Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds

Let (M,g) be a compact Riemannian manifold without boundary, and (N,g) a compact Riemannian manifold with boundary. We will prove in this paper that the integral(M) udV(g)=0, integral(sup)(M) vertical bar del u vertical bar(n)dV(g)=1 integral(M) (e alpha n vertical bar u vertical bar n/n-1dVg), integral(M) (vertical bar del vertical bar(n) + vertical bar u vertical bar)dV(g)=1 integral(M) (e alpha n vertical bar u vertical bar n/n-1dVg), and u vertical bar partial derivative N = 0, integral(sup)(M)vertical bar del u vertical bar dV(gN)=1 integral(N) (e alpha n vertical bar u vertical bar n/n-1dVgN), can be attained. Our proof uses the blow-up analysis.