As the fundamental solution to the Poisson equation, Green’s function is one of the most important objects associated with the Laplacian. This course will focus on the estimates of the Green’s function on complete Riemannian manifolds with positive spectrum. Various applications to the geometry and topology will also be discussed. We will start from the very basics including the existence of Green’s function and try to make the course as self-contained as possible.

Riemannian Geometry and some basic knowledge of geometric analysis.

Geometric Analysis by Peter Li; Cambridge Studies in Advanced Mathematics 134; Cambridge University Press, 2012.