I plan to work on four basic problems in several complex variables: (i) The continuous extension of biholomorphic mappings between general, smoothly bounded, pseudoconvex domains in euclidean space to their boundaries, (ii) Inequalities on twisted version of the Cauchy-Riemann operators, (iii) square-integrable cohomology of Cauchy-Riemann operator in the Bergman and other Kaehler metrics, and (iv) Conditions implying compactness of the d-bar Neumann operator. My attack on these problems will be through certain partial differential equations associated to the Cauchy-Riemann and Laplace operators. In our problems, we are in a situation where classical estimates do not provide enough information on the Cauchy-Riemann and Laplace operators to conclude the results we expect. Thus, our approach will be to perturb these two operators suitably, obtain strong estimates on the perturbed operators, and then connect these estimates to the unperturbed operators. The starting point is a twisted version of the Cauchy-Riemann complex which we have previously studied; but I also plan to study more general versions of the twisting idea. The initial twisted complex has already produced some powerful results in Complex analysis and it is currently being extended to other contexts by several mathematicians, including the proposer. We view its continued development as one of the very fertile ideas presently ongoing in several complex variables.
Our proposed research will make a significant contribution to the broad, general question: by how much may one perturb a system of partial differential equations and still obtain information on the unperturbed system from the perturbed system? The results we expect will have many connections to areas of mathematics outside complex analysis, especially topology (through the connection between square-integrable cohomology and ordinary homology), differential equations, and algebraic geometry (by deepening our understanding of order of contact between algebraic and analytic varieties). Additionally, however, since partial differential equations are ubiquitous throughout science and engineering, our perturbation methods will be useful outside pure mathematics. We believe that adaptations of our perturbation methods will apply to partial differential equations which arise in parts of current scientific theory and our proposed research will thereby strengthen the ties and establish new connections between complex analysis and other fields of science.
