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Kiselman’s principle, the Dirichlet problem for the Monge-Ampère equation, and rooftop obstacle problems. (English) Zbl 1353.32039

This paper describes a way of obtaining solutions to the Dirichlet problem for the homogeneous real or complex Monge-Ampère equation on special product spaces: a convex tube domain \(K^{\mathbb{C}}=K\times\mathbb{R}^k\) times a Kähler manifold \(M\) in the complex case, an interval (or more generally, a convex subset of an Euclidean space) times \(\mathbb{R}^n\) in the real case.
In the complex case, the authors look for solutions invariant with respect to the natural \(\mathbb{R}^k\)-action on the tube domain \(K^{\mathbb{C}}\), and they show that such a solution can be obtained as the partial Legendre transform of a plurisubharmonic upper envelope of a family of functions obtained starting from the boundary data and parametrized by \(M\). In the real case, the solution is instead obtained as a Legendre transform of a convex upper envelope.
A number of results contained in this paper, on the regularity of the plurisubharmonic or convex upper envelope, are both useful as a tool in the proofs and interesting per se. In particular, the authors show that the plurisubharmonic upper envelope of a finite family of functions with finite \(C^2\)-norm has finite \(C^2\)-norm, proving a suitable a-priori estimate.
Reviewer: Marco Abate (Pisa)

MSC:

32W20 Complex Monge-Ampère operators
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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References:

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