  Consider the Ising model on a centered box of side length $n$ in $\mathbb
Z^d$ with $\mp$-boundary conditions that are minus in the upper half-space and
plus in the lower half-space. Dobrushin famously showed that in dimensions
$d\ge 3$, at low-temperatures the Ising interface (dual-surface separating the
plus/minus phases) is rigid, i.e., it has $O(1)$ height fluctuations. Recently,
the authors decomposed these oscillations into pillars and identified their
typical shape, leading to a law of large numbers and tightness of their
maximum.
  Suppose we condition on a height-$h$ level curve of the interface, bounding a
set $S \subset \mathbb Z^{d-1}$, along with the entire interface outside the
cylinder $S\times \mathbb Z$: what does the interface in $S\times \mathbb Z$
look like? Many models of random surfaces (e.g., SOS and DGFF) fundamentally
satisfy the domain Markov property, whereby their heights on $S$ only depend on
the heights on $S^c$ through the heights on $\partial S$. The Ising interface
importantly does not satisfy this property; the law of the interface depends on
the full spin configuration outside $S\times \mathbb Z$.
  Here we establish an approximate domain Markov property inside the level
curves of the Ising interface. We first extend Dobrushin's result to this
setting, showing the interface in $S\times \mathbb Z$ is rigid about height
$h$, with exponential tails on its height oscillations. Then we show that the
typical tall pillars in $S\times \mathbb Z$ are uniformly absolutely continuous
with respect to tall pillars of the unconditional Ising interface. Using this
we identify the law of large numbers, tightness, and Gumbel tail bounds on the
maximum oscillations in $S\times \mathbb Z$ about height $h$, showing that
these only depend on the conditioning through the cardinality of $S$.
