On the first eigenvalue of the linearized operator of the \(r\)-th mean curvature of a hypersurface
Let \(M^ n\) be a closed submanifold in \(\mathbb{R}^{m + 1}\) and let \(H_ 1\), \(\lambda_ 1\) denote the mean curvature and the first eigenvalue of the Laplacian of \(M^ n\). \textit{R. C. Reilly} proved [Comment. Math. Helv. 52, 525-533 (1977; Zbl 0382.53038)] that \({\lambda_ 1\over n} \leq {1\over \text{Vol}(M} \int_ M H^ 2_ 1\). In this paper the authors generalize this result to the first eigenvalue of the linearized operator of the higher-order curvature. Let \(A\) be the shape operator of \(M^ n\), let \(S_ r\) be the \(r\)-th elementary symmetric function of the eigenvalues of \(A\), and let \(T_ r = S_ r I - AT_{r - 1}\) with \(T_ 0 = I\). The linearized operator \(L_ r\) of \(S_{r + 1} = \left( \begin{smallmatrix} m\\ r+1\end{smallmatrix}\right) H_{r + 1}\) is defined by \(L_ r(f) = \text{div}(T_ r \nabla f)\). They prove that if \(M\) is a closed hypersurface in \(\mathbb{R}^{n + 1}\) with \(H_{r + 1} > 0\), then the first eigenvalue \(\lambda_ 1\) of \(L_ r\) satisfies \(\lambda_ 1 \int_ M H_ r \leq c(r) \int_ M H^ 2_{r + 1}\) with \(c(r) = (m - r) \left(\begin{smallmatrix} m\\r \end{smallmatrix} \right)\), and equality holds if and only if \(M\) is a sphere. Using this inequality they give some estimates on \(\lambda_ 1\). They also obtain some results on the \(r\)- stable submanifolds in \(\mathbb{R}^{m + 1}\).
scientific article; zbMATH DE number 727102