Moser-Trudinger Inequality on Compact Riemannian Manifolds of Dimension Two

Abstract

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:$$u∈H^1(M), {\rm and} \  ||u||_{H^1(M)}=_1\int_M e^{4\pi u^2}<+∞$$ $$u∈H^1(M), ∫_M u=0, {\rm and} ∫_M|∇u|^2=_1∫_M {e^{4\pi u^2}}<+∞$$ $$u∈H^1_0(N), {\rm and} ∫_M|∇u^2|=_1∫_M e^{4\pi u^2}<+∞$$ Moreover, we shall show that there exist of extremal functions which at tain the above three inequalities.