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