On the Convergence of NonInteger Linear Hopf Flow
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 noninteger 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 nonround Hopf sphere. Conditions on the falloff 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 SturmLiouville operators to construct an eigenbasis for an appropriate space in which the evolution is shown to converge.
 Publication:

arXiv eprints
 Pub Date:
 May 2022
 arXiv:
 arXiv:2205.15978
 Bibcode:
 2022arXiv220515978G
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Analysis of PDEs;
 Mathematics  Classical Analysis and ODEs;
 Mathematics  Spectral Theory;
 Primary 35K10;
 Secondary 53A05
 EPrint:
 22 pages