A spherical flatness index and a stability inequality for harmonic pseudospheres

We introduce a new flatness index for the boundary of an open subset $\mathrm{\Omega }$ of ${\mathbb{R}}^n$, $n\ge 2$. This index provides a necessary condition for $\mathrm{\partial }\mathrm{\Omega }$ 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.