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Holomorphic discs and the complex Monge-Ampère equation. (English) Zbl 1035.53102

Continuing a sequence of his previous papers on connections between Kähler manifolds and symplectic geometry [see Fields Medallists’ Lectures, World Scientific Publishing, 384–403 (1997; Zbl 0913.01020), and Asian J. Math. 3, 1–15 (1999; Zbl 0999.53053)], the author studies a version of the Dirichlet problem for the homogeneous complex Monge-Ampère equation:
For a compact Kähler \(n\)-manifold \((V,\omega_0)\) with the lift \(\Omega_0\) of \(\omega_0\) to the product \(D\times V\), \(D=\{z\in\mathbb C: | z| \leqq 1\}\), and for a given smooth function \(F: \partial D\times V\to \mathbb R\) such that \(\Omega_0 + i\overline{\partial}\partial F\) is positive on the vertical slices over \(\partial D\), find a smooth function \(\Phi\,:\, D\times V\to \mathbb R\) which agrees with \(F\) on the boundary, with \(\Omega_0+i\overline{\partial}\partial\Phi\) positive on all vertical slices over \(D\), and which satisfies in \(D\times V\) the Monge-Ampère equation: \((\Omega_0+ i\overline{\partial} \partial\Phi)^{n+1}=0\,.\) Previously in “Symmetric spaces, Kähler geometry and Hamiltonian dynamics” [Transl., Ser. 2, Am. Math. Soc. 196, 13–33 (1999; Zbl 0972.53025)] the author proved uniqueness of a solution, if it exists. Then X. Chen [J. Differ. Geom. 56, 189–234 (2000; Zbl 1041.58003)] found \(C^{1,1}\) solutions to the problem.
The first result of the present paper shows that the set of smooth functions \(F\) for which a smooth solution to the problem exists is open in \(C^{\infty}(D\times V)\). However as the second result, the author constructs (by using a new alternative geometric interpretation of the problem) some functions \(F\) for which the problem does not have a smooth solution.

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
32W20 Complex Monge-Ampère operators
32Q15 Kähler manifolds
58J32 Boundary value problems on manifolds
53D15 Almost contact and almost symplectic manifolds
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