\(p\)-Fundamental tone estimates of submanifolds with bounded mean curvature
type: Item
instance of: scholarly article
title: \(p\)-Fundamental tone estimates of submanifolds with bounded mean curvature
zbMATH Open document ID: 1432.53081
author: Israel Evangelista
author: Keomkyo Seo
published in: Annals of Global Analysis and Geometry
publication date: 2017-10-26T00:00:00Z
review text: The \(p\)-th fundamental tone of a Riemannian manifold \(M\) is the infimum of the (normalized) integral of the \(p\)-th norm of the gradient of functions of class \(W^{1,0}_0\) on \(M\). The fundamental tone is a natural invariant which can be characterized by a variational principle of a Dirichlet eigenvalue problem for the \(p\)-Laplacian on domains in \(M\), and it behaves well with exhaustions if \(M\) is complete and non-compact. In this paper, continuing earlier results of others as well as more recent works of their own, the authors prove various estimates of the \(p\)-th fundamental tone for immersed minimal submanifolds in Cartan-Hadamard manifolds. In addition, they also derive lower estimates of the mean curvature of leaves of codimension-one foliations of an open set of a Riemannian manifold in terms of the \(p\)-th fundamental tone of the open set.
reviewed by: Gábor Tóth
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 6798472
zbMATH Keywords: fundamental tone
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: Cartan-Hadamard manifold
MaRDI profile type: Publication
full work available at URL: s10455-017-9557-1
OpenAlex ID: W2604368841
cites work: On the first variation of a varifold
cites work: On the first eigenvalue of the \(p\)-Laplacian in a Riemannian manifold
cites work: On Bernstein–Heinz–Chern–Flanders inequalities
cites work: A direct uniqueness proof for equations involving the \(p\)-Laplace operator
cites work: Eigenvalue estimates for submanifolds with locally bounded mean curvature
cites work: Q4067705
cites work: Q4882738
cites work: On the curvatures of a piece of hypersurface in Euclidean space
cites work: Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space
cites work: \(p\)-harmonic functions and connectedness at infinity of complete submanifolds in a Riemannian manifold
cites work: Q5824009
cites work: Remark on Mean Curvature
cites work: Sharp estimates for the size of balls in the complement of a hypersurface
cites work: On the fundamental tone of minimal submanifolds with controlled extrinsic curvature
cites work: Volume growth, number of ends, and the topology of a complete submanifold
cites work: Extrinsic isoperimetry and compactification of minimal surfaces in Euclidean and hyperbolic spaces
cites work: Über Flächen mit eindeutiger Projektion auf eine Ebene, deren Krümmungen durch Ungleichungen eingeschränkt sind
cites work: An Estimate for the Curvature of Bounded Submanifolds
cites work: Eigenvalue estimates for the \(p\)-Laplace operator on manifolds
cites work: On the Equation div( | ∇u | p-2 ∇u) + λ | u | p-2 u = 0
cites work: A lower estimation of the principal frequencies of simply connected membranes
cites work: An upper bound to the spectrum of \(\Delta\) on a manifold of negative curvature
cites work: A note on Hayman's theorem on the bass note of a drum
cites work: Isoperimetric Inequalities for Extrinsic Balls in Minimal Submanifolds and their Applications
cites work: Graphs with Parallel Mean Curvature
cites work: Q4838145
cites work: Isoperimetric inequalities for submanifolds with bounded mean curvature
cites work: Q3336268
cites work: Q4309836
DOI: 10.1007/S10455-017-9557-1
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description: scientific article; zbMATH DE number 6798472