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Stationary isothermic surfaces in Euclidean 3-space

Let $Ω$ be a domain in $\mathbb R^3$ with $\partialΩ= \partial\left(\mathbb R^3\setminus \overlineΩ\right)$, where $\partialΩ$ is unbounded and connected, and let $u$ be the solution of the Cauchy problem for the heat equation $\partial_t u= Δu$ over $\mathbb R^3,$ where the initial data is the characteristic function of the set $Ω^c = \mathbb R^3\setminus Ω$. We show that, if there exists a stationary isothermic surface $Γ$ of $u$ with $Γ\cap \partialΩ= \varnothing$, then both $\partialΩ$ and $Γ$ must be either parallel planes or co-axial circular cylinders . This theorem completes the classification of stationary isothermic surfaces in the case that $Γ\cap\partialΩ=\varnothing$ and $\partialΩ$ is unbounded. To prove this result, we establish a similar theorem for {\it uniformly dense domains } in $\mathbb R^3$, a notion that was introduced by Magnanini, Prajapat \& Sakaguchi in \cite{MPS2006tams}. 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.