### Abstract

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 Ω in the N-dimensional Euclidean space ℝ^{N} is said to be uniformly dense in a surface Γ ⊂ ℝ^{N} of codimension 1 if, for every small r > 0, the volume of the intersection of Ω with a ball of radius r and center x does not depend on x for x ∈ Γ. 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 ∂Ω, and we show that the principal curvatures of ∂Ω satisfy certain identities. The case in which the surface Γ coincides with ∂Ω 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.

Original language | English |
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Pages (from-to) | 4821-4841 |

Number of pages | 21 |

Journal | Transactions of the American Mathematical Society |

Volume | 358 |

Issue number | 11 |

DOIs | |

Publication status | Published - 2006 Nov 1 |

Externally published | Yes |

### Keywords

- Minimal surfaces
- Stationary surfaces
- Uniformly dense domains

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Transactions of the American Mathematical Society*,

*358*(11), 4821-4841. https://doi.org/10.1090/S0002-9947-06-04145-6