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The Monge cone at a given point (x 0, ..., x n) is the zero locus of the equation in the tangent space at the point. The Monge equation is unrelated to the (second-order) Monge–Ampère equation . References
The Monge gauge has two obvious limitations: If the average surface is not plane, then the Monge gauge only makes sense on length scales smaller than the curvature of the average surface. And the Monge gauge fails completely if the surface is so strongly bent that there are overhangs (points x,y corresponding to more than one z).
Curvature of general surfaces was first studied by Euler. In 1760 [4] he proved a formula for the curvature of a plane section of a surface and in 1771 [5] he considered surfaces represented in a parametric form. Monge laid down the foundations of their theory in his classical memoir L'application de l'analyse à la géometrie which
In mathematics, a (real) Monge–Ampère equation is a nonlinear second-order partial differential equation of special kind. A second-order equation for the unknown function u of two variables x,y is of Monge–Ampère type if it is linear in the determinant of the Hessian matrix of u and in the second-order partial derivatives of u.
The key observation is that the Legendre transform of a solution of the Monge–Ampère equation has its graph's Gaussian curvature prescribed by a simple formula depending on the "right-hand side" of the Monge–Ampère equation.
The Monge-Ampère equation is an essential component of the Monge-Kantorovich transport problem; it is used in conformal, affine, Kähler geometries, in meteorology and in financial mathematics. Pogorelov once said about the Monge–Ampère equation: this is a great equation with which I had the honor to work.
Gaspard Monge, Comte de Péluse (French pronunciation: [ɡaspaʁ mɔ̃ʒ kɔ̃t də pelyz]; 9 May 1746 [2] – 28 July 1818) [3] was a French mathematician, commonly presented as the inventor of descriptive geometry, [4] [5] (the mathematical basis of) technical drawing, and the father of differential geometry. [6]
However, Yau's analysis of the complex Monge–Ampère equation in resolving the Calabi conjecture was sufficiently general so as to also resolve the existence of Kähler–Einstein metrics of negative scalar curvature. The third and final case of positive scalar curvature was resolved in the 2010s, in part by making use of the Calabi conjecture.