On the conformal duality between constant mean curvature surfaces in $\mathbb{E}(\kappa,\tau)$ and $\mathbb{L}(\kappa,\tau)$

TL;DR

This survey explores the conformal duality between constant mean curvature surfaces in Riemannian and Lorentzian 3-manifolds, revealing geometric insights and applications in various spaces.

Contribution

It generalizes the Calabi type duality in terms of conformal immersions, linking features of mean curvature surfaces across different geometries.

Findings

Duality swaps mean curvature and bundle curvature.

Geometric interpretations of surfaces in hyperbolic and Heisenberg spaces.

Applications include gradient estimates and existence results for spacelike surfaces.

Abstract

The main aim of this survey paper is to gather together some results concerning the Calabi type duality discovered by Hojoo Lee between certain families of (spacelike) graphs with constant mean curvature in Riemannian and Lorentzian homogeneous 3-manifolds with isometry group of dimension 4. The duality is conformal and swaps mean curvature and bundle curvature, and we will revisit it by giving a more general statement in terms of conformal immersions. This will show that some features in the theory of surfaces with mean curvature $\frac{1}{2}$ in $\mathbb{H}^2\times\mathbb{R}$ or minimal surfaces in the Heisenberg space have nice geometric interpretations in terms of their dual Lorentzian counterparts. We will briefly discuss some applications such as gradient estimates for entire minimal graphs in Heisenberg space or the existence of complete spacelike surfaces, and we will also give an uniform treatment to the behavior of the duality with respect to ambient isometries. Finally, some open questions are posed in the last section.