On the lowest eigenvalue of the Hodge Laplacian on compact, negatively curved domains (Q1029594)

Let \(M\) be a complete, simply connected, oriented Riemannian \(n\)-manifold whose sectional curvature \(K\) satisfies \(-a^2\leq K\leq-b^2\), where \(a\geq b\geq0\). Let \(\Omega\) be a bounded domain in \(M\), which is starlike with respect to some \(x_0\in\Omega\), and let \(R\) denote the radius of the smallest ball centered at \(x_0\) that contains \(\Omega\). Let \(\mu_1^{|p|}\) denote the first eigenvalue of the Laplacian acting on forms \(\omega\) of degree \(p\) on \(\Omega\) with absolute boundary conditions (\(i_N\omega=i_Nd\omega=0\), where \(N\) is the inner unit vector field normal to \(\partial\Omega\)). The first main result of the paper asserts that, if \(p>\frac{(n-1)a}{2b}+1\), then \[ \mu_1^{[p]}\geq\frac{(2(p-1)b-(n-1)a)^2}{4}\,\text{coth}^2(bR)\;. \] This generalizes the McKean inequality (the case of functions). Moreover, it is shown that this inequality is sharp. The cases of the hyperbolic and Euclidean spaces are specially analyzed. In the second main result of the paper, \(M\) is required to satisfy slightly stronger curvature restrictions: the eigenvalues of the curvature operator are bounded below by \(-a^2\), and the sectional curvatures are bounded above by \(-b^2\). Moreover, instead of assuming that \(\Omega\) is starlike, it is assumed that \(\partial\Omega\) is \(p\)-convex for some \(p\in\{2,\dots,n-1\}\) with \(\frac{b^2}{a^2}\geq\frac{p(n-p)}{(p-1)^2}\). Then \(H^q(\Omega;\mathbb{R})=0\) for all \(q\geq p\), and \[ \sqrt{\mu_1^{[p]}}>\frac{(p-1)^2b^2-p(n-p)a^2}{2(p-1)b}\;. \]