A sphericity criterion for strictly pseudoconvex hyper surfaces in $\mathbb{C}^2$ via invariant curves

TL;DR

This work proves that for a strictly pseudoconvex hypersurface $M$ in $\\mathbb{C}^2$, if every chain is the boundary of a stationary disc, then $M$ is locally spherical. The authors analyze Fefferman’s Lorentzian construction by expressing the Fefferman Hamiltonian in Chern–Moser normal form, constructing a special family of chains near the origin, and applying moment conditions from stationarity. A weighted Taylor expansion reveals that any nonzero Cartan cubic obstruction $A_p$ would generate a detectable inconsistency, and the analysis shows that the relevant normal-form coefficient $a$ must vanish. Consequently, $A_p=0$ in a neighborhood, yielding local CR-equivalence to the sphere. This provides a sharp, invariant criterion linking chain and stationary-disc data to sphericity via Fefferman geometry and umbilicity.

Abstract

We prove that if every chain on a strictly pseudoconvex hypersurface $M$ in $\mathbb{C}^2$ coincides with the boundary of a stationary disc, then $M$ is locally spherical.

Theorems & Definitions (19)

• Theorem 1
• Theorem 2: Cartan:1932ws
• Definition 3
• Theorem 4: Chern-Moser Chern:1974wu, n=2
• Theorem 5: Cartan's umbilical tensor Cartan:1932wsCartan:1933ux
• Example 6
• Lemma 7
• proof
• Theorem 8
• Lemma 9