Abstract

I will discuss a new vanishing theorem on complete, Kaehler manifolds.
The result says that there are no harmonic  (p,q) - forms on a complete,
Kaehler  manifold  M  (if  p+q  is not equal to  n = dim M)  whenever  M
satisfies 2 conditions:  (i) the  metric on M  is given by a global potential,
and (ii) the gradient of this potential grows slower than (a constant times)
the potential function itself. This result extends an earlier result of Gromov.
My main interest is with (bounded) domains in C^n,equipped with the
Bergman metric, and I will give some examplesto illustrate the new theorem.