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We prove an upper bound on the systolic ratio of an orientable isometricfilling of the circle equipped with a Riemannian metric. The bound depends onlyon the genus of isometric filling. We also apply the bound to the class oforientable isometric filling with a certain lower bound on the systole. Wededuce that the Filling Area Conjecture holds true for this class when thegenus is sufficiently large.

A Systolic Inequality for the Filling Area Conjecture