Generalized Calabi correspondence and complete spacelike surfaces

2013/01/30 by Lee, Hojoo, Manzano, José M.
#35B08 #53A10 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Primary 49Q05 #Secondary 53C50

paper · doi:10.48550/arxiv.1301.7241

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 ℝ³ and maximal graphs in the Lorentz-Minkowski spacetime \mathbbL³, by allowing arbitrary prescribed mean curvature and bundle curvature. For instance, we transform the prescribed mean curvature equation in \mathbbL³ 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.

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