In this talk, we study upper bounds for eigenvalues of the sub-Laplacian on sub-Riemannian manifolds. Sub-Riemannian structures naturally occur in control theory, analysis of hypoelliptic operators, contact geometry and CR geometry. The sub-Laplacian is an intrinsic hypoelliptic operator in sub-Riemannian geometry. It is a deep relationship between its eigenvalues and the sub-Riemannian structure. I recall basic geometric properties of sub-Riemannian manifolds and discuss geometric bounds for eigenvalues of the sub-Laplacian. Further, I give examples on which eigenvalue upper bounds are independent of the geometry of the underlying manifold. This is a joint work with Gerasim Kokarev.
