Local convexity of renormalized volume for rank-1 cusped manifolds
aa r X i v : . [ m a t h . DG ] J un LOCAL CONVEXITY OF RENORMALIZED VOLUME FOR RANK-1CUSPED MANIFOLDS
FRANCO VARGAS PALLETE
Abstract.
We study the critical points of the renormalized volume for acylindrical geo-metrically finite hyperbolic 3-manifolds that include rank-1 cusps, and show that the renor-malized volume is locally convex around these critical points. We give a modified definitionof the renormalized volume that is additive under gluing, and study some local properties. Introduction
Renormalized volume is a quantity that gives a notion of volume for hyperbolic manifoldswhich have infinite volume under the classical definition. Its study for convex co-compacthyperbolic 3-manifolds can be found in [KS08], while the geometrically finite case whichincludes rank 1-cusps has been developed in [GMR17].The article is organized as follows: sections (2) and (3) give a review of parametrizationof hyperbolic metrics and renormalized volume for convex co-compact manifolds. Section(4) gives a way to complete a result of [KS08] using Ricci flow. Section (5) gives a prooffor the convex co-compact case by showing that the Hessian of the renormalized volume ispositive definite, while section (6) shows how to extend this proof when rank-1 cusps aretaken into account. Finally, in section (7) we define the corrected renormalized volume,which is additive under gluing, and we prove convexity at one of its critical points.Moroianu [Mor17] has proved independently that the Hessian is positive definite at thecritical points for the convex co-compact case, by the use of minimal surfaces. Our methodrelies on computing the Hessian for quasi-Fuchsian manifolds, and then using the skinningmap to compute it for acylindrical manifolds.
Acknowledgements:
I would like to thank Ian Agol for introducing me to the topic andfor his guidance during this project. The definition of corrected renormalized volume is dueto him. I would also like to thank David Dumas, Richard Canary and Martin Bridgemanfor their helpful comments, as well to the anonymous referee.2.
Local parametrization of hyperbolic metrics
Let R be a Riemann surface of genus g with n punctures, T its Teichmuller space, B itsspace of Beltrami differentials and Q its space of quadratic holomorphic differentials. Thereis a nice description of these two last spaces in terms of complex-valued functions in thefollowing way (it can be found, for example, in [Gar87]):Take the covering map H → R and denote by Γ ⊆ PSL(2 , R ) the group of deck trans-formations. Then elements of B are represented in H by measurable L ∞ functions µ such Research partially supported by NSF grant DMS-1406301. that(1) µ ( A ( z )) A ′ ( z ) = µ ( z ) A ′ ( z ) , ∀ A ∈ Γ . Similarly, elements of Q are represented in H by holomorphic functions φ such that(2) φ ( A ( z )) A ′ ( z ) = φ ( z ) , ∀ A ∈ Γand RR R (Γ) | φ ( z ) | dx ∧ dy < ∞ , where R (Γ) is any fundamental domain for Γ in H . Observethat the hyperbolic metric on H , y dzdz , corresponds to ρ ( z ) = y which has the property(3) ρ ( A ( z )) A ′ ( z ) A ′ ( z ) = ρ ( z ) , ∀ A ∈ PSL(2 , R ) , and hence φ ρ satisfies (1). Because RR R (Γ) | φ ( z ) | dx ∧ dy < ∞ , φ ρ is in L ∞ , and then we havea map Q → B , φ φ ρ .Thanks to the theory of the Beltrami equation ([Ahl66], [AB60], [LV65]), we can find asolution of f z ( z ) = µ ( z ) f z ( z ) for k µ k ∞ < f µ of H which extends continuously to ∂ H , fixes 0 , , ∞ and depends analytically on µ ( f solvesthe Beltrami equation in the distributional sense). With this we have a map from the unitball of B to T , and by Teichm¨uller theory, the correspondence that sends φ φ ρ and thento the solution of the Beltrami equation for φ ρ defines a local homeomorphism between aneighbourhood of 0 in Q to a neighbourhood of T .Now, the solution to the Beltrami equation for µ = φ ρ , ( f µ ) ∗ ( ρdzdz ) is a local parametriza-tion of hyperbolic metrics defined on the same space. We would like to compute the variationof this family of metrics at the origin. Because f tµ satisfies the Beltrami equation for tµ , wehave(4) ( f tµ ) ∗ ( ρdzdz )( z ) = ρ ( f tµ ( z ) ) | f tµz | | dz + tµ ( z ) dz | . Hence (here is implicit the analytic dependence of f µ with respect to µ )(5) ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( f tµ ) ∗ ( ρdzdz )( z ) = ρ ( z ) µ ( z ) dz + ρµ ( z ) dz + Edzdz, where E groups the terms of the derivative that go together with dzdz . We can then replace µ = φ ρ to obtain(6) ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( f tµ ) ∗ ( ρdzdz )( z ) = Re( φdz ) + Edzdz.
Define RQ as the space of real parts of quadratic holomorphic differentials. Since takingthe real part defines an isomorphism Q → RQ , we have a local homeomorphism from aneighbourhood of 0 in RQ to a neighbourhood of T . Moreover, if we define I v to be thehyperbolic metric ( f µ ) ∗ ( ρdzdz ), where µ = φ ρ and Re( φdz ) = v , then (6) implies that at 0(7) D I v = v + Edzdz.
Using the hyperbolic metric of R to define an inner product for tensors, D I v projects orthog-onally to v (because dz , dz are pointwise orthogonal to dzdz ). In particular, for v, w ∈ RQ c OCAL CONVEXITY OF RENORMALIZED VOLUME FOR CUSPED MANIFOLDS 3 we have(8) h D I c ( v ) , w i = h v, w i . Observe finally that if we had chosen R with the same hyperbolic metric but with oppositeorientation, the space RQ coincides with the one defined for the original Riemann surfacestructure. 3. Review of renormalized volume
Given a convex co-compact hyperbolic 3-manifold (
M, g ), Krasnov and Schlenker [KS08]defined its renormalized volume and calculated its first variation from the W -volume of acompact submanifold N ([KS08] Definition 3.1) as(9) W ( M, N ) = V ( N ) − Z ∂N Hda, where da is the area form of the induced metric.Expanding on the notation used in [KS08], denote by I the metric induced on ∂N , IIits second fundamental form (so II( x, y ) = I( x, By ), where B is the shape operator) andIII( x, y ) = I( Bx, By ) its third fundamental form.If we further assume that N has convex boundary and that the normal exponential map(pointing towards the exterior of ∂N ) defines a family of equidistant surfaces { S r } thatexhaust the complement of N ( S = ∂N ), then the W -volume of N r (points on the interiorof S r ) satisfies ([Sch13] Lemma 3.6)(10) W ( N r ) = W ( N ) − πrχ ( ∂N ) . Also, as observed in ([Sch13] Definition 3.2, Proposition 3.3), I ∗ = 4 lim r →∞ e − r I r (where I r isthe metric induced on S r , which is identified with S by the normal exponential map) existsand lies in the conformal class of the boundary. The analogous re-scaled limits for II , III , B also exist and are denoted by II ∗ , III ∗ , B ∗ . The reason to multiply by 4 is so I ∗ = g | N in thecase when N is a totally geodesic surface.For the case of convex co-compact manifolds, any metric at infinity that belongs to theconformal class given by the hyperbolic structure can be obtained as the rescaled limit of theinduced metrics of some family of equidistant surfaces. Theorem 5.8 of [KS08] describes thisby the use of Epstein surfaces (as stablished in [Eps84]), which in turn allows us to define(11) W ( M, h ) = W ( M, N r ) + πrχ ( ∂M ) , where { N r } corresponds to the equidistant surfaces given by the Epstein surfaces of h (inother words, h = I ∗ ). Then W ( M, h ) is well-defined as a consequence of (10).We can finally define the renormalized volume of M as(12) V R ( M ) = W ( M, h ) , where h is the metric in the conformal class at infinity that has constant curvature − W -volume in terms of theinput at infinity (observe that because of the description by Epstein surfaces, I ∗ determines II ∗ and III ∗ ) from the volume variation of Rivin-Schlenker [RS99]. Since for the rest of the paper FRANCO VARGAS PALLETE we are going to write everything in terms of these limit tensors, let us omit the superscript ∗ . As we did in the previous section, let us fix c ∈ T ( ∂M ) and some metric I c that representsit, so we can parametrize T ( ∂M ) by RQ c . Then for v ∈ RQ c we have the variation of V R at I c ([KS08] Corollary 6.2, Lemma 8.5)(13) DV R ( v ) = − Z ∂M h D I c ( v ) , II i da, where the metric between tensors and the area form da are defined from I c and II = II − Iis the traceless second fundamental form. This 2-form is (at each component of ∂M , aftertaking quotient by the action of π ( M )) the negative of the real part of the Schwarzianderivative of the holomorphic map between one component of the region of discontinuityand a disk ([KS08] Lemma 8.3). In particular (as we stated in (8)) h D I c ( v ) , II i = h v, II i pointwise. Then if we take c to be a critical point (i.e. DV R ( v ) = 0 at I c for every v ∈ RQ c )II must vanish at every point. This in turn implies that the holomorphic map between acomponent of the region of discontinuity and a disk has Schwarzian derivative identicallyzero, which means that the components are disks and the boundary of the convex core istotally geodesic.4. On the maximality of the W -volume among metrics of constant area In Section 7 of [KS08], Krasnov and Schlenker study the variation of the W -volume amongmetrics of the same conformal class while keeping the area constant. One way of showingthis is by observing that the Ricci flow is a gradient-like flow for the W -Volume. We includethis argument due to the connection to Ricci flow, since earlier proofs of this fact appearedin [GMR17] (Proposition 7.1, which includes the cusped case) and previously in [GMS](Proposition 3.11, convex co-compact case).From [KS08] Corollary 6.2 we know that(14) δW = − Z ∂M δH + h δ I , II i da, where we omit the superscript ∗ and H denotes the mean curvature at infinity.Now, since we are taking a conformal variation, δ I = u I for some function u : ∂M → R .Moreover, since the volume is preserved, R ∂M uda = 0.Remember that II is the traceless part of the second fundamental form, so h u I , II i = 0pointwise. Also ([KS08] Remark 5.4) H = − K and hence we have(15) δW = 14 Z ∂M δKda. But by the Gauss-Bonnet formula R ∂M δKda + R ∂M Kδ ( da ) = 0 and the equality δ ( da ) = h δ I , I i da = uda , we can reduce it to(16) δW = − Z ∂M Kuda,
OCAL CONVEXITY OF RENORMALIZED VOLUME FOR CUSPED MANIFOLDS 5 from where we can recover that K = const. is the unique critical point. If two points haddifferent values of K , we can take u supported around those points such that R ∂M uda = 0,but − R ∂M Kuda > u = − K + πχ ( ∂M )vol( ∂M ) has integral equal to 0 ( R ∂M Kda = 2 πχ ( ∂M )), and byH¨older inequality(17) (cid:18)Z ∂M K da (cid:19) . (vol( ∂M )) ≥ (cid:18)Z ∂M Kda (cid:19) , giving − R ∂M Kuda ≥
0. Hence the W -volume is no decreasing under the Ricci flow in twodimensions. It is known (see, for example, [IMS11]) that this flow converges to the metricof constant curvature, proving that this metric is a global maximum. Since K = const. isthe only critical point, the W -volume increases strictly under the flow, making this metric astrict global maximum.5. Local convexity at the geodesic class
In order to study local behavior around the critical points we want to compute the Hessianof V R at these points. Let c be a critical point (i.e. II ≡
0) and I c a metric representing thisclass. For v, w ∈ RQ c , we vary (6) with respect to w to obtainHess V R ( v, w ) = − Z ∂M h D I( v ) , D II ( w ) i da + (terms obtained by varying the other tensors w.r.t. w) , (18)and since II vanishes identically, all the terms in parenthesis get canceled, so(19) Hess V R ( v, w ) = − Z ∂M h D I( v ) , D II ( w ) i da. Let us first understand the quasi-Fuchsian case (here we are referring to hyperbolic struc-tures on the product S × R , where S is a closed orientable surface of genus g > − . Theorem 1.
Let M be a Fuchsian manifold (i.e. the conformal classes at infinty c + , c − bothequal to say a conformal class c ). Then the Hessian at M of the renormalized volume ispositive definite in the orthogonal complement of the diagonal of RQ c × RQ c ≈ T c T ( ∂M ) = T c + T ( ∂M + ) × T c − T ( ∂M − ) (where we are using the parametrization by real parts of holo-morphic quadratic differentials w.r.t. ( c, c ) and that the real parts of holomorphic quadraticdifferentials are the same for both orientations). Moreover, the Hessian agrees with themetric induced by I with a factor.Proof. Recall by (19) that since M is a critical point, then for v = ( v + , v − ) , w = ( w + , w − )tangent vectors at ( c, c )(20) 4Hess V R ( v, w ) = − Z ∂ + M h D I + ( v ) , D II +0 ( w ) i da − Z ∂ − M h D I − ( v ) , D II − ( w ) i da. FRANCO VARGAS PALLETE
As mentioned in [KS08](Lemma 8.3) we know that II +0 ( c + , c ) is equal to − Re( q + ( c + , c )),where q + ( c + , . ) : T − → Q c + is the Bers embedding. In particular D II +0 ( v,
0) lands in RQ c ,and because D II +0 ( v, v ) = 0, then the whole image of D II +0 lands in RQ c . Since an analogousargument works for D II − , we can then reduce to4Hess V R ( v, w ) = − Z ∂ + M h v + , D II +0 ( w ) i da − Z ∂ − M h v − , D II − ( w ) i da = −h v, D II ( w ) i L , (21)where h· , ·i on forms denotes the L scalar product defined by I c .Now D II is diagonalizable with orthogonal eigenvectors (is the expression of the Hessianin terms of the metric induced by I( c, c )). We can exploit this fact in the following lemma. Lemma 1. D II ( v, − v ) = − ( v, − v ) Proof.
We prove first the following claim.
Claim:
Let M be a Fuchsian manifold with associated conformal class c. Then D III + ( v, RQ c for all v ∈ RQ c .Recall from [KS08] (as stated in the comments of Definition 5.3) that almost-Fuchsianmanifolds are quasi-Fuchsian manifolds with the principal curvatures at infinity between − ,
1, and for those manifolds III is conformal to I − , so D III + ( v,
0) is I( c, c ) multiplied by somefunction (III stays conformal to I( c, c ) when we only varied c + ). To see that those tensorsare orthogonal to RQ c (note that for every tensor space we are taking the inner productinduced by I c and integration), recall that if we take conformal coordinates, elements of RQ c are expressed in terms of dz and dz , where the metric is in terms of dzdz , meaning that D III + ( v,
0) is even pointwise orthogonal to any element of RQ c .Going back to the proof of the lemma, it follows from [KS08] (Definition 5.3) that the firstvariations at M satisfy δ II = δ II − δ I = I( δB · , · )(22) δ III = 14 δ I + I( δB · , · ) , (23)hence(24) D III = D II + 14 D I . In particular, by the claim, D II ( v,
0) = − v , and so(25) D II +0 ( v, − v ) = D II +0 (2 v,
0) + D II +0 ( − v, − v ) = − D I + (2 v,
0) = − v. The lemma follows since D II is diagonalizable in the orthogonal complement of the diagonal(the diagonal is part of the 0-eigenspace). (cid:3) This lemma implies that the orthogonal complement is the − -eigenspace for D II , con-cluding the last part of the theorem. (cid:3) Now we will use this local behavior for quasi-Fuchsian manifolds to conclude our mainresult for acylindrical manifolds.
OCAL CONVEXITY OF RENORMALIZED VOLUME FOR CUSPED MANIFOLDS 7
Theorem 2.
Let M be a compact acylindrical 3-manifold with hyperbolic interior such that ∂M = ∅ . Then there is a unique critical point c for the renormalized volume of M , where c is the unique conformal class at the boundary that makes every component of the region ofdiscontinuity a disk(a.k.a. the geodesic class). The Hessian at this critical point is positivedefinite.Proof. Since DV R ( v ) = − h D I c ( v ) , II i = − h v, II i we have(26) DV R = 0 ⇔ II ≡ . Now, II ≡ ∂M = S ∪ . . . ∪ S n , and c = ( c , . . . , c n ) denote thegeodesic class. To show that the Hessian is positive definite, take parametrization RQ for S i based at c i and use the same metric to compute variations of V R for M and for both ends ofquasi-Fuchsian manifolds S i × R .Recall that, since M is hyperbolic acylindrical, the subgroups associated to the componentsof ∂M are quasifuchsian. Hence we have a map from T ( ∂M ) (corresponding to hyperbolicmetrics in M ) to T ( ∂M ) ×T ( ∂M ) (corresponding to the quasifuchsian subgroups). The firstcoordinate of this map is the identity on T ( ∂M ), while the second coordinate σ : T ( ∂M ) →T ( ∂M ) is Thurston’s skinning map . Observe then that I at S i coincides with I + ( c i , σ i ( c )),where σ i is the image of the skinning map σ corresponding to S i . From the dependenceof II in terms of I (Remark 5.9 of [KS08]) we see that II at S i coincides with II +0 ( c i , σ i ( c )),and hence D I( v ) , D II ( w ) are equal to D I + ( v i , dσ i ( v )) , D II +0 ( w i , dσ i ( w )) at S i (note that here dσ is written in terms of our charts given by RQ c , so it is essentially the conjugation ofthe derivative of the skinning map, given our remark on how our charts behave with anorientation change). Hence(27) 4Hess V R ( v, w ) = − n X i =1 h v i , D II +0 ( w i , dσ i ( w )) i = 14 n X i =1 h v i , w i − dσ i ( w ) i , which is greater than zero for v = 0 according to the following result: Theorem 3 (McMullen [McM90]) . Under the conditions above, k dσ k < , where the norm k dσ k is calculated in terms of the Teichm¨uller metric. Indeed, (27) tells us that dσ c is diagonalizable, which together with McMullen’s theoremimplies that all eigenvalues are less than 1. Then Hess V R is also diagonalizable with alleigenvalues positive. (cid:3) McMullen’s result can be sketched as follows. The skinning map is holomorphic betweenTeichm¨uller spaces, in particular sending holomorphic disks to holomorphic disks. Thisimplies that k dσ k ≤ FRANCO VARGAS PALLETE
Teichm¨uller metric. If k dσ k = 1 at a point, then σ would be an isometry on the extremalholomorphic disk, but an earlier result of Thurston states that σ is a strict contraction forthe acylindrical case. Corollary 1.
Let c ∈ T ( ∂M ) be as in the previous theorem. Then dσ admits an orthonormaleigenbasis with respect to the L -norm on RQ c , with all eigenvalues less than in absolutevalue. Observe that if we take holomorphic quadratic differentials for the tangent space, we havethe conclusion for dσ after taking a complex conjugation.This result was somehow expected thanks to a parallel between the skinning map and theThurston map for postcritically finite rational maps, since the Thurston map is contractingfor non-Latt`e maps. Moreover, if f is a postcritically finite quadratic polynomial, thereis a uniform spectral gap for the derivative at its unique critical point (more precisely, alleigenvalues are greater than 1 / V R (for acylindrical manifolds). It is also open that V R is strictly positive for quasi-Fuchsian manifolds outside the Fuchsian locus, although this is known for almost-Fuchsianmanifolds [CM].6. Remarks for hyperbolic 3-manifolds with rank-1 cusps
Consider a quasi-Fuchsian manifold M with rank-1 cusps and a compact subset K of ∂M + (the top boundary component as we denoted in (5)). For a quasi-Fuchsian manifoldsufficiently close to the Fuchsian locus, the equidistant foliation over K extends up to the0-leaf and has principal curvatures between − M and III + is a metric conformal to I − (all of this over K ). Hence, at a Fuchsian manifold, D III + ( v,
0) is a multiple of I + at anysuch compact K , and hence over all of ∂M + .On the other hand, all the formulas of [KS08] Theorem 5.8 apply for the rank 1-cuspscase, so in particular we also have that II is (at each component of ∂M after taking quotientby the action of π ( M )) the negative of the real part of the Schwarzian derivative of theholomorphic map between one component of the region of discontinuity and a disk.As we mentioned, the variation formula of the renormalized volume was proved in [GMR17].The second variation at a critical point is well defined since for the quasi-Fuchsian case it isthe inner product of quadratic holomorphic differentials, which are in L with respect to themetric. The other terms that get canceled with II in (18) do not overcome the exponentialdecay of the metric since each of them have at most polynomial growth. The general case willthen be well defined since it is also an inner product of quadratic holomorphic differentials,thanks to the skinning map argument.Finally, McMullen’s result still applies for surfaces with punctures, so the proof of ourtheorem extends to the case of geometrically finite hyperbolic acylindrical manifolds withrank 1 cusps. We summarize this as follows. OCAL CONVEXITY OF RENORMALIZED VOLUME FOR CUSPED MANIFOLDS 9
Theorem 4.
Let M be a Fuchsian manifold (i.e. the conformal classes at infinty c + , c − coincide, say to a conformal class c ) with possibly rank 1-cusps.Then the Hessian at M ofthe renormalized volume is positive definite in the orthogonal complement of the diagonal of RQ c × RQ c ≈ T c T ( ∂M ) = T c + T ( ∂M + ) × T c − T ( ∂M − ) (where we are using the parametriza-tion by real parts of holomorphic quadratic differentials w.r.t. ( c, c ) and that the real partsof holomorphic quadratic differentials are the same for both orientations). Moreover, theHessian agrees with the metric induced by I with a factor. Theorem 5.
Let M be a geometrically finite manifold with rank 1-cusps. Then there is aunique critical point c for the renormalized volume of M , where c is the unique conformalclass at the boundary that makes every component of the region of discontinuity a disk(a.k.a.the geodesic class). The Hessian at this critical point is positive definite. We can also extend the Ricci flow argument for the maximality of the W -volume. Weuse the existence and convergence properties of the Ricci flow from the conditions stated in[JMS09]. In the language of [GMR17], let a cusp be parametrized as ( v, w ) ∈ (cid:2) , R (cid:2) × R / Z ,which under the notation of [JMS09] corresponds to ( s, w ), where v = e − s .We need to show that ψ ∈ C ∞ r ( ∂M ) ⇒ e φ ∈ R + s − µ C ,α (where the first term describesthe family of conformal factors for which the renormalized volume is defined in [GMR17],and the second term the sufficient conditions for running Ricci flow in the lines of [JMS09])for some constants µ > , R ; where v = e − s , ψ ( v, w ) = φ ( s, w ). C ∞ r ( ∂M ) denotes ([GMR17]2.7) functions whose w -derivative vanishes with infinite order at v = 0. C ,α denotes theusual H¨older space with respect to the hyperbolic metric on the cusp.Since ψ ∈ C ∞ r ( M ), lim v → ψ ( v, w ) exists and does not depend on w , which we denote by R . Then we need to show that a ( s, w ) = e φ ( s,w ) − R belongs to s − µ C ,α . We will show thatbelongs to s − µ C for 0 < µ < b ( v, w ) = a ( s, w ) extends to v = 0 in C ∞ because ψ does. In particular,(28) b v ( v, w ) = 2 ψ v ( v, w ) e ψ ( v,w ) also extends to v = 0, and b v (0 , w ) does not depend on w . There is a constant K such that | b ( v, w ) | ≤ Kv µ , so | a ( s, w ) | ≤ Ke − sµ ≤ K ′ s − µ , which gives a ∈ s − µ C .Now a s ( s, w ) = 2 φ s ( s, w ) e φ ( s,w ) , and we need to show that | a s ( s, w ) | ≤ Ks − µ . This followsfrom(29) 2 φ s ( s, w ) e φ ( s,w ) = − e − s ψ v ( v, w ) e ψ ( v,w ) and the fact that ψ v ( v, w ) e ψ ( v,w ) extends to v = 0 not depending on w .Now the pattern appears for higher y -derivatives of a . When we express the derivativesin v coordinates, the expressions gain a v = e − s , so the decay follows since ψ extends C ∞ to v = 0 without w dependence.For the case of taking a w -derivative, its norm in C is e s a w (the metric to define C k,α is ds + e − s dw ). We have(30) e s φ w ( s, w ) e φ ( s,w ) = v − ψ w ( v, w ) e ψ ( v,w ) , which is bounded by some power of v because ψ w has infinite order 0 at v = 0. Similarly, weextend our argument to further w -derivatives. This shows that we can run Ricci flow withinitial condition any metric considered by [GMR17]. Finally, to observe that the logarithm of the conformal factors along the Ricci flow stillbelong to C ∞ r ( M ), note that the curvature e − ψ ( − ψ ) is in C ∞ r ( M ). This makes theflow to preserve the set of metrics with such conformal factors, which in turn allows theargument to extend to this generality.7. Corrected renormalized volume
Let M be a closed compact hyperbolic 3-manifold with an oriented incompressible surfaceΣ ( g >
2) which divides M into two components M , M . We can consider now hyperbolicstructures N , N on the interiors of M , M such that we have the inclusions M ⊂ N , M ⊂ N by taking coverings with respect to π ( M ) , π ( M ), respectively. The fact that they gluealong Σ tells us that the conformal classes at infinity of the open manifolds are each othersskinning maps.Now, for r sufficiently large, M lives inside N r , the r -leaf of the foliation defined by therenormalized volume, and hence(31) vol( M ) = V R ( M , c ) + rπχ (Σ) − vol((int N r ) \ M ) + 14 Z N r Hda, where c is the conformal class at infinity for int( M ).We can take r even larger so that we have a similar statement for M . Observe that theregions (int N r ) \ M , (int N r ) \ M glue along Σ and live inside the quasi-Fuchsian manifoldobtained by gluing N \ M , N \ M along Σ. In particular the quasi-Fuchsian manifold( N \ M ) ∪ ( N \ M ) has ( c , c ) as a conformal class at infinity and N r , N r as the r -leavesfor the renormalized volume, from where we concludevol((int N r ) \ M ∪ (int N r ) \ M ) = V R (Σ × [0 , , c , c ) + 14 Z N r Hda + 14 Z N r Hda + 2 rπχ (Σ) . (32)This gives the following proposition. Proposition 1.
For M as above, vol( M ) = V R ( M , c ) + V R ( M , c ) − V R (Σ × [0 , , c , c )In order to use this proposition to find a lower bound for the volume of M (and since c , c are related by the skinning map), we want to show that the minimum of the correctedrenormalized volume (defined as follows) is attained at the geodesic class. Definition 1.
Let M be a compact acylindrical -manifold with hyperbolic interior, connectedboundary of genus g > , and let c ∈ T ( ∂M ) the element that defines the conformal boundaryat infinity. Then the corrected renormalized volume is defined as (33) V R ( M ) = V R ( M, c ) − V R ( ∂M × [0 , , ( c, σ ( c ))) . Observe that proposition (1) implies that, under cutting, the volume of M is equal to thesum of the corrected renormalized volumes of its parts. It is straightforward to extend thisto the case when Σ has multiple components, and if M is open without cusps, we can replacevol( M ) by V R ( M ) in Proposition (1). Then if we consider the corrected renormalized volume OCAL CONVEXITY OF RENORMALIZED VOLUME FOR CUSPED MANIFOLDS 11 to be an extension of the volume for closed hyperbolic manifolds, we obtain as a corollary ofproposition 1:
Corollary 2. V R is additive under cutting. Now, from the first variation of V R at I = I + , we have8 DV R ( v ) = − Z ∂M + h D I + ( v, dσ ( v )) , II +0 ( c, σ ( c )) i da + + Z ∂M − h D I − ( v, dσ ( v )) , II − ( c, σ ( c )) i da − , (34)where da + , da − are the volume forms for I + , I − , respectively.Observe that if we take c to be the geodesic class, II +0 , II − are both zero, and hence c is alsoa critical point for the corrected renormalized volume. Moreover, the Hessian is expressedas 8Hess V R ( v, w ) = −h D I + ( v, dσ ( v )) , D II +0 ( w, σ ( w )) ii + h D I − ( v, dσ ( v )) , D II − ( w, σ ( w )) i , (35)which by Lemma 1 is equal to(36) 8Hess V R ( v, w ) = −h v, −
14 ( w − dσ ( w )) i + h dσ ( v ) ,
14 ( w − dσ ( w )) i . Then(37) 32Hess V R ( v, w ) = h v + dσ ( v ) , w − dσ ( w ) i , and since all eigenvalues of dσ are between − Theorem 6.
Let M be a compact acylindrical -manifold with hyperbolic interior, ∂M = ∅ without cusps, and c ∈ T ( ∂M ) be the geodesic class. Then c is a local minimum for thecorrected renormalized volume of M . References [AB60] Lars Ahlfors and Lipman Bers,
Riemann’s mapping theorem for variable metrics , Ann. of Math.(2) (1960), 385–404. MR 0115006 (22 Lectures on quasiconformal mappings , Manuscript prepared with the assistanceof Clifford J. Earle, Jr. Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc.,Toronto, Ont.-New York-London, 1966. MR 0200442 (34
Eigenvalues of the thurston pullback maps , Inpreparation.[CM] Corina Ciobotaru and Sergiu Moroianu,
Positivity of the renormalized volume of almost-Fuchsianhyperbolic 3-manifolds , arXiv:1402.2555v2 [math.DG].[Eps84] Charles L. Epstein,
Envelopes of horospheres and weingarten surfaces in hyperbolic 3-space , pre-print, Princeton University, 1984.[Gar87] Frederick P. Gardiner,
Teichm¨uller theory and quadratic differentials , Pure and Applied Mathe-matics (New York), John Wiley & Sons, Inc., New York, 1987, A Wiley-Interscience Publication.MR 903027 (88m:32044)[GMR17] Colin Guillarmou, Sergiu Moroianu, and Fr´ed´eric Rochon,
Renormalized volume on the Teichm¨uller space of punctured surfaces , Ann. Sc. Norm. Super.Pisa Cl. Sci. (5) (2017), no. 1, 323–384. MR 3676051[GMS] Collin Guillarmou, Sergiu Moroianu, and Jean-Marc Schlenker, The renormalized volume anduniformisation of conformal structures , arXiv:1211.6705 [math.DG]. [IMS11] James Isenberg, Rafe Mazzeo, and Natasa Sesum,
Ricci flow in two dimensions , Surveys in geo-metric analysis and relativity, Adv. Lect. Math. (ALM), vol. 20, Int. Press, Somerville, MA, 2011,pp. 259–280. MR 2906929[JMS09] Lizhen Ji, Rafe Mazzeo, and Natasa Sesum,
Ricci flow on surfaces with cusps , Math. Ann. (2009), no. 4, 819–834. MR 2545867 (2011c:53161)[KS08] Kirill Krasnov and Jean-Marc Schlenker,
On the renormalized volume of hyperbolic 3-manifolds ,Comm. Math. Phys. (2008), no. 3, 637–668. MR 2386723 (2010g:53144)[LV65] O. Lehto and K. I. Virtanen,
Quasikonforme Abbildungen , Die Grundlehren der mathematischenWissenschaften in Einzeldarstellungen mit besonderer Ber¨ucksichtigung der Anwendungsgebiete,Band, Springer-Verlag, Berlin-New York, 1965. MR 0188434 (32
Iteration on Teichm¨uller space , Invent. Math. (1990), no. 2, 425–454.MR 1031909 (91a:57008)[Mor17] Sergiu Moroianu, Convexity of the renormalized volume of hyperbolic 3-manifolds , Amer. J. Math. (2017), no. 5, 1379–1394. MR 3702501[RS99] Igor Rivin and Jean-Marc Schlenker,
The Schl¨afli formula in Einstein manifolds with boundary ,Electron. Res. Announc. Amer. Math. Soc. (1999), 18–23 (electronic). MR 1669399(2000a:53076)[Sch13] Jean-Marc Schlenker, The renormalized volume and the volume of the convex core of quasifuchsianmanifolds , Math. Res. Lett. (2013), no. 4, 773–786. MR 3188032 Department of Mathematics, University of California at Berkeley, 745 Evans Hall,Berkeley, CA 94720-3860, U.S.A.
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