Yang–Mills connections on conformally compact manifolds

Abstract

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