On the Convergence of Non-Integer Linear Hopf Flow

TL;DR

This paper investigates the convergence behavior of a non-integer linear Hopf flow on rotationally symmetric surfaces, showing conditions under which the flow converges to round or non-round spheres using spectral theory.

Contribution

It extends the understanding of linear Hopf flows to non-integer coefficient cases, providing convergence criteria and spectral analysis methods.

Findings

Flow converges to a round sphere if focal points coincide.

Flow converges to a non-round Hopf sphere otherwise.

Spectral theory is used to prove convergence properties.

Abstract

The evolution of a rotationally symmetric surface by a linear combination of its radii of curvature equation is considered. It is known that if the coefficients form certain integer ratios the flow is smooth and can be integrated explicitly. In this paper the non-integer case is considered for certain values of the coefficients and with mild analytic restrictions on the initial surface. We prove that if the focal points at the north and south poles on the initial surface coincide, the flow converges to a round sphere. Otherwise the flow converges to a non-round Hopf sphere. Conditions on the fall-off of the astigmatism at the poles of the initial surface are also given that ensure the convergence of the flow. The proof uses the spectral theory of singular Sturm-Liouville operators to construct an eigenbasis for an appropriate space in which the evolution is shown to converge.