A spherical flatness index and a stability inequality for harmonic pseudospheres

TL;DR

This paper introduces a new flatness index and a stability inequality based on the Kuran gap to characterize harmonic pseudospheres and their relation to Euclidean spheres.

Contribution

It presents a novel flatness index and stability inequality that provide necessary and sufficient conditions for harmonic pseudospheres to be Euclidean spheres.

Findings

The flatness index characterizes harmonic pseudospheres.

The stability inequality links the Kuran gap to spherical shape.

Conditions for boundary flatness are established.

Abstract

We introduce a new flatness index for the boundary of an open subset $\Omega$ of $\mathbb{R}^n$, $n\ge 2$. This index provides a necessary condition for $\partial\Omega$ to be a harmonic pseudosphere and sufficient conditions for a harmonic pseudosphere to be a Euclidean sphere. These conditions will follow from a stability inequality formulated in terms of a harmonic invariant, the Kuran gap, recently introduced by the last two authors.