Evolution of an extremum by curvature motion

Abstract

In this paper we consider the evolution of an isolated extremum of a function under the curvature motion in the plane. We define different notions of circular extrema and show that, immediately after the motion begins, the isolated extrema become circular. We also show that if the initial function is smooth, then after any small positive time, the new function will have a quadratic expansion at the extremum with equal 'eigenvalues.' © 2004 Elsevier Inc. All rights reserved.