CMC hypersurfaces with canonical principal direction in space forms

A hypersurface $M \subset \overline{M}$ of the space form $\overline{M}$ has a canonical principal direction (CPD) relative to the closed and conformal vector field $Z$ of $\overline{M}$ if the projection $Z^{\top}$ of $Z$ to $M$ is a principal direction of $M$. We show that CPD hypersurfaces with constant mean curvature are foliated by isoparametric hypersurfaces. In particular, we show that a CPD surface with constant mean curvature of space form M is invariant by the flow of a Killing vector field whose action is polar on M. As consequence we show that a compact CPD minimal surface of the sphere S^3 is a Clifford torus. Finally, we consider the case when a CPD Euclidean hypersurface has zero Gauss-Kronecker curvature.