On the biharmonic hypersurfaces with three distinct principal curvatures in space forms

TL;DR

This paper revisits a proof regarding biharmonic hypersurfaces with three distinct principal curvatures in space forms, identifying a special case where the original argument was incomplete, and confirms the hypersurface still has constant mean curvature.

Contribution

The paper corrects and completes the proof that biharmonic hypersurfaces with three principal curvatures in space forms have constant mean curvature, addressing a previously overlooked special case.

Findings

Identified a unique case where the polynomial resultant is zero

Confirmed hypersurfaces still have constant mean curvature in this case

Ensured the completeness of the original proof

Abstract

In [16] there was proved that any biharmonic hypersurface with at most three distinct principal curvatures in space forms has constant mean curvature. At the very last step of the proof, the argument relied on the fact that the resultant of two polynomials is a non-zero polynomial. In this paper we point out that, in fact, there is a case, and only one, when this resultant is the zero polynomial and therefore the original proof is not fully complete. Further, we prove that in this special case we still obtain that the hypersurface has constant mean curvature.