Let us continue the problem of prescribing Gaussian curvature. Our PDE reads as the follows
where is a compact manifold without the boundary. Today we show that if
then our PDE has unique solution.
Assume that and are solutions to the PDE, that is
By subtracting, we have
Multiplying both sides by , integrating over , and the using the integration by parts we arrive at
Since , it follows that
In particular, .