A Faber-Krahn inequality for solutions of Schrödinger's equation on Riemannian manifolds (Q1651360)

Let \((M,g)\) be a Riemannian manifold with constant sectional curvature. Let \(\Omega\) be a bounded open subset of \(M\). Let \(V\in L^\infty(\Omega)\) and let \(0\neq u\) satisfy \(-\Delta u=V u\) in \(\Omega\). If \(M\) is simply connected (i.e., if \(M\) is flat, or the sphere, or hyperbolic space), then the authors establish a sharp inequality relating the \(L^\infty\) norm of \(V\) and the first eigenvalue of the Laplacian while if \(M\) is not necessarily simply connected, a lower bound for the \(L^\infty\) norm of \(V\) is obtained in terms of the isoperimetric Cheeger constant.