Asian Journal of Mathematics

Volume 23 (2019)

Number 1

Generalized Calabi correspondence and complete spacelike surfaces

Pages: 35 – 48



Hojoo Lee (Department of Mathematical Sciences, Seoul National University, Seoul, South Korea)

José M. Manzano (Departamento de Álgebra, Geometría y Topología, Complutense University of Madrid, Spain)


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 $\mathbb{R}^3$ and maximal graphs in the Lorentz–Minkowski spacetime $\mathbb{L}^3$, by allowing arbitrary prescribed mean curvature and bundle curvature. For instance, we transform the prescribed mean curvature equation in $\mathbb{L}^3$ 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.


prescribed mean curvature, homogeneous $3$-manifolds, entire graphs, spacelike surfaces, dual correspondence

2010 Mathematics Subject Classification

Primary 49Q05, 53A10. Secondary 35B08, 53C50.

Received 17 July 2015

Accepted 13 July 2017

Published 3 May 2019