Moser-Trudinger Inequality on Compact Riemannian Manifolds of Dimension Two (Journal of Partial Differential Equations, 2001)
ln this paper, we prove Moser-Trüdinger inequality in any two dimensional manifolds. Let $(M,g_M,)$ be a two dimensional manifold without boundary and $(g, g_N)$ with boundary, we shall prove the following three inequalities:$$u∈H^1(M), {\rm and} \ ||u||_{H^1(M)}=_1\int_M e^{4π u^2}