Extremal discs and Segre varieties for real-analytic hypersurfaces in $\mathbb{C}^2$

TL;DR

This paper proves that for strictly pseudoconvex hypersurfaces in a7^2, the coincidence of Segre varieties with extremal discs for the Kobayashi metric characterizes the hypersurface as locally spherical, providing a new geometric criterion.

Contribution

It establishes a novel characterization of the unit sphere via the equivalence of Segre varieties and extremal discs in the Kobayashi metric for certain hypersurfaces.

Findings

Segre varieties are extremal discs only for the sphere

Characterization of the sphere via invariant geometric objects

New link between complex geometry and invariant metrics

Abstract

We show that if the Segre varieties of a strictly pseudoconvex hypersurface in $\mathbb{C}^2$ are extremal discs for the Kobayashi metric, then that hypersurface has to be locally spherical. In particular, this gives yet another characterization of the unit sphere in terms of two important invariant families of objects coinciding.