Minimal surface systems, maximal surface systems and special Lagrangian equations

Abstract

Abstract. We extend Calabi's correspondence between minimal graphs in Euclidean space R 3 and maximal graphs in Lorentz-Minkowski space L 3 . We establish the twin correspondence between 2-dimensional minimal graphs in Euclidean space R n+2 carrying a positive area-angle function and 2-dimensional maximal graphs in pseudo-Euclidean space R n+2 n carrying the same positive area-angle function.We generalize Osserman's Lemma on degenerate Gauss maps of entire 2-dimensional minimal graphs in R n+2 and offer several Bernstein-Calabi type theorems. A simultaneous application of Harvey-Lawson Theorem on special Lagrangian equation and our extended Osserman's Lemma yields a geometric proof of Jörgens' Theorem on 2-variables unimodular Hessian equation.We introduce the correspondence from 2-dimensional minimal graphs in R n+2 to special Lagrangian graphs in C 2 , which induces an explicit correspondence from 2-variables symplectic Monge-Ampére equations to 2-variables unimodular Hessian equation.

On the duality between rotational minimal surfaces and maximal surfaces

“…The maximal surfaces with this Weierstrass representation are the Bonnet maximal surfaces in L 3 obtained by Leite in [13]: compare the Weierstrass data (10) with the analogous surfaces in Euclidean space which will appear in (11) and the parametrization X t in (9) with the parametrization of a Bonnet minimal surface of E 3 described in (12). With the change e z → z, the isotropic curve of M is…”

“…If Min and Max denote the family of minimal surfaces of E 3 and the maximal surfaces of L 3 , respectively, we have established two maps : Min → Max, : Max → Min with the property that • and • are the identities in Max and Min respectively. We say that M (or M ) is the dual surface of M (also named in the literature as twin surface [8,12,15]). This process of duality has been generalized in other ambient spaces: see for example, [1,12,19] In this paper we are interested in a problem posed by Araujo and Leite in [3] that asks whether the dual surfaces of two congruent minimal (or maximal) surfaces are also congruent.…”

Extensions of the duality between minimal surfaces and maximal surfaces

“…We extend Calabi's correspondence to higher codimension as follows. [43]). We have the twin correspondence (up to vertical translations) between two dimensional minimal graphs having (not necessarily constant) positive area angle functions in Euclidean space R n+2 endowed with the metric dx 1 2 + · · · + dx n+2 2 and two dimensional maximal graphs having the same positive area angle functions in pseudo-Euclidean space R n+2 n equipped with the metric dx…”

“…The main aim of this survey based on author's work [39,40,41,42,43] is to illustrate geometric applications of Poincaré's Lemma to constant mean curvature equations. It implements Shiffman's 1956 proposal (section 0.3.3).…”

Generalized Calabi correspondence and complete spacelike surfaces

“…In the particular case τ = H = 0, the duality reduces to the Albujer-Alías duality [2]. It is worth mentioning that there are also extensions to higher codimension, see [25].…”