On the nullity of conformal Killing graphs in foliated Riemannian spaces

We deal with entire conformal Killing graphs, that is, graphs constructed through the flow generated by a complete conformal Killing vector field V on a Riemannian space ({overline{M}}) , and which are defined over an integral leaf of the foliation ({V^bot {rm of} overline{M}}) orthogonal to V. Under a suitable restriction on the norm of the gradient of the function z which determines such a graph Σ(z), we establish sufficient conditions to ensure that Σ(z) is totally geodesic. Afterwards, when the ambient space ({overline{M}}) has constant sectional curvature, we obtain lower estimates for the index of minimum relative nullity of Σ(z).