Stationary isothermic surfaces in Euclidean 3-space

Let $\mathrm{\Omega }$ be a domain in ${\mathbb{R}}^3$ with $\partial\Omega = \partial\left(\mathbb R^3\setminus \overline{\Omega}\right)$,where$\partial\Omega$isunboundedandconnected,andlet$u$ be the solution of the Cauchy problem for the heat equation ${\mathrm{\partial }}_{t}u=\mathrm{\Delta }u$ over $\mathbb R^3,$wheretheinitialdataisthecharacteristicfunctionoftheset$\Omega^c = \mathbb R^3\setminus \Omega$. We show that, if there exists a stationary isothermic surface $\mathrm{\Gamma }$ of $u$ with $\Gamma \cap \partial\Omega = \varnothing$,thenboth$\partial\Omega$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 $\mathrm{\Gamma }\cap \mathrm{\partial }\mathrm{\Omega }=\mathrm{\varnothing }$ and $\mathrm{\partial }\mathrm{\Omega }$ 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.