# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## Stationary isothermic surfaces and uniformly dense domainsHTML articles powered by AMS MathViewer

by R. Magnanini, J. Prajapat and S. Sakaguchi
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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• 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
• 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