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We prove local-in-time existence of smooth solutions to the semigeostrophicequations in the general setting of smooth, bounded and simply connecteddomains of $\mathbb{R}^2$ endowed with an arbitrary conformally flat metric andnon-vanishing Coriolis term. We present a construction taking place in Euleriancoordinates, avoiding the classical reformulation in dual variables, used inthe flat case with constant Coriolis force, but lacking in this generalframework.

Local existence of smooth solutions for the semigeostrophic equations on curved domains