This project involves several geometric problems giving rise to linear and nonlinear differential equations on noncompact manifolds. The PI has worked in recent years on constant scalar curvature, constant Ricci curvature, Gauss curvature flow, and equations of mean curvature type. She is above all interested in such questions in the presence of singularities - sharp corners, cones, and edges. Promising new directions generated by this work will be carried out during the grant period. Curvature conditions produce nonlinear equations, which are typically very challenging in noncompact and singular settings. Questions to be pursued include extensions of recent work carried out with R. Mazzeo and Y. Rubinstein regarding the existence and the nature of Kaehler-Einstein metrics with conical singularities on a compact, complex manifold. The PI intends likewise to pursue further work in the area of constant scalar curvature metrics with conical singularities. Earlier joint work with J. Rowlett focused on the case of negative curvature; the PI now intends to continue these investigations into the cases of zero and positive curvature. In addition to these nonlinear problems, a second class of problems involves spectral functions of the Laplacian. A unifying theme is the use of asymptotic methods. The project involves a graduate student.
The PI proposes to work on various problems in the area of partial differential equations with the unifying theme of using asymptotic methods. The development of techniques and the exploration of their applicability to problems on noncompact and singular spaces is expected to enhance the range of settings in which such problems of physical and geometric origin may be studied. These topics have been the subject of intense research activity. The PI has been very active in speaking in conferences and research programs, and funding will support such travel. Funding will also allow the PI to visit collaborators and to support the travel of collaborators to her home institution. Very importantly, it will also allow her to support a graduate student. The PI is involved in education at all levels.
