Yang-Mills connections on conformally compact manifolds

We study a boundary value problem for Yang-Mills connections on Hermitian vector bundles over a conformally compact manifold (M) over bar. Our main result is the following: for every Yang-Mills connection A that satisfies an appropriate nondegeneracy condition, and for every sufficiently small deformation gamma of A(|partial derivative(M) over bar), there is a Yang-Mills connection (unique modulo gauge if sufficiently close to A) whose restriction to the boundary is A(|partial derivative(M) over bar) +gamma. This result can be interpreted as theYang-Mills analogue of the celebrated theorem of Graham and Lee, on the existence of Poincare-Einstein metrics with prescribed conformal infinity (Graham and Lee in Adv Math 87(2):186-225, 1991). As a corollary, we confirm an expectation ofWitten, mentioned in his foundational paper on holography (Witten in Adv Theor Math Phys 2:253-291, 1998): if (M) over bar satisfies the topological condition H-1 ((M) over bar, partial derivative(M) over bar) = 0, and A is the trivial connection on a trivial Hermitian vector bundle, then every connection on the boundary sufficiently close to A(|partial derivative(M) over bar) extends to a Yang-Mills connection in the interior, unique modulo gauge in a neighborhood of A.