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The behavior of differentials with real periods under degeneration of the Riemann surfaceGiven a Riemann surface with marked points and prescribed singular parts at those points, there exists a unique differential with such singularities and real

2D hydrodynamics of ideal fluid with free surface is considered. A time-dependent conformal transformation is used which maps a free fluid surface into the real line with fluid domain mapped into the lower complex half-plane. The fluid dynamics is