Extensions of the duality between minimal surfaces and maximal surfaces

Abstract

As a generalization of the classical duality between minimal graphs in E 3 and maximal graphs in L 3 , we construct the duality between graphs of constant mean curvature H in Bianchi-Cartan-Vranceanu space E 3 (κ, τ ) and spacelike graphs of constant mean curvature τ in Lorentzian Bianchi-Cartan-Vranceanu space L 3 (κ, H ).

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

“…The proof of Theorem 6 will rely on the following gradient estimate for entire spacelike graphs in Lorentz-Minkowski 3-space L 3 with constant positive mean curvature, via the Calabi-type correspondence by Lee [18,Corollary 2]. Lemma 6.…”

“…Proof of Theorem 6. By taking into account the Calabi-type correspondence in [18] we can associate to Σ a function v ∈ C ∞ (R 2 ) such that (x, y) → (x, y, v(x, y)) defines an entire spacelike graph in L 3 with constant mean curvature τ , and v satisfies the relation (1 − |∇v| 2 )(1 + |Gu| 2 ) = 1. From Lemma 6 we get that there exists A > 0 such that 1 + |Gu| 2 = (1 − |∇v| 2 ) −1 ≤ A −1 (1 + r 2 ) 2 < 1 + A −1 (1 + r 2 ) 2 , and we get item (a) by just taking B = A −1/2 .…”

Dual quadratic differentials and entire minimal graphs in Heisenberg space

“…If Σ v has constant mean curvature τ , then this includes the totally umbilical hyperboloid (f (x) = τ −2 +x 2 ), the hyperbolic cylinder (f (x) = 1 4 τ −2 ), and the semitrough given by the parametrization (x, y) → 1 H x − 1 2 coth(x), 1 2 coth(x) sinh(y), 1 2 coth(x) cosh(y) . The dual to τ -graphs of the form v(x, y) = f (x) 2 + y 2 in L 3 are minimal graphs in Nil 3 (τ ) of the form u(x, y) = yg(x) for some g ∈ C ∞ (R), as noticed by Lee [21,Example 4]. Some of them have also been studied by Daniel [9], and include umbrellas and some invariant surfaces, which will be discussed in Section 3.2.…”

“…They include the product spaces H 2 (κ) × R and S 2 (κ) × R, the Heisenberg space Nil 3 (τ ), and SU (2) and Sl 2 (R) endowed with special left-invariant metrics. There are homogeneous Lorentzian counterparts L(κ, τ ) enjoying the same description, but such that the fibers of the submersion are timelike, see [21,22]. It is important to notice that some space forms live also among them, namely the Euclidean space Other milestones in the theory of H-surfaces in E(κ, τ )-spaces were the discovery by Daniel [8] of an isometric Lawson type correspondence, and the solution to the Bernstein problem in Heisenberg space by Fernández and Mira [16].…”

Slant and Legendre curves in Bianchi-Cartan-Vranceanu geometry

“…They form a two parameters family (with parameters denoted as l and m) containing, among others, some remarkable 3-manifolds: R 3 , S 3 , S 2 × R, H 2 × R and the 3-dimensional Heisenberg group Nil 3 . Recently, several studies have been devoted to special submanifolds in these spaces: parallel surfaces [2], biharmonic curves [6], [12] and [13], constant angle surfaces [15], graphs of constant mean curvature [19], biharmonic surfaces [21], higher order parallel and totally umbilical surfaces [23].…”