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This was first considered by Lagrange (1762), who raised the question of existence of surfaces of least area having a given closed curve in three-space as the boundary. He derived the differential equation that must be satisfied by a function of two variables whose graph minimizes area among surfaces with a given contour.
Later Meusnier discovered that this is equivalent to the vanishing of the mean curvature, and the study of the differential geometry of these surfaces was started.
the plane in $R^3$, helicoid, catenoid, Scherk surface, and Enneper surface
representation formula of Enneper and Weierstrass which expresses a given minimal surface in terms of integrals involving a holomorphic function μ and a meromorphic function ν

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