In this short note, we shall discuss a very beautiful identity named after Jerry L. Kazdan and F. W. Warner published in the Ann. of Math. (2) long time ago [here].
To be precise, let us consider the following partial differential equation
over a -sphere where is a function and is a constant. We shall prove the following
Theorem (Kazdan-Warner). It holds
where are coordinates.
For any we first multiple both sides of PDE with and then integrate over the sphere to arrive at
On the other hand,
The proof follows.
There are lots of variations and generalization in the literature so far. We are going to discuss later.