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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Stationary isothermic surfaces and uniformly dense domains


Authors: R. Magnanini, J. Prajapat and S. Sakaguchi
Journal: Trans. Amer. Math. Soc. 358 (2006), 4821-4841
MSC (2000): Primary 35K05, 35K20; Secondary 53A10, 58J70
DOI: https://doi.org/10.1090/S0002-9947-06-04145-6
Published electronically: April 11, 2006
MathSciNet review: 2231874
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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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Additional Information

R. Magnanini
Affiliation: Dipartimento di Matematica U. Dini, Università di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy
Email: magnanin@math.unifi.it

J. Prajapat
Affiliation: Indian Statistical Institute, Stat-Math Unit, 8th Mile, Mysore Road, R.V.C.E. Post, Bangalore 560 059, India
Address at time of publication: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400 097, India
Email: jyotsna@isibang.ac.in

S. Sakaguchi
Affiliation: Department of Mathematics, Faculty of Science, Ehime University, 2-5 Bunkyo-cho, Matsuyama-shi, Ehime 790-8577 Japan
Email: sakaguch@dpc.ehime-u.ac.jp

Keywords: Stationary surfaces, uniformly dense domains, minimal surfaces
Received by editor(s): August 11, 2004
Published electronically: April 11, 2006
Additional Notes: This research was partially supported by a Grant-in-Aid for Scientific Research (B) ($\sharp$ 12440042) and (B) ($\sharp$ 15340047) of the Japan Society for the Promotion of Science and by a Grant of the Italian MURST
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.