Lectures on the boundary Yamabe problem
Abstract：In this series of three talks, we will present some detailed proofs of the boundary Yamabe problem, involving the works of Escobar, Han-Li, Marques, Almaraz, and our recent work in this direction. To demonstrate the techniques used in this problem, we select some natural geometric assumptions imposed on compact manifolds with boundary to show the existence results. Here is the outline: Lecture I. 1. Generalized Yamabe constants and relationships among them; 2. Three types of boudary Yamabe problems and three distinct boundary bubbles; 3. The criterions of existence of minimizers of the related conformal covariant functionals. Lecture II. Escobar’s work: 1. A global test function: Locally conformal flat manifolds with umbilic boundary; 2. A local test function: The boundary admits a non-umbilic point. Lecture III. After Escobar: 1. Han-Li conjecture: Background and recent advances; 2. Han-Li conjecture in dimension seven under the assumption that the boundary is umbilic, the Weyl tensor of the manifold is nonzero at some boundary point. Our test function is also a local one, however containing such test functions used by Marques, Almaraz et al.