The Complex Monge-Amp\`ere Equation, Zoll Metrics and Algebraization

TL;DR

This paper proves that the only Zoll metric on a sphere with a global adapted complex structure is the round sphere and explores algebraization of Stein manifolds via solutions to the homogeneous complex Monge-Ampère equation.

Contribution

It establishes the uniqueness of the Zoll metric on the sphere with an entire adapted complex structure and addresses a specific algebraization problem using HCMA solutions.

Findings

Uniqueness of the Zoll metric on the sphere with entire adapted complex structure

Characterization of certain Stein manifolds as affine algebraic via HCMA solutions

Extension of complex structures on tangent bundles of real analytic manifolds

Abstract

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-Amp\`ere (HCMA) equation.