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We extend the notion of Epstein maps to conformal metrics on submanifolds of the unit sphere $\mathbb{S}^n=\partial_\infty\mathbb{H}^{n+1}$. Using this construction for curves in $\mathbb{S}^2$, we define the W-volume for conformal metrics on domains in $\overline{\mathbb{C}}=\mathbb{S}^2$ with round circles as boundaries. We show that the W-volume is a realization in $\mathbb{H}^3$ of the determinant of the Laplacian. We use this and work of Osgood, Phillips and Sarnak to show that a classical Schottky uniformization of a genus g Riemann surface has renormalized volume bounded by $(6g-8)\pi$, and by $-2\pi$ under further assumptions. This gives a partial answer to a question of Maldacena. We also then provide a $\mathbb{H}^3$ realization of the Loewner energy of a $C^{2,\alpha}$ Jordan curve., Comment: 16 pages, 1 figure. Comments are welcome!

Effective control and readout of qubits form the technical foundation of next-generation, transformative quantum information sciences and technologies. The nitrogen-vacancy (NV) center, an intrinsic three-level spin system, is naturally relevant in this context due to its excellent quantum coherence, high fidelity of operations, and remarkable functionality over a broad range of experimental conditions. It is an active contender for the development and implementation of cutting-edge quantum technologies. Here, we report magnetic domain wall motion driven local control and measurements of NV spin properties. By engineering the local magnetic field environment of an NV center via nanoscale reconfigurable domain wall motions, we show that NV photoluminescence, spin level energies, and coherence time can be reliably controlled and correlated to the magneto-transport response of a magnetic device. Our results highlight the electrically tunable dipole interaction between NV centers and nanoscale magnetic structures, providing an attractive platform to realize interactive information transfer between spin qubits and non-volatile magnetic memory in hybrid quantum spintronic systems., Comment: 13 pages, 5 figures

We have performed an experimental and modeling-based study of the spin-orbit torque-induced growth of magnetic stripe domains in heavy metal/ferromagnet thin-film heterostructures that possess chiral N\'eel-type domain walls due to an interfacial Dzyaloshinskii-Moriya interaction. In agreement with previous reports, the stripe domains stabilized in these systems exhibit a significant transverse growth velocity relative to the applied current axis. This behavior has previously been attributed to the Magnus force-like skyrmion Hall effect of the stripe domain spin topology, which is analogous to that of a half-skyrmion. However, through analytic modeling of the in-plane torques generated by spin-orbit torque, we find that a dynamical reconfiguration of the domain wall magnetization profile is expected to occur - promoting motion with similar directionality and symmetry as the skyrmion Hall effect. These results further highlight the sensitivity of spin-orbit torque to the local orientation of the domain wall magnetization profile and its contribution to domain growth directionality., Comment: 21 pages, 4 figures, 3 supplementary figures

Time-resolved ultrafast EUV magnetic scattering was used to test a recent prediction of >10 km/s domain wall speeds by optically exciting a magnetic sample with a nanoscale labyrinthine domain pattern. Ultrafast distortion of the diffraction pattern was observed at markedly different timescales compared to the magnetization quenching. The diffraction pattern distortion shows a threshold-dependence with laser fluence, not seen for magnetization quenching, consistent with a picture of domain wall motion with pinning sites. Supported by simulations, we show that a speed of $\approx$ 66 km/s for highly curved domain walls can explain the experimental data. While our data agree with the prediction of extreme, non-equilibrium wall speeds locally, it differs from the details of the theory, suggesting that additional mechanisms are required to fully understand these effects., Comment: 5 pages, 4 figures; Supplemental Material: 9 pages, 9 figures

Novel noncollinear antiferromagnets with spontaneous time-reversal symmetry breaking, nontrivial band topology, and unconventional transport properties have received immense research interest over the past decade due to their rich physics and enormous promise in technological applications. One of the central focuses in this emerging field is exploring the relationship between the microscopic magnetic structure and exotic material properties. Here, the nanoscale imaging of both spin-orbit-torque-induced deterministic magnetic switching and chiral spin rotation in noncollinear antiferromagnet Mn3Sn films using nitrogen-vacancy (NV) centers is reported. Direct evidence of the off-resonance dipole-dipole coupling between the spin dynamics in Mn3Sn and proximate NV centers is also demonstrated with NV relaxometry measurements. These results demonstrate the unique capabilities of NV centers in accessing the local information of the magnetic order and dynamics in these emergent quantum materials and suggest new opportunities for investigating the interplay between topology and magnetism in a broad range of topological magnets., Comment: 16 pages, 5 figures

Symmetry is a powerful concept in physics, but its applicability to far-from-equilibrium states is still being understood. Recent attention has focused on how far-from-equilibrium states lead to spontaneous symmetry breaking. Conversely, ultrafast optical pumping can be used to drastically change the energy landscape and quench the magnetic order parameter in magnetic systems. Here, we find a distinct symmetry-dependent ultrafast behaviour by use of ultrafast x-ray scattering from magnetic patterns with varying degrees of isotropic and anisotropic symmetry. After pumping with an optical laser, the scattered intensity reveals a radial shift exclusive to the isotropic component and exhibits a faster recovery time from quenching for the anisotropic component. These features arise even when both symmetry components are concurrently measured, suggesting a correspondence between the excitation and the magnetic order symmetry. Our results underline the importance of symmetry as a critical variable to manipulate the magnetic order in the ultrafast regime.

We report the experimental observation of micron-scale magnetic skyrmions at room temperature in several Pt/Co-based thin film heterostructures designed to possess a low exchange stiffness, perpendicular magnetic anisotropy, and a modest interfacial Dzyaloshinskii-Moriya interaction (iDMI). We find both experimentally and by micromagnetic and analytic modeling that the combined action of low exchange stiffness and modest iDMI eliminates the energetic penalty associated with forming domain walls in thin film heterostructures. When the domain wall energy density approaches negative values, the remanent domain morphology transitions from a uniform state to a labyrinthian stripe phase. A low exchange stiffness, indicated by a reduction in the Curie temperature below 400 K, is achieved in Pt/Co, Pt/Co/Ni, and Pt/Co/Ni/Re structures by reducing the Co thickness to the ultrathin limit (< 0.3 nm). A similar effect occurs in thicker Pt/Co/NixCu1-x structures when the Ni layer is alloyed with Cu. At this transition in domain morphology, skyrmion phases are stabilized when a small (< 1 mT) perpendicular magnetic field is applied and current-induced skyrmion motion including the skyrmion Hall effect is observed. The temperature and thickness-induced morphological phase transitions observed are similar to the well-studied spin reorientation transition that occurs in the ultrathin limit, but we find that the underlying energy balances are substantially modified by the presence of an iDMI., Comment: 22 pages, 6 figures, 3 supplementary figures

The Dzyaloshinskii-Moriya interaction (DMI) in magnetic systems stabilizes spin textures with preferred chirality, applicable to next-generation memory and computing architectures. In perpendicularly magnetized heavy-metal/ferromagnet films, the interfacial DMI originating from structural inversion asymmetry and strong spin-orbit coupling favors chiral N\'eel-type domain walls (DWs) whose energetics and mobility remain at issue. Here, we characterize a new effect in which domains expand unidirectionally in response to a combination of out-of-plane and in-plane magnetic fields, with the growth direction controlled by the in-plane field strength. These growth directionalities and symmetries with applied fields cannot be understood from static treatments alone. We theoretically demonstrate that perpendicular field torques stabilize steady-state magnetization profiles highly asymmetric in elastic energy, resulting in a dynamic symmetry breaking consistent with the experimental findings. This phenomenon sheds light on the mechanisms governing the dynamics of N\'eel-type DWs and expands the utility of field-driven DW motion to probe and control chiral DWs., Comment: 20 pages, 4 figures, 10 supplementary figures

We have studied the effects of electrical current pulses on skyrmion formation in a series of Co/Ni/Pt-based multilayers. Transmission X-ray microscopy reveals that by applying electrical current pulses of duration and current density on the order of $\tau$=50 $\mu$s and j=1.7x10$^1$$^0$ A/m$^2$, respectively, in an applied magnetic field of $\mu$$_0$Hz=50 mT, stripe-to-skyrmion transformations are attained. The skyrmions remain stable across a wide range of magnetic fields, including zero field. The skyrmions then remain stable across a wide range of magnetic fields, including zero field. We primarily attribute the transformation to current-induced Joule heating on the order of ~125 K. Reducing the magnetic moment and perpendicular anisotropy using thin rare-earth spacers dramatically reduces the pulse duration, current density, and magnetic field necessary to 25 $\mu$s, 2.4x10$^9$ A/m$^2$, and 27 mT, respectively. These findings show that energetic inputs allow for the formation of skyrmion phases in a broad class of materials and that material properties can be tuned to yield more energy-efficient access to skyrmion phases., Comment: 32 pages, 7 figures, 9 supplemental figures

In this paper, we use the Weil-Petersson gradient flow for renormalized volume to study the space $CC(N;S,X)$ of convex cocompact hyperbolic structures on the relatively acylindrical 3-manifold $(N;S)$. Among the cases of interest are the deformation space of an acylindrical manifold and the Bers slice of quasi-Fuchsian space associated to a fixed surface. To treat the possibility of degeneration along flow-lines to peripherally cusped structures, we introduce a surgery procedure to yield a surgered gradient flow that limits to the unique structure $M_{\rm geod} \in CC(N;S,X)$ with totally geodesic convex core boundary facing $S$. Analyzing the geometry of structures along a flow line, we show that if $V_R(M)$ is the renormalized volume of $M$, then $V_R(M)-V_R(M_{\rm geod})$ is bounded below by a linear function of the Weil-Petersson distance $d_{\rm WP}(\partial_c M, \partial_c M_{\rm geod})$, with constants depending only on the topology of $S$. The surgered flow gives a unified approach to a number of problems in the study of hyperbolic 3-manifolds, providing new proofs and generalizations of well-known theorems such as Storm's result that $M_{\rm geod}$ has minimal volume for $N$ acylindrical and the second author's result comparing convex core volume and Weil-Petersson distance for quasifuchsian manifolds.

In this paper we construct examples of Weil-Petersson geodesics with nonminimal ending laminations which have 1-dimensional limit sets in the Thurston compactification of Teichm\"{u}ller space., Comment: 27 pages, 4 figures; the paper was revised, in particular some pictures were added to section 3 to illustrate the construction. Also, Theorem 2.3 about the combinatorial estimate for the hyperbolic length of a curve on a Riemann surface is included. The paper will appear in J. Topol. Anal

We prove that the deformation space $AH(M)$ of marked hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold $M$ with incompressible boundary is locally connected at quasiconformally rigid points.

To a complex projective structure $\Sigma$ on a surface, Thurston associates a locally convex pleated surface. We derive bounds on the geometry of both in terms of the norms $\|\phi_\Sigma\|_\infty$ and $\|\phi_\Sigma\|_2$ of the quadratic differential $\phi_\Sigma$ of $\Sigma$ given by the Schwarzian derivative of the associated locally univalent map. We show that these give a unifying approach that generalizes a number of important, well known results for convex cocompact hyperbolic structures on 3-manifolds, including bounds on the Lipschitz constant for the nearest-point retraction and the length of the bending lamination. We then use these bounds to begin a study of the Weil-Petersson gradient flow of renormalized volume on the space $CC(N)$ of convex cocompact hyperbolic structures on a compact manifold $N$ with incompressible boundary, leading to a proof of the conjecture that the renormalized volume has infimum given by one-half the simplicial volume of $DN$, the double of $N$., Comment: 20 pages, 0 figures

In this paper we prove that the limit set of any Weil-Petersson geodesic ray with uniquely ergodic ending lamination is a single point in the Thurston compactification of Teichm\"uller space. On the other hand, we construct examples of Weil-Petersson geodesics with minimal nonuniquely ergodic ending laminations and limit set a circle in the Thurston compactification., Comment: 46 pages, 3 figures, v3: some minor additions and corrections were made. The paper will appear in Int. Math. Res. Not. (IMRN)

We give a generalization of Thurston's Bounded Image Theorem for skinning maps, which applies to pared 3-manifolds with incompressible boundary that are not necessarily acylindrical. Along the way we study properties of divergent sequences in the deformation space of such a manifold, establishing the existence of compact cores satisfying a certain notion of uniform geometry., Comment: 41 pages, 5 figures. v2 contains new Theorem 4.9 which gives a finer model core statement in the convergent case

Given a sequence of curves on a surface, we provide conditions which ensure that (1) the sequence is an infinite quasi-geodesic in the curve complex, (2) the limit in the Gromov boundary is represented by a nonuniquely ergodic ending lamination, and (3) the sequence divides into a finite set of subsequences, each of which projectively converges to one of the ergodic measures on the ending lamination. The conditions are sufficiently robust, allowing us to construct sequences on a closed surface of genus $g$ for which the space of measures has the maximal dimension $3g-3$, for example. We also study the limit sets in the Thurston boundary of Teichm\"uller geodesic rays defined by quadratic differentials whose vertical foliations are obtained from the constructions mentioned above. We prove that such examples exist for which the limit is a cycle in the $1$-skeleton of the simplex of projective classes of measures visiting every vertex., Comment: 71 pages, 4 figures. v2: Some revisions and additions have been made. An example before the description of property P in section 3 was added. Section 6 was rewritten to address a mistake in the proof of Theorem 6.5 (called Theorem 6.7 in v1). Accepted for publication in J. Reine Angew Math

We study the relationship between two norms on the first cohomology of a hyperbolic 3-manifold: the purely topological Thurston norm and the more geometric harmonic norm. Refining recent results of Bergeron, \c{S}eng\"un, and Venkatesh as well as older work of Kronheimer and Mrowka, we show that these norms are roughly proportional with explicit constants depending only on the volume and injectivity radius of the hyperbolic 3-manifold itself. Moreover, we give families of examples showing that some (but not all) qualitative aspects of our estimates are sharp. Finally, we exhibit closed hyperbolic 3-manifolds where the Thurston norm grows exponentially in terms of the volume and yet there is a uniform lower bound on the injectivity radius., Comment: 29 pages, 2 figures. V2: Added Section 5.3 on related work of Kronheimer and Mrowka. V3: Minor improvements. V4: To appear in Inventiones Mathematicae V5: Theorem and equation numbering changed to match published version

We prove 3-dimensional hyperbolic cone-manifolds are geometrically inflexible: a cone-deformation of a hyperbolic cone-manifold determines a bi-Lipschitz diffeomorphism between initial and terminal manifolds in the deformation in the complement of a standard tubular neighborhood of the cone-locus whose pointwise bi-Lipschitz constant decays exponentially in the distance from the cone-singularity. Estimates at points in the thin part are controlled by similar estimates on the complex lengths of short curves., Comment: 15 pages. This paper was originally part of our paper `Geometric inflexibility and 3-manifolds that fiber over the circle,' arXiv:0901.3870, appearing in the Journal of Topology, 4 (2011) pp. 1-38

A recent preprint of S. Kojima and G. McShane [KM] observes a beautiful explicit connection between Teichm\"uller translation distance and hyperbolic volume. It relies on a key estimate which we supply here: using geometric inflexibility of hyperbolic 3-manifolds, we show that for $S$ a closed surface, and $\psi \in \text{Mod}(S)$ pseudo-Anosov, the double iteration $Q(\psi^{-n}(X),\psi^n(X))$ has convex core volume differing from $2n \text{vol}(M_\psi)$ by a uniform additive constant, where $M_\psi$ is the hyperbolic mapping torus for $\psi$. We combine this estimate with work of Schlenker, and a branched covering argument to obtain an explicit lower bound on Weil-Petersson translation distance of a pseudo-Anosov $\psi \in \text{Mod}(S)$ for general compact $S$ of genus $g$ with $n$ boundary components: we have $$ \text{vol}(M_\psi) \le 3 \sqrt{\pi/2(2g - 2 +n)} \, \| \psi \|_{WP}.$$ This gives the first explicit estimates on the Weil-Petersson systoles of moduli space, of the minimal distance between nodal surfaces in the completion of Teichm\"uller space, and explicit lower bounds to the Weil-Petersson diameter of the moduli space via [CP]. In the process, we recover the estimates of [KM] on Teichm\"uller translation distance via a Cauchy-Schwarz estimate (see [Lin])., Comment: 17 pages, 2 figures - incorporated corrections and comments

We construct Weil-Petersson (WP) geodesic rays with minimal filling non-uniquely ergodic ending lamination which are recurrent to a compact subset of the moduli space of Riemann surfaces. This construction shows that an analogue of the Masur's criterion for Teichm\"uller geodesics does not hold for WP geodesics., Comment: 34 pages, 5 figures

We characterize sequences of Kleinian surface groups with convergent subsequences in terms of the asymptotic behavior of the ending invariants of the associated hyperbolic 3-manifolds. Asymptotic behavior of end invariants in a convergent sequence predicts the parabolic locus of the algebraic limit as well as how the algebraic limit wraps within the geometric limit under the natural locally isometric covering map.

Bounded-type 3-manifolds arise as combinatorially bounded gluings of irreducible 3-manifolds chosen from a finite list. We prove effective hyperbolization and effective rigidity for a broad class of 3-manifolds of bounded type and large gluing heights. Specifically, we show the existence and uniqueness of hyperbolic metrics on 3-manifolds of bounded type and large heights, and prove existence of a bilipschitz diffeomorphism to a combinatorial model described explicitly in terms of the list of irreducible manifolds, the topology of the identification, and the combinatorics of the gluing maps.

We construct hyperbolic integer homology 3-spheres where the injectivity radius is arbitrarily large for nearly all points of the manifold. As a consequence, there exists a sequence of closed hyperbolic 3-manifolds which Benjamini-Schramm converge to H^3 whose normalized Ray-Singer analytic torsions do not converge to the L^2-analytic torsion of H^3. This contrasts with the work of Abert et. al. who showed that Benjamini-Schramm convergence forces convergence of normalized betti numbers. Our results shed light on a conjecture of Bergeron and Venkatesh on the growth of torsion in the homology of arithmetic hyperbolic 3-manifolds, and we give experimental results which support this and related conjectures., Comment: 29 pages, 11 figures. v2: Incorporates referee's comments. To appear in Geometry and Topology

We prove a continuity property for ending invariants of convergent sequences of Kleinian surface groups. We also analyze the bounded curve sets of such groups and show that their projections to non-annular subsurfaces lie a bounded Hausdorff distance from geodesics joining the projections of the ending invariants., Comment: 40 pages

We use ending laminations for Weil-Petersson geodesics to establish that bounded geometry is equivalent to bounded combinatorics for Weil-Petersson geodesic segments, rays, and lines. Further, a more general notion of non-annular bounded combinatorics, which allows arbitrarily large Dehn-twisting, corresponds to an equivalent condition for Weil-Petersson geodesics. As an application, we show the Weil-Petersson geodesic flow has compact invariant subsets with arbitrarily large topological entropy., Comment: 39 Pages, 3 figures. Minor revisions

We prove that the deformation space AH(M) of marked hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M with incompressible boundary is locally connected at minimally parabolic points. Moreover, spaces of Kleinian surface groups are locally connected at quasiconformally rigid points. Similar results are obtained for deformation spaces of acylindrical 3-manifolds and Bers slices.

We prove hyperbolic 3-manifolds are geometrically inflexible: a unit quasiconformal deformation of a Kleinian group extends to an equivariant bi-Lipschitz diffeomorphism between quotients whose pointwise bi-Lipschitz constant decays exponentially in the distance form the boundary of the convex core for points in the thick part. Estimates at points in the thin part are controlled by similar estimates on the complex lengths of short curves. We use this inflexibility to give a new proof of the convergence of pseudo-Anosov double-iteration on the quasi-Fuchsian space of a closed surface, and the resulting hyperbolization theorem for closed 3-manifolds that fiber over the circle with pseudo-Anosov monodromy., Comment: 52 pages. Final version. Appeared in Journal of Topology. Material in original version on inflexibility of hyperbolic cone-manifolds has been rewritten into a new paper with identifier arXiv:1412.4635

We define an ending lamination for a Weil-Petersson geodesic ray. Despite the lack of a natural visual boundary for the Weil-Petersson metric, these ending laminations provide an effective boundary theory that encodes much of its asymptotic CAT(0) geometry. In particular, we prove an ending lamination theorem (Theorem 1.1) for the full-measure set of rays that recur to the thick part, and we show that the association of an ending lamination embeds asymptote classes of recurrent rays into the Gromov-boundary of the curve complex. As an application, we establish fundamentals of the topological dynamics of the Weil-Petersson geodesic flow, showing density of closed orbits and topological transitivity., Comment: To appear, Geom. & Funct. Anal., referee's comments incorporated for final version

We analyze the coarse geometry of the Weil-Petersson metric on Teichm\"uller space, focusing on applications to its synthetic geometry (in particular the behavior of geodesics). We settle the question of the strong relative hyperbolicity of the Weil-Petersson metric via consideration of its coarse quasi-isometric model, the "pants graph." We show that in dimension~3 the pants graph is strongly relatively hyperbolic with respect to naturally defined product regions and show any quasi-flat lies a bounded distance from a single product. For all higher dimensions there is no non-trivial collection of subsets with respect to which it strongly relatively hyperbolic; this extends a theorem of [BDM] in dimension 6 and higher into the intermediate range (it is hyperbolic if and only if the dimension is 1 or 2 [BF]). Stability and relative stability of quasi-geodesics in dimensions up through 3 provide for a strong understanding of the behavior of geodesics and a complete description of the CAT(0)-boundary of the Weil-Petersson metric via curve-hierarchies and their associated "boundary laminations.", Comment: References added apropos of equivalence of the notion of asymptotically tree-graded and strong relative-hyperbolicity in the sense of Drutu-Sapir. We thank Jason Behrstock for bringing this to our attention. Proof of thickness in higher dimension streamlined, some comments, questions and references added

We extend a theorem of Masur and Wolf which says that given a hyperbolic surface S, every isometry of the Teichmuller space for S with the Weil-Petersson metric is induced by an element of the mapping class group for S. Our argument handles the previously untreated cases of the four-holed sphere, the one-holed torus, and the two-holed torus., Comment: 11 pages, 2 figures

Thurston's Ending Lamination Conjecture states that a hyperbolic 3-manifold N with finitely generated fundamental group is uniquely determined by its topological type and its end invariants. In this paper we prove this conjecture for Kleinian surface groups; the general case when N has incompressible ends relative to its cusps follows readily. The main ingredient is the establishment of a uniformly bilipschitz model for a Kleinian surface group. The first half of the proof appeared in math.GT/0302208, and a subsequent paper will establish the Ending Lamination Conjecture in general., Comment: 143 pages. Comprehensive revision and inclusion of the incompressible ends case

The density conjecture of Bers, Sullivan and Thurston predicts that each complete hyperbolic 3-manifold M with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We prove that the conjecture obtains for each complete hyperbolic 3-manifold with no cusps and incompressible ends., Comment: 59 pages, 2 figures

Let N be a complete hyperbolic 3-manifold that is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We show N is homeomorphic to the interior of a compact 3-manifold, or tame, if one of the following conditions holds: (1) N has non-empty conformal boundary, (2) N is not homotopy equivalent to a compression body, or (3) N is a strong limit of geometrically finite manifolds. The first case proves Ahlfors' measure conjecture for Kleinian groups in the closure of the geometrically finite locus: given any algebraic limit G of geometrically finite Kleinian groups, the limit set of G is either of Lebesgue measure zero or all of the Riemann sphere. Thus, Ahlfors' conjecture is reduced to the density conjecture of Bers, Sullivan, and Thurston., Comment: New revised version, 22 pages. To appear, Publ. I.H.E.S. This version represents a fairly substantial reorganization of the logical structure of the paper

We give an expository account of our proof that each cusp-free hyperbolic 3-manifold M with finitely generated fundamental group and incompressible ends is an algebraic limit of geometrically finite hyperbolic 3-manifolds., Comment: 19 Pages, 2 figures; to appear, proceedings of the Warwick Conference: Kleinian Groups and Hyperbolic 3-Manfiolds, September 2001

Given a closed hyperbolic 3-manifold T_\psi that fibers over the circle with monodromy \psi : S -> S, the monodromy $\psi$ determines an isometry of Teichmuller space with its Weil-Petersson metric whose translation distance ||\psi||_WP is positive. We show there is a constant K >= 1 depending only on the topology of S so that the volume of T_\psi satisfies ||\psi||_WP/K <= vol(T_\psi) <= K ||\psi||_WP., Comment: 10 pages, 2 figures

We introduce a coarse combinatorial description of the Weil-Petersson distance d_WP(X,Y) between two finite area hyperbolic Riemann surfaces X and Y. The combinatorics reveal a connection between Riemann surfaces and hyperbolic 3-manifolds conjectured by Thurston: the volume of the convex core of the quasi-Fuchsian manifold Q(X,Y) with X and Y in its boundary is comparable to the Weil-Petersson distance d_WP(X,Y). Applications include a connection of the Weil-Petersson distance with the Hausdorff dimension of the limit set and the lowest eigenvalue of the Laplacian as well as a new finiteness criterion for geometric limits., Comment: 48 pages, 14 figures. Revised final version. To appear, J. Amer. Math. Soc

Let S be a surface with genus g and n boundary components and let d(S) = 3g-3+n denote the number of curves in any pants decomposition of S. We employ metric properties of the graph of pants decompositions CP(S) prove that the Weil-Petersson metric on Teichmuller space Teich(S) is Gromov-hyperbolic if and only if d(S) <= 2. When d(S) >= 3 the Weil-Petersson metric has higher rank in the sense of Gromov (it admits a quasi-isometric embedding of R^k, k >= 2); when d(S) <= 2 we combine the hyperbolicity of the complex of curves and the relative hyperbolicity of CP(S) prove Gromov-hyperbolicity. We prove moreover that Teich(S) admits no geodesically complete Gromov-hyperbolic metric of finite covolume when d(S) >= 3, and that no complete Riemannian metric of pinched negative curvature exists on Moduli space M(S) when d(S) >= 2., Comment: 23 pages, 1 figure