We prove an upper bound for the volume-normalized second nonzero eigenvalue of the Laplace operator on closed Riemannian manifold, in terms of the conformal volume. This bound provides effective upper bound for a large class of manifolds, thereby generalizing many known results.
The proof uses the spherical cap folding mechanism originating in work of Nadirashvili in combination with the definition of the conformal volume of Li and Yau. This leads to very convenient admissible trial functions for the min-max characterisation of the second non-zero eigenvalue.
This is based on joint work with Mehdi Eddaoudi.
