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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 adomain in N-dimensional Euclidean space with S as a boundary. Consider in D theheat flow with initial temperature 0 and boundary temperature 1. The problem weconsider is to characterize S in such a way that there exists a stationaryisothermic surface in D. We show that S must be a hyperplane under some generalconditions on S. This is related to Liouville or Bernstein-type theorems forsome elliptic Monge-Amp\`ere-type equation.

Stationary isothermic surfaces and some characterizations of the hyperplane in the N-dimensional Euclidean space