The complex Monge–Ampère equation, Zoll metrics and algebraization

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Let M be a real analytic Riemannian manifold. An adapted complex structure on TM is a complex structure on a neighborhood of the zero section such that the leaves of the Riemann foliation are complex submanifolds. This structure is called entire if it may be extended to the whole of TM. We prove here that the only real analytic Zoll metric on the n-sphere with an entire adapted complex structure on TM is the round sphere. Using similar ideas, we answer a special case of an algebraization question raised by the first author, characterizing some Stein manifolds as affine algebraic in terms of plurisubharmonic exhaustion functions satisfying the homogeneous complex Monge–Ampere equation. The result presented here is an extension to higher dimensions of an observation attributed to W. Stoll for the case of Riemann surfaces.