  We derive extrinsic GJMS operators and $Q$-curvatures associated to a
submanifold of a conformal manifold. The operators are conformally covariant
scalar differential operators on the submanifold with leading part a power of
the Laplacian in the induced metric. Upon realizing the conformal manifold as
the conformal infinity of an asymptotically Poincar\'e--Einstein space and the
submanifold as the boundary of an asymptotically minimal submanifold thereof,
these operators arise as obstructions to smooth extension as eigenfunctions of
the Laplacian of the induced metric on the minimal submanifold. We derive
explicit formulas for the operators of orders 2 and 4. We prove factorization
formulas when the original submanifold is a minimal submanifold of an Einstein
manifold. We also show how to reformulate the construction in terms of the
ambient metric for the conformal manifold, and use this to prove that the
operators defined by the factorization formulas are conformally invariant for
all orders in all dimensions.
