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In this paper we discuss the stability of geodesic spheres in$\mathbb{S}^{n+1}$ under constrained curvature flows. We prove that under somestandard assumptions on the speed and weight functions, the spheres are stableunder perturbations that preserve a volume type quantity. This extends resultsby Escher and Simonett, 1998, and the author, 2015, to a Riemannian manifoldsetting.

Stability of Geodesic Spheres in $\mathbb{S}^{n+1}$ under Constrained Curvature Flows