The following theorem is well-known
Theorem (Liouville). Let
be a simply connected domain in
. Then all real solutions of
in
where
a constant, are of the form
where
is a locally univalent meromorphic function in
.
In geometry, our PDE
says that under the case , it holds
where denotes the standard metric on
with constant curvature
. Thus we have
Corollary. All solutions of the PDE in
with
and
are of the form
.