Bisectors in the Heisenberg group I

TL;DR

This paper characterizes metric bisectors in the Heisenberg group with the Korányi metric as spinal spheres and computes their horizontal mean curvature explicitly.

Contribution

It establishes a precise geometric correspondence between bisectors and spinal spheres in the Heisenberg group and provides explicit curvature calculations.

Findings

Bisectors are spinal spheres in the Heisenberg group.

Explicit formulas for horizontal mean curvature of bisectors.

Bidirectional characterization between bisectors and spinal spheres.

Abstract

We show that metric bisectors with respect to the Kor\'anyi metric in the Heisenberg group are spinal spheres and vice versa. We also calculate explicitly their horizontal mean curvature.