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.

我们研究了保形流形\（\ overline {M} \）上Hermitian向量束上的Yang-Mills连接的边值问题。我们的主要结果是：为每个杨-米尔斯连接阿其满足一个适当的非退化状态，并为每足够小变形\（\伽马\）的\（A_ {| \局部\划线{M}} \） ，有一个Yang-Mills连接（如果足够接近A，则为唯一模量规），其对边界的限制为\（A_ {| \ partial \ overline {M}} + \ gamma \）。该结果可以解释为Graham和Lee著名定理的Yang-Mills类似物，关于存在规定的保形无穷大的Poincaré-Einstein度量（Graham和Lee在Adv Math 87（2）：186-225，1991年）。 。作为推论，我们确认对Witten的期望，这在他的全息基础论文中提到（Witten in Adv Theor Math Phys 2：253–291，1998）：如果\（\ overline {M} \）满足拓扑条件\（ H ^ 1 \ left（\ overline {M}，\ partial \ overline {M} \ right）= 0 \），并且A是平凡的Hermitian向量束上的平凡连接，则边界上的每个连接都足够接近\ （A_ {| \ partial \ overline {M}} \）延伸到内部A附近唯一模量规中的Yang-Mills连接。