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ANCIENT SOLUTIONS OF CODIMENSION TWO SURFACES WITH CURVATURE PINCHING - RETRACTED
Part of
- Parabolic equations and systems
- Global differential geometry
ZHENGCHAO JI
Journal:

Bulletin of the Australian Mathematical Society / Volume 102 / Issue 1 / August 2020

Published online by Cambridge University Press:

06 March 2020, pp. 162-171

Print publication:

August 2020
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We prove rigidity theorems for ancient solutions of geometric flows of immersed submanifolds. Specifically, we find conditions on the second fundamental form that characterise the shrinking sphere among compact ancient solutions for the mean curvature flow in codimension two surfaces.

Ancient multiple-layer solutions to the Allen–Cahn equation
Manuel del Pino, Konstantinos T. Gkikas
Journal:

Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 148 / Issue 6 / December 2018

Published online by Cambridge University Press:

18 December 2017, pp. 1165-1199

Print publication:

December 2018
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We consider the parabolic one-dimensional Allen–Cahn equation
The steady state connects, as a ‘transition layer’, the stable phases –1 and +1. We construct a solution u with any given number k of transition layers between –1 and +1. Mainly they consist of k time-travelling copies of w, with each interface diverging as t → –∞. More precisely, we find
where the functions ξj (t) satisfy a first-order Toda-type system. They are given by
for certain explicit constants γjk.