On the first eigenvalue of the linearized operator of the \(r\)-th mean curvature of a hypersurface
type: Item
instance of: scholarly article
title: On the first eigenvalue of the linearized operator of the \(r\)-th mean curvature of a hypersurface
zbMATH Open document ID: 0816.53031
DOI: 10.1007/BF00773553
author: Hilário Alencar
author: Harold Rosenberg
author: Manfredo P. do Carmo
published in: Annals of Global Analysis and Geometry
publication date: 1995-03-01T00:00:00Z
review text: 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 \(\mathbb{R}^{m + 1}\).
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 53A07
zbMATH DE Number: 727102
zbMATH Keywords: Reilly's inequality
zbMATH Keywords: first eigenvalue
zbMATH Keywords: Laplacian
zbMATH Keywords: higher-order curvature
zbMATH Keywords: \(r\)-stable submanifolds
reviewed by: Changping Wang
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 727102