Stationary isothermic surfaces and uniformly dense domains

Magnanini, R.; Prajapat, J.; Sakaguchi, S.

doi:10.1090/S0002-9947-06-04145-6

Stationary isothermic surfaces and uniformly dense domains
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by R. Magnanini, J. Prajapat and S. Sakaguchi PDF

Trans. Amer. Math. Soc. 358 (2006), 4821-4841 Request permission

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 $\Omega$ 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.

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