Hyperbolic Anti-de Sitter Research Articles

Wild globally hyperbolic maximal anti‐de Sitter structures

Let $\Sigma$ be a connected, oriented surface with punctures and negative Euler characteristic. We introduce wild globally hyperbolic anti-de Sitter structures on $\Sigma \times \mathbb{R}$ and provide two parameterisations of their deformation space: as a quotient of the product of two copies of the Teichm\"uller space of crowned hyperbolic surfaces and as the bundle over the Teichm\"uller space of $\Sigma$ of meromorphic quadratic differentials with poles of order at least $3$ at the punctures.

Regular globally hyperbolic maximal anti‐de Sitter structures

Let Σ be a connected, oriented surface with punctures and negative Euler characteristic. We introduce regular globally hyperbolic anti-de Sitter structures on Σ × R and provide two parameterisations of their deformation space: as an enhanced product of two copies of the Fricke space of Σ and as the bundle over the Teichmüller space of Σ whose fibre consists of meromorphic quadratic differentials with poles of order at most 2 at the punctures.

Degeneration of globally hyperbolic maximal anti-de Sitter structures along pinching sequences

Let S be a closed oriented surface of genus at least 2. Using the parameterisation of the deformation space of globally hyperbolic maximal anti-de Sitter structures on S×R by the cotangent bundle over the Teichmüller space of S, we study the behaviour of these geometric structures along pinching sequences. We show, in particular, that the regular globally hyperbolic anti-de Sitter structures introduced in [15] naturally appear as limiting points.

Symplectic Wick rotations between moduli spaces of 3-manifolds

Given a closed hyperbolic surface $S$, let $\cQF$ denote the space of quasifuchsian hyperbolic metrics on $S\times\R$ and $\cGH_{-1}$ the space of maximal globally hyperbolic anti-de Sitter metrics on $S\times\R$. We describe natural maps between (parts of) $\cQF$ and $\cGH_{-1}$, called "Wick rotations", defined in terms of special surfaces (e.g. minimal/maximal surfaces, CMC surfaces, pleated surfaces) and prove that these maps are at least $C^1$ smooth and symplectic with respect to the canonical symplectic structures on both $\cQF$ and $\cGH_{-1}$. Similar results involving the spaces of globally hyperbolic de Sitter and Minkowski metrics are also described. These 3-dimensional results are shown to be equivalent to purely 2-dimensional ones. Namely, consider the double harmonic map $\cH:T^*\cT\to\cTT$, sending a conformal structure $c$ and a holomorphic quadratic differential $q$ on $S$ to the pair of hyperbolic metrics $(m_L,m_R)$ such that the harmonic maps isotopic to the identity from $(S,c)$ to $(S,m_L)$ and to $(S,m_R)$ have, respectively, Hopf differentials equal to $i q$ and $-i q$, and the double earthquake map $\cE:\cT\times\cML\to\cTT$, sending a hyperbolic metric $m$ and a measured lamination $l$ on $S$ to the pair $(E_L(m,l), E_R(m,l))$, where $E_L$ and $E_R$ denote the left and right earthquakes. We describe how such 2-dimensional double maps are related to 3-dimensional Wick rotations and prove that they are also $C^1$ smooth and symplectic.

On the volume of Anti-de Sitter maximal globally hyperbolic three-manifolds

We study the volume of maximal globally hyperbolic Anti-de Sitter manifolds containing a closed orientable Cauchy surface S, in relation to some geometric invariants depending only on the two points in Teichmüller space of S provided by Mess’ parameterization—namely on two isotopy classes of hyperbolic metrics h and h′ on S. The main result of the paper is that the volume coarsely behaves like the minima of the L 1-energy of maps from (S, h) to (S, h′). The study of L p -type energies had been suggested by Thurston, in contrast with the well-studied Lipschitz distance. A corollary of our result shows that the volume of maximal globally hyperbolic Anti-de Sitter manifolds is bounded from above by the exponential of (any of the two) Thurston’s Lipschitz asymmetric distances, up to some explicit constants. Although there is no such bound from below, we provide examples in which this behavior is actually realized. We prove instead that the volume is bounded from below by the exponential of the Weil–Petersson distance. The proof of the main result uses more precise estimates on the behavior of the volume, which is proved to be coarsely equivalent to the length of the (left or right) measured geodesic lamination of earthquake from (S, h) to (S, h′), and to the minima of the holomorphic 1-energy.

Astronomical time-of-flight photon speedometer.

A dual-band, fiber-optic, photon time-of-flight instrument was developed. Its design was optimized for measuring the velocity of visible photons emanating from relatively dim astronomical sources (apparent magnitude m>12), such as distant galaxies and quasars. We report the first direct photon group velocity measurements for extragalactic objects. The photon group velocity is found to be 3.00±0.03×108 ms-1 and is invariant, within experimental error, over the range of redshifts measured (0≤z≤1.33). This measurement provides additional validation of general relativity and is consistent with the Friedmann-Lemaître-Robertson-Walker and hyperbolic anti-de Sitter metrics but not with the elliptical de Sitter metric.

Fixed points of compositions of earthquakes

Let S be a closed surface of genus at least 2, and let λ and μ be two laminations that fill S. Let Erλ and Erμ be the right earthquakes on λ and μ, respectively. We show that the composition Erλ∘Erμ has a fixed point in the Teichmüller space of S. Another way to state this result is that it is possible to prescribe any two measured laminations that fill a surface as the upper and lower measured bending laminations of the convex core of a globally hyperbolic anti-de Sitter (AdS) manifold. The proof uses some estimates from the geometry of those AdS manifolds.

Minimal surfaces and particles in 3-manifolds

We use minimal (or CMC) surfaces to describe 3-dimensional hyperbolic, anti-de Sitter, de Sitter or Minkowski manifolds. We consider whether these manifolds admit ``nice'' foliations and explicit metrics, and whether the space of these metrics has a simple description in terms of Teichm\uller theory. In the hyperbolic settings both questions have positive answers for a certain subset of the quasi-Fuchsian manifolds: those containing a closed surface with principal curvatures at most 1. We show that this subset is parameterized by an open domain of the cotangent bundle of Teichm\uller space. These results are extended to ``quasi-Fuchsian'' manifolds with conical singularities along infinite lines, known in the physics literature as ``massive, spin-less particles''. Things work better for globally hyperbolic anti-de Sitter manifolds: the parameterization by the cotangent of Teichm\uller space works for all manifolds. There is another description of this moduli space as the product two copies of Teichm\uller space due to Mess. Using the maximal surface description, we propose a new parameterization by two copies of Teichm\uller space, alternative to that of Mess, and extend all the results to manifolds with conical singularities along time-like lines. Similar results are obtained for de Sitter or Minkowski manifolds. Finally, for all four settings, we show that the symplectic form on the moduli space of 3-manifolds that comes from parameterization by the cotangent bundle of Teichm\uller space is the same as the 3-dimensional gravity one.