Stationary isothermic surfaces and uniformly dense domains

We establish a relationship between stationary isothermic surfaces and uniformly dense domains. A stationary isothermic surface is a level surface of temperature which does not evolve with time. A domain $\Omega$ in the $N$-dimensional Euclidean space $\mathbb{R}^N$ is said to be uniformly dense in a surface $\Gamma\subset\mathbb{R}^N$ of codimension $1$ if, for every small $r>0,$ the volume of the intersection of $\Omega$ with a ball of radius $r$ and center $x$ does not depend on $x$ for $x\in\Gamma.$

We prove that the boundary of every uniformly dense domain which is bounded (or whose complement is bounded) must be a sphere. We then examine a uniformly dense domain with unbounded boundary $\partial\Omega$, and we show that the principal curvatures of $\partial\Omega$ satisfy certain identities.

The case in which the surface $\Gamma$ coincides with $\partial\Omega$ is particularly interesting. In fact, we show that, if the boundary of a uniformly dense domain is connected, then (i) if $N=2$, it must be either a circle or a straight line and (ii) if $N=3,$ it must be either a sphere, a spherical cylinder or a minimal surface. We conclude with a discussion on uniformly dense domains whose boundary is a minimal surface.