Following the previous post, we are interested in solving the following equation

where (with , ) is a conformal metric conformally to . In this entry, we introduce the Hidehiko Yamabe approach. His approach is variational. To keep his notation used, we rewrite the PDE as the following

Yamabe tried to minimize the following

over the Sobolev space where . Let us say

In the first stage, he showed that

**Theorem B**. For any , there exists a positive function satisfying