A Riemannian manifold with maximal \(L^p\) spectrum (Q998814)

For any complete Riemannian manifold \(M\) and for every \(p \in [1, \infty)\), there is a Laplacian operator \(\Delta_{M,p}\) defined as the generator of a heat semigroup on \(L^p(M)\). The \(L^p\)-spectrum of \(M\) is defined to be the spectrum of \(\Delta_{M,p}\), denoted by \(\sigma(\Delta_{M,p})\). It is known from the general semigroup theory that the \(\sigma(\Delta_{M,p})\) is contained in a sector \(\Sigma_p\) with vertex at the origin and containing the half positive \(x\)-axis of the complex plane. The purpose of this paper is to show by an explicitly constructed example of a complete Riemannian manifold \(M\) that such an upper bound on the \(L^p\)-spectrum can be achieved, i.e., \(\sigma(\Delta_{M,p})=\Sigma_p\).