Evolving to Non-round Weingarten Spheres: Integer Linear Hopf Flows

Abstract

In the 1950’s Hopf gave examples of non-round convex 2-spheres in Euclidean 3-space with rotational symmetry that satisfy a linear relationship between their principal curvatures. In this paper, we investigate conditions under which evolving a smooth convex rotationally symmetric sphere by a linear combination of its radii of curvature yields a Hopf sphere. When the coefficients of the flow have certain integer values, the fate of an initial sphere is entirely determined by the local geometry of its isolated umbilic points. A variety of behaviours is uncovered: convergence to round spheres and non-round Hopf spheres, as well as divergence to infinity. The critical quantity is the rate of vanishing of the astigmatism—the difference of the radii of curvature—at the isolated umbilic points. It is proven that the size of this quantity versus the coefficient in the flow function determines the fate of the evolution. The geometric setting for the equation is Radius of Curvature space, viewed as a pair of hyperbolic/AdS half-planes joined along their boundary, the umbilic horizon. A rotationally symmetric sphere determines a parameterized curve in this plane with end-points on the umbilic horizon. The slope of the curve at the umbilic horizon is linked by the Codazzi–Mainardi equations to the rate of vanishing of astigmatism, and for generic initial conditions can be used to determine the outcome of the flow. The slope can jump during the flow, and a number of examples are given: instant jumps of the initial slope, as well as umbilic circles that contract to points in finite time and ‘pop’ the slope. Finally, we present soliton-like solutions: curves that evolve under linear flows by mutual hyperbolic/AdS isometries (dilation and translation) of Radius of Curvature space. A forthcoming paper will apply these geometric ideas to non-linear curvature flows.