A remark on the inequality of McKean
For a connected, simply connected, complete non-compact Riemannian manifold \((M,g)\) of dimension \(m\) let \(\lambda_1(M,g)\) be its first eigenvalue, i.e. the infimum by \(\delta\) among the spectral parameters \(\lambda\) for which the following Dirichlet problem is solvable: \(\Delta\phi+\lambda\phi=0\) on \(B(x,\delta)\), \(\phi=0\) on \(\partial B(x,\delta)\) (the choice of point \(x\in M\) is inessential in the limit). Let \(M={\mathbb R}\times N\), where \((N,g_0)\) is a complete Riemannian manifold of dimension \(m-1\). Equip \(M\) with the metric \(g=dt^2+e^{2w(t)}g_0\), where \(t\) is the parameter on \({\mathbb R}\). The first theorem states that if \(w'(t)\geq\kappa>0\), then \[ \lambda_1(M,g)\geq(m-1)^2\kappa^2/4. \] The result is similar to the inequality of McKean, who proved the above formula for open Riemannian manifolds of dimension \(m\) with sectional curvature \(\leq-\kappa^2<0\). Another theorem of the paper concerns bounds of \(\lambda_1(M',g| _{M'})\) for a submanifold \(M'\subset M\) via its mean curvature.
scientific article; zbMATH DE number 2038414