Curvature contraction flows in the sphere

We show that convex surfaces in an ambient three-sphere contract to round points in finite time under fully nonlinear, degree one homogeneous curvature flows, with no concavity condition on the speed. The result extends to convex axially symmetric hypersurfaces of the (n+1)-dimensional sphere. Using a different pinching function we also obtain the analogous results for contraction by Gauss curvature.