## Geometry of the ∂ ¯ -Neumann Problem

A fundamental theorem in the subject of function theory of several complex variables is the solution of the Levi problem, establishing the identity of domains of holomorphy and pseudoconvex domains. The main point is that a subelliptic estimate holds on forms if and only if the ideal of subelliptic multipliers is the full ring of germs of smooth functions. Kohn discovered the geometric meaning of the ideals J_{k}, when the defining function is real analytic. the key analytic idea is the construction of plurisubharmonic functions, smooth on the closed domain and bounded by unity there, but with arbitrarily large Hessians at the boundary. Catlin observed earlier that the existence of such plurisubharmonic functions implies global regularity of the Neumann operator. Then the radical of the original ideal would be the maximal ideal; the next step would yield the constant function unity as a Jacobian determinant.