A spectral isoperimetric inequality on the n-sphere for the Robin-Laplacian with negative boundary parameter

For every given $\beta <0$, we study the problem of maximizing the first Robin eigenvalue of the Laplacian ${\lambda }_{\beta }(\mathrm{\Omega })$ among convex (not necessarily smooth) sets $\mathrm{\Omega }\subset {\mathbb{S}}^n$ with fixed perimeter. In particular, denoting by ${\sigma }_n$ the perimeter of the $n$-dimensional hemisphere, we show that for fixed perimeters $P<{\sigma }_n$, geodesic balls maximize the eigenvalue. Moreover, we prove a quantitative stability result for this isoperimetric inequality in terms of volume difference between $\mathrm{\Omega }$ and the ball $D$ of the same perimeter.