We establish a relationship between stationary isothermic surfaces and what we call uniformly dense domains. A stationary isothermic surface is a level surface of temperature which does not evolve with time. A domain in the Euclidean space is said to be uniformly dense in a surface of co-dimension one (e.g. the boundary of the domain) if the volume of the intersection of the domain with a ball of fixed radius does not depend on its center, if this lies on the surface. We prove that the bounded boundary of every uniformly dense domain which is bounded must be a sphere. We then examine a uniformly dense domain with unbounded boundary and show that the principal curvatures of its boundary satisfy certain identities. The case in which the surface coincides with the boundary is particularly interesting. In fact, we show that, if the boundary of a uniformly dense domain is connected, then (i) it must be either a circle or a straight line, if the domain is planar, and (ii) it must be either a sphere, a spherical cylinder or a minimal surface, in three-space. We conclude with a discussion on uniformly dense domains whose boundary is a minimal surface and discover a new type of stationary isothermic surface: the helicoid.
Stationary isothermic surfaces and uniformly dense domains / R. MAGNANINI; J. PRAJAPAT; S. SAKAGUCHI. - In: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9947. - STAMPA. - 385:(2006), pp. 4821-4841. [10.1090/S0002-9947-06-04145-6]
Stationary isothermic surfaces and uniformly dense domains
MAGNANINI, ROLANDO;
2006
Abstract
We establish a relationship between stationary isothermic surfaces and what we call uniformly dense domains. A stationary isothermic surface is a level surface of temperature which does not evolve with time. A domain in the Euclidean space is said to be uniformly dense in a surface of co-dimension one (e.g. the boundary of the domain) if the volume of the intersection of the domain with a ball of fixed radius does not depend on its center, if this lies on the surface. We prove that the bounded boundary of every uniformly dense domain which is bounded must be a sphere. We then examine a uniformly dense domain with unbounded boundary and show that the principal curvatures of its boundary satisfy certain identities. The case in which the surface coincides with the boundary is particularly interesting. In fact, we show that, if the boundary of a uniformly dense domain is connected, then (i) it must be either a circle or a straight line, if the domain is planar, and (ii) it must be either a sphere, a spherical cylinder or a minimal surface, in three-space. We conclude with a discussion on uniformly dense domains whose boundary is a minimal surface and discover a new type of stationary isothermic surface: the helicoid.File | Dimensione | Formato | |
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