Tohoku Mathematical Journal
2026
June
SECOND SERIES VOL. 78, NO. 2

Tohoku Math. J.
78 (2026), 197-212

Title NOTES ON THE CLASSIFICATION OF HYPERSURFACES WITH CONSTANT MÖBIUS RICCI CURVATURE IN $\MATHBB{S}^{n+1}$

Author Tongzhu Li and Bingxin Xie

(Received June 18, 2024, revised August 30, 2024)
Abstract. Let $x: M^n\to \mathbb{S}^{n+1}$ be an $n$-dimensional immersed hypersurface without umbilical points, one can define the Möbius metric $g$, which is invariant under the Möbius transformation group of $\mathbb{S}^{n+1}$. The Ricci curvature with respect to the Möbius metric $g$ is called Möbius Ricci curvature. A hypersurface is called a Willmore hypersurface if it is the critical point of the volume functional of $M^n$ with respect to the Möbius metric $g$. In this paper, sharpening a theorem of Z. Guo et al (Tohoku Math. J. 67: 383-403, 2015), we first give the classication theorem for hypersurfaces with constant Möbius Ricci curvature in the unit sphere, then as its application we classify the Willmore hypersurfaces with constant Möbius Ricci curvature in $\mathbb{S}^{n+1}$.

Mathematics Subject Classification. Primary 53A31; Secondary 53B25.
Key words and phrases. Möbius metric, Möbius Ricci curvature, Möbius sectional curvature, Willmore hypersurface.

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