Entire Grauert Tubes with Closed Monge–Ampère Leaves

We study Monge–Amp`ere models with maximal-dimensional compact center, equivalently entire Grauert tubes in the global radial normal form, under the hy- pothesis that all Monge–Amp`ere leaves are closed. We show that closed leaves make the center a Besse manifold, and that each completed leaf is a spherical football whose area and cone angle determine the corresponding prime geodesic period. Ex- ceptional geodesics correspond to cone angles strictly below the maximal one. Two families of examples show how sharp this is. Finite free quotients of the hyper- quadric (lens spaces) have closed leaves and exceptional geodesics, so closedness alone does not imply Zoll. Sz˝oke’s rotationally symmetric spheres have leaves that are closed only for the round metric, so closedness is a genuine restriction. For spherical centers in the dimensions covered by Berger-type theorems, closed leaves force the hyperquadric.