Minimal Conformally Flat Hypersurfaces

Abstract

We study conformally flat hypersurfaces $$f:M^{3} \rightarrow {\mathbb {Q}}^{4}(c)$$ 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 $$H\ne 0$$ . Our main results are for the minimal case $$H=0$$ . If $$c\ne 0$$ , we prove that if $$f:M^{3} \rightarrow {\mathbb {Q}}^{4}(c)$$ is a minimal conformally flat hypersurface with three distinct principal curvatures then $$f(M^3)$$ is an open subset of a generalized cone over a Clifford torus in an umbilical hypersurface $${\mathbb {Q}}^{3}({\tilde{c}})\subset {\mathbb {Q}}^4(c)$$ , $$\tilde{c}>0$$ , with $${\tilde{c}}\ge c$$ if $$c>0$$ . For $$c=0$$ , we show that, besides the cone over the Clifford torus in $${{\mathbb {S}}}^3\subset {{\mathbb {R}}}^4$$ , there exists precisely a one-parameter family of (congruence classes of) minimal isometric immersions $$f:M^3 \rightarrow {{\mathbb {R}}}^4$$ with three distinct principal curvatures of simply connected conformally flat Riemannian manifolds.