Closed Minimal Willimore Hypersurfaces of <b>S</b>⁵(1) with Constant Scalar Curvature

Abstract

We consider minimal closed hypersurfaces M 4 ⊂ S 5 (1) with constant scalar curvature. We prove that, if M 4 is additionally a Willmore hypersurface, then it is isoparametric. This gives a positive answer to the question made by Chern about the pinching of the scalar curvature for closed minimal Willmore hypersurfaces in dimension 4.

Closed Hypersurfaces of $$\mathbb{S}^4$$ with Two Constant Curvature Functions

“…By using exactly the same methods of proof as in Lusala-Scherfner-Sousa (cf. [13]) we have the following inequality…”

“…We want to point out that Chern's conjecture mentioned above is still unsolved for n ≥ 4. An interesting result, recently published (see [13]), gives a partial answer for the case n = 4. It is due to T. Lusala, M. Scherfner and L. A. M. de Sousa Jr. and reads: Theorem 1.4.…”

On the Chern conjecture for isoparametric hypersurfaces

“…The method of [dB90] was taken to deal with 4 and 6 dimensional cases. In the case n = 4, Lusala-Scherfner-Sousa [LSS05] showed that a closed, minimal, Willmore hypersurface M 4 of S 5 with non-negative constant scalar curvature is isoparametric. Denoting the r-th power sum of principal curvatures by f r , the Willmore condition for minimal hypersurfaces with constant scalar curvature is equivalent to the condition that f 3 = 0.…”

An intrinsic rigidity theorem for closed minimal hypersurfaces in 5-dimensional Euclidean sphere with constant nonnegative scalar curvature

“…For n 4, the Chern conjecture remains open, although some partial result exist for low dimensions and with additional conditions for the curvature functions, such as: Theorem 1.1. [8] Let M 4 be a closed minimal Willmore hypersurface in S 5 with constant nonnegative scalar curvature. Then M 4 is isoparametric.…”