Stationary isothermic surfaces and uniformly dense domains

R. Magnanini, J. Prajapat, S. Sakaguchi

Research output: Contribution to journalArticle

14 Citations (Scopus)

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 ℝN is said to be uniformly dense in a surface Γ ⊂ ℝN 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 languageEnglish
Pages (from-to)4821-4841
Number of pages21
JournalTransactions of the American Mathematical Society
Volume358
Issue number11
DOIs
Publication statusPublished - 2006 Nov 1
Externally publishedYes

Keywords

  • Minimal surfaces
  • Stationary surfaces
  • Uniformly dense domains

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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