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Suppose $(M,g)$ is a compact Riemannian manifold with boundary. I am interested about existence of surfaces $\Gamma$ embedded in $M$ with the following property:

$\Gamma$ is foliated by geodesics (here I mean geodesics with respect to the induced metric on $\Gamma$) that are all at the same time asymptotic lines.

Do these surfaces exist in abundance? For example given curves on boundary of manifold can I construct a special $\Gamma$ enclosed by that boundary curve? Thanks!

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If a surface is foliated by geodesics (in the induced metric) that are also asymptotic lines, then these curves are also geodesics in the ambient manifold (just look at the definitions). Conversely, if a surface is a $1$-parameter family of geodesics in an ambient manifold, then these curves are also geodesics in the induced metric and are asymptotic lines. Thus, your surfaces do exist in abundance; they are just the surfaces that are $1$-parameter families of geodesics in the ambient manifold, i.e., they are simply the surfaces that are ruled by geodesics. There are lots of such surfaces locally, of course.

As far as your global question is concerned, I think that one can probably come up with examples of compact manifolds with boundary and (closed) curves in the boundary that are not the boundary of any geodesically ruled surface in the manifold. For example, there might be a closed curve in the boundary that is not homologous to zero in the ambient manifold. Then the desired surface won't exist for topological reasons.

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