  We consider an entire graph S of a continuous real function over
(N-1)-dimensional Euclidean space with N larger than or equal to 3. Let D be a
domain in N-dimensional Euclidean space with S as a boundary. Consider in D the
heat flow with initial temperature 0 and boundary temperature 1. The problem we
consider is to characterize S in such a way that there exists a stationary
isothermic surface in D. We show that S must be a hyperplane under some general
conditions on S. This is related to Liouville or Bernstein-type theorems for
some elliptic Monge-Amp\`ere-type equation.
