Contact information: jguoap at.connect.ust.hk

In Fall 2019, we are organizing a reading seminar on Yamabe’s problem. here is our ongoing progress:

 Date+Time Progress 16/09/2019 14:00-15:30 Formulation of Yamabe’s PDE and variational problem. We also gave the upper bound for $$\lambda(M)$$ for compact Riemannian manifold $$M$$ of dimension $$\geq3$$. 27/09/2019 14:00-16:00 Yamabe’s problem is solvable only when $$\lambda(M)<\lambda(\mathbb{S}^n)$$; the failure of Yamabe’s theorem without the condition. 23/10/2019 14:00-16:40 Conformal normal coordinates: brief introduction. Prove that $$\lambda(M)<\lambda(\mathbb{S}^n)$$ when $$(M,g)$$ is not conformally flat and $$\text{dim}M\geq 6$$. 1/11/2019 Lions’s approach using concentration of compactness.

Some references used in the course MATH 5230:

The Schouten-Nijenhuis bracket and interior products: Extension of Lie bracket

Reference for sympletic geometry: Géométrie symplectique et géométrie de Poisson by Charles-Michel Marle. Here is the table of content