Some Differential Operators in Real and Complex Geometry

The famous Laplace-Beltrami operator △ acting on differential forms on a Riemannian manifold M determines in some sense the geometry of M. For example the Hodge decomposition theorem implies that in the compact case $$ X(M) = Trace {e^{{ - t{\Delta_{{even}}}}}} - Trace {e^{{ - t{\Delta_{{odd}}}}}} $$ where X(M) is the Euler characteristic of M and $$ {\Delta_{\text{even}}} = \Delta {\left| {_{p{even}}^{ \oplus }\Lambda P(M),{ }{\Delta_{\text{odd}}} = \Delta } \right|_p}_{\text{odd}}^{ \oplus }\Lambda P $$ Of course the theory of the operator in the complex case is much richer. We are going to give a short review of the theory of the Laplace-Beltrami operator on compact complex manifolds. In particular, the Hodge decomposition and its applications will be given. The case of a compact Kähler manifold will also be mentioned. Some other elliptic operators essentially connected with the geometry of M will be introduced. One of them is the so-called Ahlfors-Laplacian S*S acting on 1-forms. S is the Ahlfors’ operator which arises naturally in the theory of quasiconformal deformations of M. S*S is strongly influenced by the geometry of M. It behaves nicely both in the real and in the complex cases. Before passing to the operators some necessary information from the theory of real and complex geometry will be given.