# Minimal conformally flat hypersurfaces

## TL;DR

This paper classifies minimal conformally flat hypersurfaces with three distinct principal curvatures in space forms, showing they are generalized cones over Clifford tori or belong to a unique family in Euclidean space.

## Contribution

It extends classification results for conformally flat hypersurfaces, especially characterizing minimal cases with three principal curvatures in space forms.

## Key findings

- No such hypersurfaces exist if mean curvature H is non-zero.
- Minimal conformally flat hypersurfaces with three principal curvatures are generalized cones over Clifford tori when c≠0.
- In Euclidean space, a unique one-parameter family of such hypersurfaces exists.

## Abstract

We study conformally flat hypersurfaces $f\colon M^{3} \to \Q^{4}(c)$ 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 $H\neq 0$. Our main results are for the minimal case $H=0$. If $c\neq 0$, we prove that if $f\colon M^{3} \to \Q^{4}(c)$ 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 $\Q^{3}(\tilde c)\subset \Q^4(c)$, $\tilde c>0$, with $\tilde c\geq c$ if $c>0$. For $c=0$, we show that, besides the cone over the Clifford torus in $\Sf^3\subset \R^4$, there exists precisely a one-parameter family of (congruence classes of) minimal isometric immersions $f\colon M^3 \to \R^4$ with three distinct principal curvatures of simply-connected conformally flat Riemannian manifolds.