Abstract.
In this short course, I will discuss the prescribed scalar curvature problem on the unit sphere via the negative gradient flow. One should start with the ba- sic knowledge regarding the unit sphere as embedded simplest manifold in the Euclidean space. It has a natural metric and easy to calculate the curvature tensor, including the sectional curvature, Ricci curvature as well as the scalar curvature. Then we consider the change of metrics, in particular, the conformal changes of metric, and induce the scalar curvature equation for a conformal metric. So-called the prescribed scalar curvature problem is the reverse proce- dure, that is, given a smooth function on the sphere, can we find a conformal metric such that its scalar curvature equals the given function? Such equation has a natural variational structure and it is natural to consider its negative gradient flow. We then discuss its local existence, global existence, some kind of convergent properties as well as its blow-up analysis. We also need to study the spectral decomposition and finally the infinitely dimensional Morse theory. At the end, we will summary up what we can claim on the prescribed scalar curvature problem. The main reference is our recent paper (joint with Dr. Chen Xuezhang): The scalar curvature flow on Sn—-perturbation theorem revisited, Invent. Math., 187(2012) 395-506 with an Erratum: 187(2012), 507-509.