LEAPS-MPS: Higher order partial differential equations in conformal and Riemannian geometry.

The central idea in Differential Geometry is to study how much a space is curved and the connection with its topology, that is, the study of its shape. A significant example is the Gauss-Bonnet Theorem which links the Gaussian curvature of a surface with a topological invariant. Differential Geometry has many applications in diverse fields such as general relativity, mechanics, thermodynamics, imaging processing and computer science, to name a few. In this project, the PI will investigate geometric and analytic problems related to several higher order curvature quantities, which are natural generalizations of Gaussian curvature in a space of dimensions four and higher. The PI will use this award to train undergraduate and graduate students at Wichita State University in Differential Geometry. The PI will also organize events and outreach activities to attract more students, especially individuals from underrepresented groups, into STEM fields. There has been growing interest in studying higher order curvature quantities with similar properties as Gaussian and scalar curvatures. This project consists of two major objectives. First, the PI will study the Q-curvature Yamabe problem for manifolds with corners or with boundaries in dimensions four and higher. This involves solving a certain type of fourth-order partial differential equations with appropriately chosen geometric assumptions, in hope of finding better geometric characterizations of Q-curvature. Second, the PI will investigate geometric properties and fully nonlinear Sobolev-type inequalities for a larger class of scalar invariants, which arise as the gradient of a Riemannian functional. The proposed approach is to study the relationship of the invariants and the associated multilinear operators through Ovisenko-Redou operators. As an outcome, the PI will to develop better analytic tools for higher order nonlinear geometric analysis problems. This project is jointly funded by LEAPS-MPS and the Established Program to Stimulate Competitive Research (EPSCoR). This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.