On the asymptotic Plateau problem in Cartan-Hadamard manifolds

In [18], Labourie defines what he calls the asymptotic Plateau problem for immersed surfaces of constant extrinsic curvature in 3-dimensional Cartan-Hadamard manifolds. We present a complete solution of this problem. Classification AMS: 53C42, 53C20, 53C21, 53C40, 53C38, 53D15, 53C26 1 Introduction. 1.1 The asymptotic Plateau problem. A standard problem in geometry is that of prescribing constant curvature submanifolds in terms of geometric data. In this paper, we will be concerned with the asymptotic Plateau problem, introduced by Labourie in [18], which asks whether immersed surfaces in 3dimensional Cartan-Hadamard manifolds, of constant extrinsic curvature and complete in a suitable sense, can be uniquely prescribed in terms of their asymptotic Gauss maps (defined below). Labourie’s pioneering work on constant extrinsic curvature surfaces has yielded deep results in the studies of hyperbolic geometry, general relativity and Teichmüller theory (see, for example, [3], [4], [5], [15], [16] and [22]) and, within this work, the above-mentioned asymptotic Plateau problem has played a significant role. Furthermore, Labourie showed in [18] and [19] that, when the ambient manifold is cocompact, the space of such surfaces may be viewed as a dynamical system which inherits the hyperbolic properties of the geodesic flow. The mathematical implications of this remarkable result still remain to be fully investigated. Let X be a 3-dimensional Cartan-Hadamard manifold and let ∂∞X denote its ideal boundary. Let SX denote the unit sphere bundle over X and define the horizon map Hor : SX → ∂∞X by Hor(ξ) := Lim t→+∞ γξ(t), (1.1) where, for all ξ ∈ X, γξ : R→ X denotes the unique geodesic such that γ̇ξ(0) = ξ. (1.2) Before formally introducing immersed surfaces, it is worth observing that the concept of smooth immersion e : Y → X is well-defined even in the absence of a differential structure over the domain Y . Indeed, given a topological surface Y , we will say that a function e : Y → X is a smooth immersion whenever there exists a smooth atlas over Y with respect to which e has this property. Trivially, any two such atlases are contained within the same maximal atlas, which we call the atlas induced by e. We define an immersed surface in X to be a pair (Y, e), where Y is a topological surface and e : Y → X is a smooth immersion. When working with immersed surfaces, we will always use the induced atlas without comment. In addition, we will always assume that the surfaces we work with are oriented. Let (Y, e) be an immersed surface in X. Let νe denote its unit normal vector field compatible with its orientation, let Ie, IIe and IIIe denote its three fundamental forms, let Ae denote its Weingarten operator, and let Ke := Det(Ae) denote its extrinsic curvature. We will say that (Y, e) is quasicomplete whenever it is complete with respect to the metric Ie + IIIe and we will say that it is infinitesimally strictly convex (ISC) whenever IIe is everywhere positive-definite. For k > 0, we say that (Y, e) is a k-surface whenever it is quasicomplete, ISC, and of constant extrinsic curvature equal to k.† When (Y, e) is a k-surface, we define its asymptotic Gauss map by φe := Hor ◦ νe, (1.3) ∗ Instituto de Matemática, Universidade Federal do Rio de Janerio, Av. Athos da Silveira Ramos 149, Centro de Tecnologia Bloco C, Cidade Universitária Ilha do Fundão, Caixa Postal 68530, 21941-909, Rio de Janeiro, RJ BRAZIL † The ISC condition is here almost redundant since, in the case of surfaces of positive extrinsic curvature, it is always satisfied upon reversing the orientation if necessary.