\(p\)-Fundamental tone estimates of submanifolds with bounded mean curvature
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.
scientific article; zbMATH DE number 6798472
scientific article