Sharp Sobolev inequalities in the presence of a twist

Abstract

Let ( M , g ) (M,g) be a smooth compact Riemannian manifold of dimension n ≥ 3 n \ge 3 . Let also A A be a smooth symmetrical positive ( 0 , 2 ) (0,2) -tensor field in M M . By the Sobolev embedding theorem, we can write that there exist K , B > 0 K,B>0 such that for any u ∈ H 1 2 ( M ) u \in H_1^2(M) , \[ ( ∫ M | u | 2 ⋆ d v g ) 2 / 2 ⋆ ≤ K ∫ M A x ( ∇ u , ∇ u ) d v g + B ∫ M u 2 d v g \left (\int _M\vert u\vert ^{2^\star }dv_g\right )^{2/2^\star }\le K \int _MA_x(\nabla u, \nabla u)dv_g + B\int _Mu^2dv_g \] where H 1 2 ( M ) H_1^2(M) is the standard Sobolev space of functions in L 2 L^2 with one derivative in L 2 L^2 . We investigate in this paper the value of the sharp K K in the equation above, the validity of the corresponding sharp inequality, and the existence of extremal functions for the saturated version of the sharp inequality.