Sub-Laplacian Eigenvalues on Heisenberg Manifolds

A Heisenberg manifold is a manifold with a non-integrable hyperplane distribution. This definition covers many examples such as the Heisenberg group, CR and contact Manifolds. There is a natural and important hypoelliptic operator on Heisenberg manifolds called the sub-Laplacian. We introduce a new upper bound for its eigenvalues which are asymptotically optimal. Moreover, it is invariant under any conformal change of the metric on the hyperplane distribution. We discuss examples on which upper bounds are independent of the geometry of Heisenberg manifolds. This talk is based on joint work with Gerasim Kokarev.