where , , and are given smooth functions of . To be precise, we obtain the following
Theorem. Suppose , , and . Let . Assume that
Then there exists no positive solution to the above Lichnerowicz equation.
Let us denote the integral
by . We call the average of on the sphere of radius , or sphere mean of a function around the origin.
Proof. Note that a simple calculation shows us that
Since on the sphere , is also the outer normal vector, therefore
Thus by the divergence theorem, one gets
Differentiating once more yields
Therefore, taking this average operation we have
Since for each fixed ,
Then by using the general Cauchy inequality, one gets
It turns out that
which implies that
after an integration. This is because, by definition of the sphere mean,
Dividing both sides by and integrating this inequality over , we have
Sending we have
which is impossible by our assumption. The proof is complete.