Upper bounds for invariant eigenvalues of the Laplacian
by Emily Dryden (Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, IST)
To every Riemannian metric g on a Riemannian manifold M, we can associate the Laplacian Δ = -div grad, and its spectrum. The kth eigenvalue in this spectrum, suitably normalised, can be seen as a functional on the space of Riemannian metrics. The search for critical or extremal metrics for this functional has motivated much work in spectral geometry. We will review this history, then examine the case where we require the metrics and eigenfunctions to be invariant under the action of a group.