New curvature inequalities for hypersurfaces in the Euclidean ambient space (Q390062)

The authors investigate curvature inequalities for hypersurfaces in the Euclidean ambient space. Let \(\sigma: U\subset \mathbb{R}^{n}\longrightarrow \mathbb{R}^{n+1}\) be a hypersurface with second fundamental form \(h\). They prove that the spread \(s(L)\) of the shape operator \(L\) satisfies the inequalities NEWLINE\[NEWLINE \frac{2}{n}\sqrt{\delta}\leq s(L)\leq \sqrt{\frac{2}{n}}\sqrt{\delta}, NEWLINE\]NEWLINE where \(\delta= n \|h\|^{2} - n^{2} H^{2}\) and \(\|h\|=\sum_{ij} (h_{ij})^{2}\). They also prove that the amalgamatic curvature \(A(p)\) and the absolute mean curvature \(\overline{H}(p)\) at any point \(p\) satisfy NEWLINE\[NEWLINE \overline{H}(p)\geq A(p)NEWLINE\]NEWLINE with equality at all absolutely umbilical points. Here NEWLINE\[NEWLINEA(p)= \frac{2}{n(n-1)}\sum_{1\leq i\leq j\leq n}\frac{2|k_{i}||k_{j}|}{|k_{i}|+ |k_{j}|}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\overline{H}(p)=\frac{1}{n}\{|k_{1}|+ |k_{2}|+\cdots +|k_{n}|\}.NEWLINE\]