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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]
Pogorelov's method of a priori estimates was used by S.-T. Yau to obtain a priori estimates for solutions of complex Monge-Ampere equations. This was the main step in the proof of the existence of Calabi-Yau manifolds, which play an important role in theoretical physics. A Monge-Ampère equation has the form
Calabi transformed the Calabi conjecture into a non-linear partial differential equation of complex Monge–Ampère type, and showed that this equation has at most one solution, thus establishing the uniqueness of the required Kähler metric. Yau proved the Calabi conjecture by constructing a solution of this equation using the continuity ...
This partial differential equation is similar to a real Monge–Ampere equation, and is known as a complex Monge–Ampere equation, and subsequently can be studied using tools from convex analysis. Its behaviour is highly sensitive to the sign of the topological constant λ = − 1 , 0 , 1 {\displaystyle \lambda =-1,0,1} .
Name Dim Equation Applications Landau–Lifshitz model: 1+n = + Magnetic field in solids Lin–Tsien equation: 1+2 + = Liouville equation: any + = Liouville–Bratu–Gelfand equation
As a consequence of their resolution of the Minkowski problem, Cheng and Yau were able to make progress on the understanding of the Monge–Ampère equation. [CY77a] 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 ...
With Caffarelli, they studied the Yamabe equation on Euclidean space, proving a positive mass-style theorem on the asymptotic behavior of isolated singularities. In 1974, Spruck and David Hoffman extended a mean curvature -based Sobolev inequality of James H. Michael and Leon Simon to the setting of submanifolds of Riemannian manifolds . [ 4 ]
Lee's research has focused on the Yamabe problem, geometry of and analysis on CR manifolds, and differential geometry questions of general relativity (such as the constraint equations in the initial value problem of Einstein equations and existence of Einstein metrics on manifolds). [2]