The renormalized volume of quasifuchsian 3-manifolds with applications I

Transcription

1 of quasifuchsian 3-manifolds with applications I University of Luxembourg Cortona, June 5, /15

2 The minicourse The WP metric Goal : present the main properties of the renormalized volume of quasifuchsian manifolds (resp. convex co-compact hyperbolic 3-manifolds) and show some applications to the Weil-Petersson geometry of Teichmüller space. 1 Background material, denition of the renormalized volume, comparison with volume of convex core. (JMS) 2 Variational formula for the renormalized volume, relation to the WP distance. (Ken Bromberg.) 3 Some applications to WP geometry. (Je Brock.) 2/15

3 The Weil-Petersson metric on Teichmüller space S is a closed, oriented surface of genus g 2. Denition T S space of complex structures on S (up to isotopy). T S is a complex manifold of (complex) dimension 3g 3, homeomorphic to a ball. T S is also the space of hyperbolic structures on S, up to isotopy. (Poincaré/Riemann uniformization theorem.) T X T = Q X, holomorphic quadratic dierentials q = fdz 2 on X. The hyperbolic metric on X is h = ρdzd z. Denition WP pairing for q, q Q X : q, q WP = S qq /h = S f f ρ dzd z 3/15

4 Basic facts on the WP metric WP is Kähler (Weil, Ahlfors) it has negative sectional curvature (Tromba, Wolpert) it is non-complete, but geodesically convex (Wolpert) Main goal here : show that the renormalized volume of quasifuchsian mds is useful to understand the geometry of (T, WP) (Brock-Bromberg), the geometry of hyperbolic 3-mds that ber over the circle (Brock-Bromberg, Kojima-McShane). (Use comparison to volume of the convex core.) 4/15

5 of quasifuchsian mds Two origins Liouville functional (Takhtajan, Zograf, Teo...) the renormalized volume of conformally compact Einstein mds (Graham, Witten,...) Denes a function V R : T T R such that V R (, h) is a Kähler potential for WP on T V R (h, h) 3 π(g 1)d WP (h, h) V R (h, h) V C Lm(l) 4 V R (h, h) + C g where V C is the volume, and m and l are the induced metric and measured bending lamination on the boundary of the convex core of M(h, h), a quasifuchsian md determined by h, h. 5/15

6 Quasifuchsian groups The WP metric Let Γ = π 1 (S). Denition A morphism ρ : Γ PSL(2, C) is quasifuchsian if it is conjugate to a Fuchsian morphism ρ 0 : Γ PSL(2, R) PSL(2, C) by a quasiconformal homeo u : CP 1 CP 1. The limit set Λ ρ CP 1 is then a Jordan curve (quasicircle). CP 1 \ Λ ρ = D D +, each with a holo. action ρ ± of Γ So D ± /ρ ± (Γ) have complex structures c ±. D ± /ρ ± (Γ) are endowed with CP 1 structures σ ± C. CP 1 -structure : locally modeled on CP 1, changes of charts in PSL(2, C). 6/15

7 The WP metric Limit sets of quasifuchsian groups (pictures Je Brock) 7/15

8 Denition A quasifuchsian metric g on M = S R is a complete hyperbolic metric g s.t. (M, g) contains a non-empty compact convex subset. The holonomy representation of g is then a quasifuchsian morphism. QF = space of quasifuchsian metrics/morphisms. M = S S +, where S ± = D ± /ρ ± (Γ). M equiped with c = (c, c + ) and σ = (σ, σ + ). Theorem (Bers simultaneous uniformization) Any (c, c + ) T T is obtained for a unique g QF. Extension to convex co-compact hyperbolic mds : complete hyperbolic mds containing a non-empty compact convex subset. Bers theorem extends (Ahlfors-Bers). 8/15

9 The convex core Let M = M(c, c + ) is a quasifuchsian md. If K, K M are convex, then K K is convex. Therefore M contains a unique smallest (compact) non-empty convex subset C(M). C(M) = CH(Λ ρ ) H 3 C(M) is a pleated surface in M (pleated along lines) The induced metric on C(M) is hyperbolic m, m + T The pleating denes a measured lamination l, l + ML Denition V C (c, c + ) = Vol(C(M)). Theorem (Bonahon-Schläi formula) V C = 1 2 L m( l). 9/15

10 The convex hull of a limit set (picture Je Brock) 10/15

11 Equidistant foliations from a convex subset Let N M be convex, and N r := {x M d(x, N) r}. Lemma V (N r ) = v 2 e 2r + v 1 r + W (N) + o(1), where W (N) = V (N) 1 4 N Hda. Let I r be the induced metric on N r. Then I r = I ((cosh(r)e + sinh(r)b), (cosh(r)e + sinh(r)b) ) = 1 2 (e2r I + 2II + e 2r III ), where I = 1 2 (I + 2II + III ), II = I (B, ), III = I (B, B ), B = (E + B) 1 (E B). I c = (c, c + ), and I ( N r ) = e 2r I ( N). Lemma (Gauss-Codazzi equations at innity) d D B = 0, tr(b ) = K. (I, II ) σ (see talk II by Ken Bromberg). 11/15

12 Variational formula for W Let N be a subset of a hyperbolic md M. Theorem (Rivin-S, 1999) ) 2 V (N) = N (Ḣ + 1 İ, II da. 2 Consequence : Ẇ = 1 4 N II H İ, I da. 2 d Eg, dr W (N r) = πχ( M), and W (N r ) = W (N) πrχ( M). Theorem (C. Epstein) Let h be any metric in the conformal class at innity. For ρ large enough,!n ρ M such that I = e 2ρ h. Denition Let h be any metric in the c. W (M, h)= W (N ρ ) + πρχ( M), where I = e 2ρ h (for any ρ large enough). 12/15

13 Variational formula from innity There is perfect symmetry between I, II, III, B and I, II, III, B : I = 1 2 (I + 2II + III ), II = I (B, ), III = I (B, B ), B = (E + B ) 1 (E B ). Lemma (Krasnov-S, 2008) Ẇ = 1 4 M II H İ, I da. 2 Example : W (e 2r h) = W (h) rπχ( M). 13/15

14 Variation within a conformal class Let I = e 2u I 0, then K = e 2u ( u + K 0 ), İ = 2 ui. II, I = ( II, I ) + H I, İ = Ḣ + 2 uh, so 4Ẇ = M K da = M (2 u( u + K 0 ) u)e 2u da so 4Ẇ = M 2 u( u + K 0 ) uda 0, and 4W (e 2u I 0 ) = 4W (I 0 ) du 2 + 2K 0 uda 0 for xed area of I, W is maximal at the constant curvature metric. M relation to CFT on S : conformal invariance OK, but invariance only under isotopies Denition V R (c) = W (h), where h c has constant curvature 2. 14/15

15 Comparing V R (M) with V C (M) Theorem (S, 2013) V R (h, h) V C Lm(l) 4 V R (h, h) + C g Fact : if h and h have K 0 and h h, then W (h) W (h ) W (C(M)) = V (C(M)) 1 4 L m(l) = V C (M) 1 4 L m(l) W (C(M)) = W (h Th /2), where h Th is the Thurston metric. h Th h 1, so W (C(M)) = W (h Th /2) W (h 2 ) = V R 1st. area(h Th ) = area(m) + L m (l) c g area(h 2 ) because L m (l) uniformly bounded. ( ) W (h Th /2) area(hth /2) W area(h 2 ) h 2 = ) W (h 2 ) πχ( M) log W (h 2 ) + C g ( area(hth /2) area(h 2 ) and the 2nd inequality follows. 15/15