Dual quadratic differentials and entire minimal graphs in Heisenberg space

Abstract

We define holomorphic quadratic differentials for spacelike surfaces with constant mean curvature in the Lorentzian homogeneous spaces L(κ, τ ) with isometry group of dimension 4, which are dual to the Abresch-Rosenberg differentials in the Riemannian counterparts E(κ, τ ), and obtain some consequences. On the one hand, we give a very short proof of the Bernstein problem in Heisenberg space, and provide a geometric description of the family of entire graphs sharing the same differential in terms of a 2-parameter conformal deformation. On the other hand, we prove that entire minimal graphs in Heisenberg space have negative Gauss curvature.

Conjugate Plateau constructions in product spaces

“…4.6.3] and the references therein. Note that, by the Fernández and Mira's solution to the Bernstein problem [25] (see also [65]), there are plenty of entire graphs with critical mean curvature, generically a two-parameter family of them for each choice of a holomorphic quadratic differential on the complex plane C (not identically zero) or on the unit disk D ⊆ C.…”

Isoparametric surfaces in $E(\kappa,\tau)$-spaces

“…The modulus of Q yields a geometric function given by q(p) = 1 4 |Q p (u, u)| 2 , see [18]. The function q ∈ C ∞ (Σ) does not depend upon the choice of the unitary vector u ∈ T p M , p ∈ M , and can be expressed in terms of the angle function ν = N, ξ , its gradient, and det(A) as follows (see [14,…”

A duality for prescribed mean curvature graphs in Riemannian and Lorentzian Killing submersions

“…This yields a geometric connection between two apparently different theories which helps understand some geometric features. For instance, Fernández and Mira's classification [11] of entire minimal graphs in Heisenberg space Nil 3 = 𝔼(0, , see [18]. Also, the third author and Nelli [19] showed that gradient estimates for entire minimal graphs in Nil 3 are related to the Cheng and Yau's estimates [8] for the dual graphs in 𝕃 3 .…”

“…This yields a geometric connection between two apparently different theories which helps understand some geometric features. For instance, Fernández and Mira's classification [11] of entire minimal graphs in Heisenberg space $$\mathrm{Nil}_3=\mathbb {E}(0,\frac{1}{2})$$ becomes transparent by considering the dual entire space‐like graphs in $$\mathbb {L}^3$$ with constant mean curvature $$\frac{1}{2}$$, see [18]. Also, the third author and Nelli [19] showed that gradient estimates for entire minimal graphs in $$\mathrm{Nil}_3$$ are related to the Cheng and Yau's estimates [8] for the dual graphs in $$\mathbb {L}^3$$.…”