Nadirashvili presented a beautiful example showing that the Poincaré recurrence does not occur near a particular solution to the 2D Euler equation of inviscid incompressible fluids. Unfortunately, Nadirashvili’s setup of the phase space is not appropriate, and details of the proof are missing. This note fixes that.

We prove that second-order Monge-Ampère equations for one function of two variables are connected to the wave equation by a Bäcklund transformation if and only if they are integrable by the method of Darboux at second order.

In this paper, we study the singular Monge–Ampère equations on a quasi–projective manifold with a Poincaré metric. As a consequence, we construct Poincaré Kähler–Einstein metrics which degenerate or grow upward at most like a pole along a given effective divisor.