I will provide a brief introduction to the canonical metric problem in Kähler geometry and related objects. Then I'll explain how generalizations of these objects naturally appear in the context of partition functions of determinantal point processes on polarized Kähler manifolds. The talk will be aimed at beginning geometry students and I will be rather pedagogical. Especially, I will focus on geometric aspects of the topic, so probabilistic or physical discussion will be postponed or omitted. This is based on my recent preprint.

Serrin’s overdetermined problem is a famous result in mathematics that deals with the uniqueness and symmetry of solutions to certain boundary value problems. It is called "overdetermined" because it has more boundary conditions than usually required to determine a solution, which leads to strong restrictions on the shape of the domain. In this talk, we discuss overdetermined boundary value problems in a Riemannian manifold and discuss a Serrin-type symmetry result to the solution to an overdetermined Steklov eigenvalue problem on a domain in a Riemannian manifold with nonnegative Ricci curvature and it will be discussed about an overdetermined problems with a nonconstant Neumann boundary condition in a warped product manifold.