  Let $(X_1, \bar g_1)$ and $(X_2, \bar g_2)$ be two compact Riemannian
manifolds with boundary $(M_1,g_1)$ and $(M_2,g_2)$ respectively. The Escobar
problem consists in prescribing a conformal metric on a compact manifold with
boundary with zero scalar curvature in the interior and constant mean curvature
of the boundary. The present work is the construction of a connected sum $X=X_1
\sharp X_2$ by excising half ball near points on the boundary. The resulting
metric on $X$ has zero scalar curvature and a CMC boundary. We fully exploit
the nonlocal aspect of the problem and use new tools developed in recent years
to handle such kinds of issues. Our problem is of course a very well-known
problem in geometric analysis and that is why we consider it but the results in
the present paper can be extended to other more analytical problems involving
connected sums of constant fractional curvatures.
