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We show that the infinite-dimensional space of Zoll Finsler metrics on theprojective plane strongly deformation retracts to the canonical round metric.In particular, this space of Zoll Finsler metrics is connected. Moreover, thestrong deformation retraction arises from a deformation of the geodesic flow ofevery Zoll Finsler projective plane to the geodesic flow of the round metricthrough a family of smooth free circle actions induced by the curvature flow ofthe canonical round projective plane. This construction provides a descriptionof the geodesics of the Zoll Finsler metrics along the retraction.

Strong deformation retraction of the space of Zoll Finsler projective planes