## 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.

## References

- A. D. Aleksandrov,
*Uniqueness theorems for surfaces in the large. V*, Vestnik Leningrad. Univ.**13**(1958), no. 19, 5–8 (Russian, with English summary). MR**0102114** - Giovanni Alessandrini,
*Matzoh ball soup: a symmetry result for the heat equation*, J. Analyse Math.**54**(1990), 229–236. MR**1041182**, DOI 10.1007/BF02796149 - Marc Chamberland and David Siegel,
*Convex domains with stationary hot spots*, Math. Methods Appl. Sci.**20**(1997), no. 14, 1163–1169. MR**1468407**, DOI 10.1002/(SICI)1099-1476(19970925)20:14<1163::AID-MMA885>3.0.CO;2-7 - Isaac Chavel and Leon Karp,
*Movement of hot spots in Riemannian manifolds*, J. Analyse Math.**55**(1990), 271–286. MR**1094719**, DOI 10.1007/BF02789205 - David Gilbarg and Neil S. Trudinger,
*Elliptic partial differential equations of second order*, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR**737190**, DOI 10.1007/978-3-642-61798-0 - R. Gulliver and N. B. Willms, A conjectured heat flow problem, in Solutions, SIAM Review 37 (1995), 100–104.
- David Hoffman and Hermann Karcher,
*Complete embedded minimal surfaces of finite total curvature*, Geometry, V, Encyclopaedia Math. Sci., vol. 90, Springer, Berlin, 1997, pp. 5–93. MR**1490038**, DOI 10.1007/978-3-662-03484-2_{2} - Nikolaos Kapouleas,
*Complete embedded minimal surfaces of finite total curvature*, J. Differential Geom.**47**(1997), no. 1, 95–169. MR**1601434** - B. Kawohl, A conjectured heat flow problem, in Solutions, SIAM Review 37 (1995), 104–105.
- M. S. Klamkin, A conjectured heat flow problem, in Problems, SIAM Review 36 (1994), 107.
- Rémi Langevin and Harold Rosenberg,
*A maximum principle at infinity for minimal surfaces and applications*, Duke Math. J.**57**(1988), no. 3, 819–828. MR**975123**, DOI 10.1215/S0012-7094-88-05736-5 - Francisco J. López and Francisco Martín,
*Complete minimal surfaces in $\mathbf R^3$*, Publ. Mat.**43**(1999), no. 2, 341–449. MR**1744617**, DOI 10.5565/PUBLMAT_{4}3299_{0}1 - R. Magnanini,
*On symmetric invariants of level surfaces near regular points*, Bull. London Math. Soc.**24**(1992), no. 6, 565–574. MR**1183313**, DOI 10.1112/blms/24.6.565 - Rolando Magnanini and Shigeru Sakaguchi,
*Matzoh ball soup: heat conductors with a stationary isothermic surface*, Ann. of Math. (2)**156**(2002), no. 3, 931–946. MR**1954240**, DOI 10.2307/3597287 - Rolando Magnanini and Shigeru Sakaguchi,
*On heat conductors with a stationary hot spot*, Ann. Mat. Pura Appl. (4)**183**(2004), no. 1, 1–23. MR**2044330**, DOI 10.1007/s10231-003-0077-1 - R. Magnanini and S. Sakaguchi, Stationary isothermic surfaces for unbounded domains, in preparation.
- Rolando Magnanini and Shigeru Sakaguchi,
*The spatial critical points not moving along the heat flow*, J. Anal. Math.**71**(1997), 237–261. MR**1454253**, DOI 10.1007/BF02788032 - Johannes C. C. Nitsche,
*Characterizations of the mean curvature and a problem of G. Cimmino*, Analysis**15**(1995), no. 3, 233–245. MR**1357961**, DOI 10.1524/anly.1995.15.3.233 - Joaquín Pérez and Antonio Ros,
*Properly embedded minimal surfaces with finite total curvature*, The global theory of minimal surfaces in flat spaces (Martina Franca, 1999) Lecture Notes in Math., vol. 1775, Springer, Berlin, 2002, pp. 15–66. MR**1901613**, DOI 10.1007/978-3-540-45609-4_{2} - Robert C. Reilly,
*On the Hessian of a function and the curvatures of its graph*, Michigan Math. J.**20**(1973), 373–383. MR**334045** - Richard M. Schoen,
*Uniqueness, symmetry, and embeddedness of minimal surfaces*, J. Differential Geom.**18**(1983), no. 4, 791–809 (1984). MR**730928** - Martin Traizet,
*An embedded minimal surface with no symmetries*, J. Differential Geom.**60**(2002), no. 1, 103–153. MR**1924593** - Lawrence Zalcman,
*Some inverse problems of potential theory*, Integral geometry (Brunswick, Maine, 1984) Contemp. Math., vol. 63, Amer. Math. Soc., Providence, RI, 1987, pp. 337–350. MR**876329**, DOI 10.1090/conm/063/876329

## 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
- 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
- © Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2231874