The spread of the shape operator as a curvature invariant for a smooth hypersurface (Q6551728)

Let \(A \in M_{n}(\mathbb{C})\) be an \(n\)-by-\(n\) matrix. Let \(\kappa_{1}, \dots, \kappa_{n}\) be the roots of its characteristic polynomial. The spread of \(A\) is defined as\N\[\Ns(A) = \mathrm{max}_{i,j}|\kappa_{i}-\kappa_{j}| \ .\N\]\NThe paper studies the spread of the shape operator \(B\) of an hypersurface \(\sigma : U \subset \mathbb{R}^{n} \rightarrow \mathbb{R}^{n+1}\) around an umbilical point \(p_{0}\). Note that, in this case, the roots \(\kappa_{1}, \dots, \kappa_{n}\) are the principal curvatures of the hypersurface. More precisely, the author proves that\N\[\N\lim_{p \to p_{0}} \frac{s(B)}{\sqrt{\|II\|^{2}-nH^{2}}} \in \left[\frac{2}{\sqrt{n}}, \sqrt{2} \right] ,\N\]\Nwhere \(II\) denotes the second fundamental form and \(H\) the mean curvature of \(\sigma\), assuming that such limit exists.