Smooth Convergence of Inverse Gauss Curvature Flow on the Two-Sphere

We prove smooth convergence of the normalized inverse Gauss curvature flow on the two-sphere for every smooth initial conformal metric of positive Gauss curvature. No smallness or symmetry assumption is imposed, and convergence takes place in the original conformal coordinates. The curvature-weighted distance estimate of Gursky and Streets, together with Darvas's endpoint comparison, controls background energy. Curvature entropy then controls the conformal factor through the spherical Green kernel. Two maximum-principle estimates give time-uniform upper and lower curvature bounds. An exact change of variables to inverse Monge-Ampere flow supplies higher regularity, and distance contraction selects a single round limit.

This result answers the explicit two-dimensional component of AIM workshop Problem 3.3, recorded as AIM-GEOMETRY-0171 in UnsolvedMath v1.6.0. The source's separate higher-dimensional question is not resolved. Established estimates retain attribution; no absolute priority claim is made. This is an AI-assisted, self-audited and unrefereed preprint, without independent review or formal proof-assistant verification. The author is responsible for the claims.

The source package includes a portable exact-arithmetic checker and its two matching 20,057-check outputs. These verify finite algebraic identities and boundary examples only; the general convergence theorem is the written analytic proof.