Mathematical Sciences: Nonlinear Elliptic Equations in Differential Geometry

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9704861 Chen This project lies in the area of Riemannian geometry and nonlinear elliptic partial differential equations. The investigator wishes to pursue the problem of prescribing scalar curvature on spheres: given a function on a sphere, can one find a metric conformal to the standard metric whose scalar curvature is the given function? This problem can be stated in terms of a quasi-linear elliptic partial differential equation; the current project is to study various aspects of this underlying differential equation. Rimannian manifolds are higher dimensional generalizations of curved surfaces. Such manifolds have well-known applications in theoretical physics. A Riemannian manifold possesses a notion of distance, or a metric. And by taking the derivative of the metric one can measure its curvature. The problem of prescribing curvature then is to see if one can reverse this process: one specifies the curvature property first, and then look for a suitable metric.