Suppose that is a compact Riemannian manifold with boundary . Let be an outer unit normal vector field to the boundary .
Using notation introduced in a previous note, the unnormalized mean curvature computed using the trace of the associated second fundamental form obeys the following conformal change rule
where the conformal metric in terms of the background metric is defined to be .
Following the same strategy for the closed case, Escobar found the following invariant, still named Yamabe invariant, as follows
The original Yamabe invariant for the closed manifolds is simply the following
which can be rewritten in terms of the conformal metric as follows
The purpose of this note is to obtain a short form for in terms of instead of the conformal factor . To do so, let us recall that obeys the following rule
we can write
Therefore, the numerator of is nothing but
Thanks to the conformal change rule for the mean curvature, we can further write the numerator of as below
which is equal to
Thus, the short form we obtain is the following
To the best of my knowledge, I have not seen this type of presentation before.