Prépublication n° 154
We define a non-degenerated Monge-Ampère structure on a $6$-manifold associated with a Monge-Ampère equation as a couple $(\Omega,\omega)$, such that $\Omega$ is a symplectic form and $\omega$ is a $3$-differential form which satisfies $\omega\wedge\Omega=0$ and which is non-degenerated in the sense of Hitchin. We associate with such a couple an almost (pseudo) Calabi-Yau structure and we study its integrability from the point of view of Monge-Ampère operators theory. The result we prove appears as an analogue of Lychagin and Roubtsov theorem on integrability of the almost complex or almost product structure associated with an elliptic or hyperbolic Monge-Ampère equation in the dimension $4$. We study from this point of view the example of the Stenzel metric on $T^*S^3$.
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