CMC hypersurfaces with canonical principal direction in space forms

Abstract

A hypersurface M⊂M¯ of the space form M¯ has a canonical principal direction (CPD) relative to the closed and conformal vector field Z of M¯ if the projection Z⊤ 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 S3 is a Clifford torus. Finally, we consider the case when a CPD Euclidean hypersurface has zero Gauss–Kronecker curvature.

Geometry of submanifolds with respect to ambient vector fields

“…The principal direction property with respect to the radial vector field was investigated in [17] for surfaces in R 3 and with respect to the vector field Z = ∂ ∂t tangent to the factor R for hypersurfaces of a warped product R × ρ P in [10]. Constant mean curvature hypersurfaces of R × ρ P with this property were characterized in [9], yielding a description of constant mean curvature hypersurfaces with the principal direction property with respect to closed and conformal vector fields in Q n ǫ . Our first main result is a description of the isometric immersions f : M m → R n , for arbitrary values of the dimension m and the codimension n − m, that have the constant ratio property with respect to a constant vector field, as well as of the isometric immersions f : M m → Q n ǫ × R that have the constant ratio property with respect to the unit vector field Z = ∂ ∂t tangent to the factor R. Several of the remaining results of this article rely on the elementary but useful observation that, for an isometric immersion f : M m → N n , both the principal direction and constant ratio properties with respect to a vector field Z on N n are invariant under conformal changes of the metric of the ambient space.…”