Moser-Trudinger inequality on compact Riemannian manifolds of dimension two (Q2745442)

The author studies the Moser-Trudinger type inequality concerning the Orlicz space embeddings of the Sobolev space \(H^{1,2} (M)\) where \(M\) is a two-dimensional Riemannian manifold without boundary. He determines that the best possible constant \(\alpha\) in the inequality NEWLINE\[NEWLINE\sup \Bigl\{\int_M \exp\bigl(\alpha|u|^2\bigr) :\|u\|_{H^{1,2} (M)} \leq 1\Bigr\} <+\inftyNEWLINE\]NEWLINE is equal to \(4\pi\) and shows the existence of an extremal function for which the supremum is attained. The similar results concerning the above inequality with the unit ball in \(H^{1,2}(M)\) replaced by \(\{u\in H^{1,2}(M): \int_Mu=0\), \(\int_M|\nabla u|^2\leq 1\}\) or by \(\{u \in H_0^{1,2}(N): \int_N|\nabla u|^2\leq 1\}\) in case \(N\) is a two-dimensional Riemannian manifold with boundary. The blow-up analysis technique is applied.