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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
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

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

S. Sakaguchi
Affiliation: Department of Mathematics, Faculty of Science, Ehime University, 2-5 Bunkyo-cho, Matsuyama-shi, Ehime 790-8577 Japan

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) ($♯$ 12440042) and (B) ($♯$ 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.