This is a sequel to this topic where we have recalled several properties of the stereographic projection . Recall that by the following transformation
we know that
are equivalent in the weak sense.
The way to see it comes from the following identities
Now, we would like to mention the fact that this projection can be used to classify solutions to the following fourth-order elliptic equation coming from the -curvature problem
on the -sphere where
and is a given function defined on . To be exact, up to a constant, the following function
will solve the PDE where
Similarly, the following function
will solve the following PDE
on the -sphere.