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The most complete results so far have been obtained when the equation is elliptic. Monge–Ampère equations frequently arise in differential geometry, for example, in the Weyl and Minkowski problems in differential geometry of surfaces. They were first studied by Gaspard Monge in 1784 [1] and later by André-Marie Ampère in 1820. [2]
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
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
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 ).
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.
The normal curvature, k n, is the curvature of the curve projected onto the plane containing the curve's tangent T and the surface normal u; the geodesic curvature, k g, is the curvature of the curve projected onto the surface's tangent plane; and the geodesic torsion (or relative torsion), τ r, measures the rate of change of the surface ...
The product k 1 k 2 of the two principal curvatures is the Gaussian curvature, K, and the average (k 1 + k 2)/2 is the mean curvature, H. If at least one of the principal curvatures is zero at every point, then the Gaussian curvature will be 0 and the surface is a developable surface. For a minimal surface, the mean curvature is zero at every ...
The scalar curvature is the total trace of the Riemannian curvature tensor, a smooth function on the manifold (,), and in the Kähler case the condition that the scalar curvature is constant admits a transformation into an equation similar to the complex Monge–Ampere equation of the Kähler–Einstein setting.