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# Asian Journal of Mathematics

## Volume 22 (2018)

### Number 1

### Hypersurfaces with closed Möbius form and three distinct constant Möbius principal curvatures in $\mathbb{S}^{m+1}$

Pages: 181 – 210

DOI: https://dx.doi.org/10.4310/AJM.2018.v22.n1.a5

#### Authors

#### Abstract

Let $x$ be an $m$-dimensional umbilic-free hypersurface in an $(m+1)$-dimensional unit sphere $\mathbb{S}^{m+1} (m \geq 4)$. There are four basic Möbius invariants of $x$, i.e. Möbius metric $\mathbf{g}$, Möbius form $\Phi$, Blaschke tensor $\mathbf{A}$ and Möbius second fundamental form $\mathbf{B}$. The eigenvalues of $\mathbf{B}$ are called Möbius principal curvatures. In this paper, we study hypersurfaces with closed Möbius form and three distinct constant Möbius principal curvatures, and give the Classification Theorem. Moreover, we give new Willmore hypersurfaces, which can be seen that they aren’t Cartan minimal or Möbius isoparametric hypersurfaces.

#### Keywords

Möbius geometry, Möbius form, Möbius principal curvature

#### 2010 Mathematics Subject Classification

53A30, 53C21, 53C40

Received 11 April 2015

Accepted 22 August 2017

Published 10 May 2018