Stationary isothermic surfaces in Euclidean 3-space

Let $Omega$ be a domain in $mathbb R^3$ with $partialOmega = partialleft(mathbb R^3setminus overline{Omega} ight)$, where $partialOmega$ is unbounded and connected, and let $u$ be the solution of the Cauchy problem for the heat equation $partial_t u= Delta u$ over $mathbb R^3,$ where the initial data is the characteristic function of the set $Omega^c = mathbb R^3setminus Omega$. We show that, if there exists a stationary isothermic surface $Gamma$ of $u$ with $Gamma cap partialOmega = arnothing$, then both $partialOmega$ 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 $GammacappartialOmega=arnothing$ and $partialOmega$ 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.