Stationary isothermic surfaces in Euclidean 3-space

Let Omega be a domain in R-3 with partial derivative Omega = partial derivative(R-3\(Omega) over bar), where partial derivative Omega is unbounded and connected, and let u be the solution of the Cauchy problem for the heat equation partial derivative(t)u = Delta u over R-3, where the initial data is the characteristic function of the set Omega(c) = R-3\Omega. We show that, if there exists a stationary isothermic surface Gamma of u with Gamma boolean AND partial derivative Omega = empty set , then both partial derivative Omega and Gamma must be either parallel planes or co-axial circular cylinders. This theorem completes the classification of stationary isothermic surfaces in the case that Gamma boolean AND partial derivative Omega = empty set and partial derivative Omega is unbounded. To prove this result, we establish a similar theorem for uniformly dense domains in R-3, a notion that was introduced by Magnanini et al. (Trans Am Math Soc 358:4821-4841, 2006). In the proof, we use methods from the theory of surfaces with constant mean curvature, combined with a careful analysis of certain asymptotic expansions and a surprising connection with the theory of transnormal functions.