Conformal Geometry and Prescribed Scalar Curvature on S^2
In this dissertation, we seek to understand prescribed scalar curvature through the gradient flow of conformal metrics. On S^2, we will define a modified Liouville energy and derive a geometric flow equation related to the energy functional. We will prove longtime existence for solutions of this equation with arbitrary data through the methods used by Gursky and Streets. We will then show that Gauss curvature retains its regularity under evolution through the flow assuming bounds on the Gauss curvature. We will finally show that the solution is stable when converging to constant curvature if the initial curvature is close to the geometry of S^2.