Riemannian foliations and eigenvalue comparison (Q1271518)

The authors provide several lower and upper bounds for eigenvalues of the basic Laplacian \(\Delta_B\) on a Riemannian manifold \(M\) equipped with a Riemannian foliation \(\mathcal F\) (and a bundle-like metric). Here are some of their results: (1) \(\lambda^B_j\geq\lambda_j\) for all \(j > 0\) (here, \(\lambda^B_j\) are eigenvalues of \(\Delta_B\) while \(\lambda_j\) eigenvalues of \(\Delta\), the standard Laplace operator on \(M\)), (2) \(\lambda^B\geq\frac{1}{4}h_B^2\), where \(\lambda_B\) is the smallest positive eigenvalue of \(\Delta_B\) and \(h_B\) is the Cheeger constant obtained while considering splittings of \(M\) into two components bounded by \(\mathcal F\)-saturated submanifolds only, (3) if the sectional curvature \(K\) of \(M\) satisfies \(K\geq\alpha\), then \(\lambda^B(L, R)\leq\lambda^B_{\alpha , h}\), where \(\lambda^B(L, R)\) is the first Dirichlet eigenvalue of \(\Delta_B\) on the tubular neighbourhood of a compact leaf closure \(L\), \(R\) is the radius of the tube, \(h\) is a sufficiently large constant and \(\lambda^R_{\alpha , h}\) is the first eigenvalue of a suitable (described explicitly in the paper) 1-dimensional eigenvalue problem with parameters \(\alpha\), \(h\) and \(R\).