The Laplace–Beltrami Operator on Conic and Anti-conic Surfaces

We consider the evolution of a free particle on a two-dimensional manifold endowed with the degenerate Riemannian metric $ds^2=dx^2+|x|^{2\alpha}d\theta^2$, where $x\in R$, $\theta\in S^1$ and the parameter $\alpha\in R$. For $\alpha$ smaller or equal to $-1$ this metric describes cone-like manifolds (for $\alpha=-1$ it is a flat cone). For $\alpha=0$ it is a cylinder. For $\alpha$ bigger or equal to $1$ it is a Grushin-like metric.
In particular, we discuss whether a free particle or the heat can cross the singular set ${x=0}$ or not, and in which cases the singularity absorbs the heat. (The latter problem is known as the stochastic completeness problem.)
In the last part of the talk we will present some recent results regarding the spectrum of the Laplace–Beltrami operator associated with these metrics and the Aharonov-Bohm effect in the Grushin case.
This is a joint work with U. Boscain and M. Seri.