An Explicit Isometric Reduction of the Unit Sphere into an Arbitrarily Small Ball

Abstract

Spheres are known to be rigid geometric objects: they cannot be deformed isometrically, i.e. while preserving the length of curves, in a twice differentiable way. An unexpected result by J. Nash (Ann. of Math. 60: 383-396, 1954) and N. Kuiper (Indag. Math. 17: 545-555, 1955) shows that this is no longer the case if one requires the deformations to be only continuously differentiable. A remarkable consequence of their result makes possible the isometric reduction of a unit sphere inside an arbitrarily small ball. In particular, if one views the Earth as a round sphere the theory allows to reduce its diameter to that of a terrestrial globe while preserving geodesic distances. Here we describe the first explicit construction and visualization of such a reduced sphere. The construction amounts to solve a non-linear PDE with boundary conditions. The resulting surface consists of two unit spherical caps joined by a C 1 fractal equatorial belt. An intriguing question then arises about the transition between the smooth and the C 1 fractal geometries. We show that this transition is similar to the one observed when connecting a Koch curve to a line segment.