Characterizations of the mean curvature and a problem of G. Cimmino
It is shown that the plane and the helicoid axe the only surfaces possessing vanishing mean curvature in a sense discussed by G. Cimmino. Selected historical remarks introduce the question. AMS 1991 Classification: Primary 49Q05, 53A05, 53A10; Secondary 35J60, 53C42. 1 Let Τ be a differential geometric surface immersed in Euclidean space i î 3 , p an interior point of Τ and <Sp(r) the sphere of radius r centered at p. For sufficiently small values of r, <Sp(r) intersects Τ in a simple closed curve Cp(r) dividing Sp(r) into two parts. Denote by <7p(r) · r the area of that part which is penetrated by the normal vector of 7* at p. CTp(r) is approximately equal to lit. If Τ is a plane, then obviously <7p(r) — 2π. One may ask, as Gianfranco Cimmino did sixty-two years ago, to find all surfaces which share this property with the plane ([4], p. 288, [10], §875, [11], A7.3): Problem: To determine the class of all surfaces Τ which have the property that for every interior point ρ G Τ there is a number r p > 0 such that a p ( r ) = 2π for 0 < r < r p .