Alejandro Rivera

On a smooth compact manifold M one can fix different possible Riemannian metrics. Intuitively, M defines the topology while the metric defines the "shape" of the space. A choice of shape also determines a Laplace operator, defined for instance as the divergence of the gradient. Compactness makes the spectrum of the Laplacian discrete. The eigenvalues have a strong influence on the behavior of solutions to such PDEs as the heat equation and wave equation. Of course, these eigenvalues depend on the metric, or shape, of M. However, the topology of M imposes some restrictions on these eigenvalues. In this talk we will focus on a result due to Cheng (1976) about the multiplicity of the first eigenvalue of the Laplacian on the sphere. Along the way we will also talk about nodal lines and Courant's theorem on the number of nodal domains.