  The paper raises a question about the optimal critical nonlinearity for the
Sobolev space in two dimensions, connected to loss of compactness, and
discusses the pertinent concentration compactness framework. We study
properties of the improved version of the Trudinger-Moser inequality on the
open unit disk $B\subset\R^2$, recently proved by G. Mancini and K. Sandeep.
Unlike the original Trudinger-Moser inequality, this inequality is invariant
with respect to M\"obius automorphisms of the unit disk, and as such is a
closer analogy of the critical nonlinearity $\int |u|^{2^*}$ in the higher
dimension than the original Trudinger-Moser nonlinearity.
