ABSTRACT

The subject of interest in this book is singular manifolds with differential operators defined on them. For an arbitrary “singular” topological space, it is obviously not an easy task even to define reasonably what differential operators are, let alone study their properties. Hence we do not deal with general singular spaces; on the contrary, we restrict our attention to spaces that possess rich additional structures permitting one to define differential operators in a natural way. Most generally speaking, we consider spacesM with a given C∞ structure on a dense open subset M◦. Then the algebra Diff(M◦) of all differential operators with smooth coefficients onM◦ is well defined, and we single out some subalgebraD ⊂ Diff(M◦) by imposing conditions on the behavior of operators near the singularity set M\M◦. For example, the operators should “degenerate” there or, on the opposite, their coefficients are allowed to go to infinity at a specified rate. It is the pair (M◦,D) rather than M alone that is of interest to us. From the viewpoint of differential equations, two pairs (M◦,D) and (M◦, D˜), where D = D˜, represent different manifolds with singularities and certainly should be distinguished in the study. The idea of defining manifolds with singularities as ringed spaces (understood as pairs of this sort) goes back to Schulze, Sternin and Shatalov (1997, 1998a). Note that in the C∞ case (where the singularity set M \M◦ is empty) the introduction of the ring of differential operators is unnecessary, since it can be reconstructed from C∞(M): all differentiations are just derivations of C∞(M). In the singular case this is, however, meaningful since we are not interested in all differential operators but deal with some subalgebra.