Minimal conformally flat hypersurfaces

Original · EN

We study conformally flat hypersurfaces f M³ → ⁴(c) with three distinct principal curvatures and constant mean curvature H in a space form with constant sectional curvature c. First we extend a theorem due to Defever when c=0 and show that there is no such hypersurface if H≠ 0. Our main results are for the minimal case H=0. If c≠ 0, we prove that if f M³ → ⁴(c) is a minimal conformally flat hypersurface with three distinct principal curvatures then f(M³) is an open subset of a generalized cone over a Clifford torus in an umbilical hypersurface ³(c)⊂ ⁴(c), c>0, with c≥ c if c>0. For c=0, we show that, besides the cone over the Clifford torus in ³⊂ ⁴, there exists precisely a one-parameter family of (congruence classes of) minimal isometric immersions f M³ → ⁴ with three distinct principal curvatures of simply-connected conformally flat Riemannian manifolds.