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The main goal of this survey is to illustrate geometric applications of thePoincar\'{e} Lemma to constant mean curvature equations. In 1970, Calabiintroduced the duality between minimal graphs in three dimensional Euclideanspace and maximal graphs in three dimensional Lorentz space. We construct twoextensions of Calabi's correspondence.

Extensions of Calabi's correspondence between minimal surfaces and maximal surfaces