On the first eigenvalue of the linearized operator of the \(r\)-th mean curvature of a hypersurface

DOI10.1007/BF00773553zbMath0816.53031MaRDI QIDQ1345047

Harold Rosenberg, Hilário Alencar, Manfredo Perdigão do Carmo

Publication date: 1 March 1995

Published in: Annals of Global Analysis and Geometry (Search for Journal in Brave)

zbMATH Keywords

first eigenvalueLaplacian\(r\)-stable submanifoldshigher-order curvatureReilly's inequality

Mathematics Subject Classification ID

Higher-dimensional and -codimensional surfaces in Euclidean and related (n)-spaces (53A07) Spectral problems; spectral geometry; scattering theory on manifolds (58J50) Global submanifolds (53C40)

Related Items (27)

Eigenvalue estimates for some natural elliptic operators on hypersurfaces ⋮ Optimal upper estimates for the first eigenvalue of a Jacobi type operator in spherical and hyperbolical spaces ⋮ Stable hypersurfaces with constant scalar curvature ⋮ Stability of hypersurfaces with constant \((r+1)\)-th anisotropic mean curvature ⋮ On the 𝐿ᵣ-operators penalized by (𝑟+1)-mean curvature ⋮ Stable hypersurfaces as minima of the integral of an anisotropic mean curvature preserving a linear combination of area and volume ⋮ On scalar curvature of \(r\)-almost Yamabe solitons ⋮ Reilly type inequality for the first eigenvalue of the \(L_{r;F}\) operator ⋮ Extrinsic eigenvalues estimates for hypersurfaces in product spaces ⋮ Sharp Reilly-type inequalities for a class of elliptic operators on submanifolds ⋮ Extrinsic eigenvalues upper bounds for submanifolds in weighted manifolds ⋮ Some new characterizations of the Wulff shape ⋮ Estimates for eigenvalues of ℒr operator on self-shrinkers ⋮ On almost stable linear Weingarten hypersurfaces ⋮ Reilly-type inequalities for Paneitz and Steklov eigenvalues ⋮ Unnamed Item ⋮ Stability of area-preserving variations in space forms ⋮ Sharp upper bounds for \(\lambda _1^{L_r}\) of immersed hypersurfaces and their stability in space forms ⋮ Estimates for eigenvalues of the operator \(L_r\) ⋮ Anisotropic eigenvalues upper bounds for hypersurfaces in weighted Euclidean spaces ⋮ Pinching of the first eigenvalue for second order operators on hypersurfaces of the Euclidean space ⋮ A NEW RESULT ABOUT ALMOST UMBILICAL HYPERSURFACES OF REAL SPACE FORMS ⋮ Inequalities for eigenvalues of elliptic operators in divergence form on Riemannian manifolds ⋮ A new stability notion of closed hypersurfaces in the hyperbolic space ⋮ Fundamental tone estimates for elliptic operators in divergence form and geometric applications ⋮ A Reilly inequality for some natural elliptic operators on hypersurfaces ⋮ An extension of Hsiung-Minkowski formulas and some applications

Cites Work

• Extrinsic upper bounds for \(\lambda _ 1\)
• Sphere theorems via Alexandrov for constant Weingarten curvature hypersurfaces. Appendix to a note of A. Ros
• Stability of hypersurfaces of constant mean curvature in Riemannian manifolds
• An inequality of ``Reilly-type for submanifolds of the hyperbolic space
• On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space
• Stable hypersurfaces with constant scalar curvature
• Variational properties of functions of the mean curvatures for hypersurfaces in space forms
• Some integral formulas for closed hypersurfaces
• Unnamed Item

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