The quermassintegral-preserving mean curvature flow in the sphere

We introduce a mean curvature flow with global term of convex hypersurfaces in the sphere, for which the global term can be chosen to keep any quermassintegral fixed. Then, starting from a strictly convex initial hypersurface, we prove that the flow exists for all times and converges smoothly to a geodesic sphere. This provides a workaround to an issue present in the volume-preserving mean curvature flow in the sphere introduced by Huisken (1987). We also classify solutions for some constant curvature-type equations in space forms, as well as solitons in the sphere and in the upper branch of the De Sitter space.