The Cauchy problem for the homogeneous Monge-Ampere equation, II. Legendre transform

Abstract

We continue our study of the Cauchy problem for the homogeneous (real and complex) Monge-Ampere equation (HRMA/HCMA). In the prequel a quantum mechanical approach for solving the HCMA was developed, and was shown to coincide with the well-known Legendre transform approach in the case of the HRMA. In this article---that uses tools of convex analysis and can be read independently---we prove that the candidate solution produced by these methods ceases to solve the HRMA, even in a weak sense, as soon as it ceases to be differentiable.

v2: added references

References in corpus (3)

• Kahler geometry on toric manifolds, and some other manifolds with large symmetry
• The Cauchy problem for the homogeneous Monge-Ampere equation, II. Legendre transform
• The Cauchy problem for the homogeneous Monge-Ampere equation, III. Lifespan

Cited by in corpus (8)

• Random Kähler Metrics
• The Cauchy problem for the homogeneous Monge-Ampere equation, II. Legendre transform
• Kiselman's principle, the Dirichlet problem for the Monge-Ampere equation, and rooftop obstacle problems
• The Cauchy problem for the homogeneous Monge-Ampere equation, III. Lifespan
• Analysis of polarity
• The exponential map of the complexification of {\em Ham} in the real-analytic case
• Chebyshev potentials, Fubini--Study metrics, and geometry of the space of Kähler metrics
• A new approximation method for geodesics on the space of Kähler metrics using complexified symplectomorphisms and Gröbner Lie series