Upper bounds for the first eigenvalue of a Jacobi type operator via anisotropic mean curvatures. (English) Zbl 1359.53008
Summary: Our purpose in this article is to obtain sharp upper estimates for the first positive eigenvalue of a Jacobi type operator, which is a suitable extension of the linearized operators of the higher order mean curvatures of a closed hypersurface immersed in the Euclidean space, through its anisotropic mean curvatures.
MSC:
| 53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
| 53C20 | Global Riemannian geometry, including pinching |
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
Keywords:
first positive eigenvalue of Jacobi operator; \(k\)th anisotropic mean curvatures; geodesic sphere; Wulff shapeReferences:
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