Generalized Calabi correspondence and complete spacelike surfaces

Abstract

We construct a twin correspondence between graphs with prescribed mean curvature in three-dimensional Riemannian Killing submersions and spacelike graphs with prescribed mean curvature in three-dimensional Lorentzian Killing submersions. Our duality extends the Calabi correspondence between minimal graphs in the Euclidean space R 3 and maximal graphs in the Lorentz-Minkowski spacetime L 3 , by allowing arbitrary prescribed mean curvature and bundle curvature. For instance, we transform the prescribed mean curvature equation in L 3 into the minimal surface equation in the generalized Heisenberg space with prescribed bundle curvature. We present several applications of the twin correspondence to the study of the moduli space of complete spacelike surfaces in certain Lorentzian spacetimes.2010 MSC: Primary 49Q05, 53A10; Secondary 53C50, 35B08.

Height and Area Estimates for Constant Mean Curvature Graphs in $$\mathbb {E}(\kappa ,\tau )$$ E ( κ , τ ) -Spaces

“…and {φ t } t∈R the 1-parameter group of isometries associated to ξ. It turns out that the mean curvature H(u) of F u , as a function on M , satisfies (3.1), where Gu = ∇u + Z for some vector field Z on Ω [19]. In that sense, some of our results extend without changes to the Killing-submersion setting (see Lemmas 4 and 5).…”

The Lorentzian version of a theorem of Krust

“…Between minimal surfaces and maximal surfaces there is a correspondence, called duality, that assigns to each minimal surface in E 3 a maximal surface in L 3 and viceversa (see [10,11] for generalizations in other ambient spaces). It was Calabi the first who realized of this correspondence when the surfaces are expressed as graphs on simply connected domains ( [3]).…”

On the geometry of non-degenerate surfaces in Lorentzian homogeneous $3$-manifolds

“…Since both spaces represent the same topological manifold, it makes sense to consider the same topological surface endowed with two different metrics, and to complete the classification of simultaneously minimal and maximal surfaces in those spaces started in [9,7,13]. Moreover, it is known that L 3 (κ, τ ) does not admit complete spacelike surfaces if κ + 4τ 2 > 0, see [11]. Thus, it seems reasonable to consider not only spacelike, but in general non-degenerate surfaces in L 3 (κ, τ ) such that both mean curvature functions related to the induced metrics from E 3 (κ, τ ) and L 3 (κ, τ ) vanish.…”

“…, we obtain the so-called Lorentz-Bianchi-Cartan-Vranceanu spaces (LBCV -spaces), which have been denoted in the literature by L 3 (κ, τ ) (see for instance [10,11]). In an analogous way to the Riemannian situation, it holds that ξ = ∂ z is a timelike unit Killing vector field in L 3 (κ, τ ), which is tangent to the fibers of the submersion π.…”