Let be a closed (compact without boundary) Riemannian manifold of dimension with a metric of arbitrary Yamabe class. Suppose be an open subset of with regular boundary and with non-empty and open in .
in for some constant to be specify later. The novelty of this result is that although the metric may be of positive Yamabe class which tells us that it is impossible to construct a conformal metric which is everywhere negative, it is possible to make it negative in a proper subset of . A proof for this result goes as follows:
(1) First, since and involves up to second order derivatives, it is clear that is continuous in . This and the compactness of tell us that is bounded from above, say by some constant , i.e.
everywhere in .
(2) We now conclude that the constant in the preceding estimate can be any constant sitting in for suitable choice of conformal metric. Indeed, recall that for any number (i.e. vanishing Laplacian) , the metric obeys the following
Hence, choosing , one can easily check that
(3) We now consider the following Dirichlet problem on
The existence, uniqueness, and positivity of is clear. In addition, the solution is at least continuous in . Hence, there is some constant such that in .
(4) We now choose a conformal factor in such a way that in . Since is open, such a exists. Finally, we simply set and do some calculation as shown below:
Thanks to and in , there holds