  Monotonic surfaces spanning finite regions of $Z^d$ arise in many contexts,
including DNA-based self-assembly, card-shuffling and lozenge tilings. One
method that has been used to uniformly generate these surfaces is a Markov
chain that iteratively adds or removes a single cube below the surface during a
step. We consider a biased version of the chain, where we are more likely to
add a cube than to remove it, thereby favoring surfaces that are "higher" or
have more cubes below it. We prove that the chain is rapidly mixing for any
uniform bias in $Z^2$ and for bias $\lambda > d$ in $Z^d$ when $d>2$. In $Z^2$
we match the optimal mixing time achieved by Benjamini et al. in the context of
biased card shuffling, but using much simpler arguments. The proofs use a
geometric distance function and a variant of path coupling in order to handle
distances that can be exponentially large. We also provide the first results in
the case of fluctuating bias, where the bias can vary depending on the location
of the tile. We show that the chain continues to be rapidly mixing if the
biases are close to uniform, but that the chain can converge exponentially
slowly in the general setting.
