof of Proposition 5.10 .
Let M = max (cid:110) | T | (cid:88) s =1 V ( α s , . . . , α s , ξ s ) (cid:12)(cid:12)(cid:12) ( α, ξ ) ∈ ∂ D H ∪ (cid:0) D A (cid:114) D H (cid:1)(cid:111) Then by [2, Section 4],
M < | T | v = 2Vol( M E (0 ,..., ); and by continuity, if θ is sufficiently closed to (0 , . . . , , then M < M E θ ) . Now by Lemma 5.11 and the continuity, for (cid:15) (cid:48) = M Eθ ) − M , we can choose a sufficiently small δ > so that if (cid:0) π a r , π k r (cid:1) / ∈ D δ H , then (cid:12)(cid:12)(cid:12) g (cid:15)r ( a , k ) (cid:12)(cid:12)(cid:12) < O (cid:16) e r π ( M + (cid:15) (cid:48) ) (cid:17) = O (cid:16) e r π (Vol( M Eθ ) − (cid:15) (cid:48) ) (cid:17) . Let ψ be the bump function supported on (D H , D δ H ) . Then the error term in Proposition 4.3 is less than O (cid:0) e r π (Vol( M Eθ ) − (cid:15) (cid:48) ) (cid:1) . Proof of Theorem 1.4.
Let (cid:15) > be sufficiently small so that the conditions of Propositions 5.5, 5.9 and5.10 and of Corollary 5.7 are satisfied, and suppose { β , . . . , β | E | } are all in ( π − (cid:15), π + (cid:15) ) . By Propositions 4.2, 4.3, 5.5, 5.9 and 5.10, TV r ( M, E, b )= ( − | E | (cid:0) r +1 (cid:1) rankH( M ; Z ) −| T | { } | E | + | T | (cid:16) (cid:88) (cid:15) ∈{ , − } | E | (cid:98) f (cid:15)r (0 , . . . , (cid:17)(cid:16) O (cid:0) e r π ( − (cid:15) (cid:48) ) (cid:1)(cid:17) = ( − | E | (cid:0) r +1 (cid:1) rankH( M ; Z ) −| T | { } | E | + | T | (cid:18) (cid:88) (cid:15) ∈{ , − } | E | C (cid:15) ( z (cid:15) ) (cid:112) − det Hess W (cid:15) ( z (cid:15) ) (cid:19) e r π Vol( M Eθ ) (cid:16) O (cid:16) r (cid:17)(cid:17) ; and by Corollary 5.7, (cid:88) (cid:15) ∈{ , − } | E | C (cid:15) ( z (cid:15) ) (cid:112) − det Hess W (cid:15) ( z (cid:15) ) (cid:54) = 0 , which completes the proof. References [1] X. Bao and F. Bonahon,
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