Extensions of the duality between minimal surfaces and maximal surfaces

As a generalization of the classical duality between minimal graphs in E3 and maximal graphs in L3, we construct the duality between graphs of constant mean curvature H in Bianchi-Cartan-Vranceanu space E3(κ, τ) and spacelike graphs of constant mean curvature τ in Lorentzian Bianchi-Cartan-Vranceanu space L3(κ, H).

[1]  B. Palmer.  Spacelike constant mean curvature surfaces in pseudo-Riemannian space forms , 1990 .

[2]  J. Nitsche.  Elementary Proof of Bernstein's Theorem on Minimal Surfaces , 1957 .

[3]  Bennett Palmer,et al.  A duality result between the minimal surface equation and the maximal surface equation , 2001 .

[4]  R. López,et al.  Maximal surfaces of Riemann type in Lorentz-Minkowski space L3. , 2000 .

[5]  J. Inoguchi,et al.  Helicoids and axially symmetric minimal surfaces in 3-dimensional homogeneous spaces , 2007 .

[6]  M. L. Leite,et al.  How many maximal surfaces do correspond to one minimal surface? , 2009, Mathematical Proceedings of the Cambridge Philosophical Society.

[7]  Robert Osserman,et al.  Lectures on Minimal Surfaces. , 1991 .

[8]  R. Osserman.  A survey of minimal surfaces , 1969 .

[9]  Alma L. Albujer,et al.  Calabi–Bernstein results for maximal surfaces in Lorentzian product spaces , 2007, 0709.4363.

[10]  M. Shiffman.  ON SURFACES OF STATIONARY AREA BOUNDED BY TWO CIRCLES, OR CONVEX CURVES, IN PARALLEL PLANES* , 1956 .

[11]  Renato H. L. Pedrosa,et al.  Invariant surfaces of the Heisenberg groups , 1999 .

[12]  P. Piu,et al.  SO(2)-invariant minimal and constant mean curvature surfaces in 3-dimensional homogeneous spaces , 1995 .

[13]  Hojoo Lee.  Maximal surfaces in Lorentzian Heisenberg space , 2011 .

[14]  The Dirichlet problem for constant mean curvature surfaces in Heisenberg space , 2006, math/0612712.

[15]  H. Choi,et al.  Gauss maps of spacelike constant mean curvature hypersurfaces of Minkowski space , 1990 .

[16]  Andrejs E. Treibergs,et al.  Harmonic maps from the complex plane into surfaces with nonpositive curvature , 1995 .

[17]  Isometric immersions into 3-dimensional homogeneous manifolds , 2005, math/0503500.