Complete hypersurfaces with constant Möbius scalar curvature (Q2816973)

The authors study complete umbilical free hypersurfaces \(M^n\) in the unit sphere \(S^{n+1}\) with constant normalized Möbius scalar curvature \(R \geq \frac{n-2}{n^2}\) and vanishing Möbius form \(\Phi.\) By computing the Laplacian of the function \(|\sim{\mathbf{A}}| ^2,\) where \(\sim{\mathbf{A}}=\mathbf{A}-\frac{1}{n} \text{tr}(\mathbf{A}) g,\) with \( \mathbf{A}\) the Blaschke tensor and \( g\) the Möbius metric and by applying the well-known generalized maximum principle of Omori-Yau, they obtain the main result:NEWLINENEWLINE``\(M^n\) must be either Möbius-equivalent to a minimal hypersurface with constant Möbius scalar curvature, when \(R=\frac{n-2}{n^2};\) \(S^1(\sqrt{1-r^2}) \times S^{n-1}(r)\) in \(S^{n+1},\) when \(\frac{n-2}{n^2} < R < \frac{n-2}{n};\) the pre-image of the stereographic projection \(\sigma \) of the circular cylinder \(\mathbb R^1 \times S^{n-1}\) in \(\mathbb R^{n+1},\) when \(R=\frac{n-2}{n};\) or the pre-image of the projection \(\tau\) of the hypersuperface \(H^1(\sqrt{1+r^2}) \times S^{n-1}(r)\) in \(H^{n+1},\) when \(R> \frac{n-2}{n}.\)''