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On the first eigenvalue of the linearized operator of the higher order mean curvature for closed hypersurfaces in space forms. (English) Zbl 1038.53061

Let \(M^n\) be a closed hypersurface immersed into an \((n+1)\)-dimensional Riemannian space form \(M^{n+1}\). The authors give sharp upper bounds for the first positive eigenvalue of the linearized operator of the higher order mean curvature of \(M^n\). Their bounds are extrinsic in the sense that they are given in terms of the higher order mean curvatures and the center(s) of gravity of \(M^n\). The bounds found in the paper under review extend the bounds for the first positive eigenvalue of the Laplacian operator recently found in the paper [A. R. Veeravalli, Comment. Math. Helv. 76, 155–160 (2001; Zbl 0998.53035)].

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C40 Global submanifolds
53C20 Global Riemannian geometry, including pinching

Citations:

Zbl 0998.53035
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