Type II ancient compact solutions to the Yamabe flow

Panagiota Daskalopoulos, Manuel Del Pino, Natasa Sesum

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Abstract

We construct new type II ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as t → -∞, to a tower of two spheres. Their curvature operator changes sign. We allow two time-dependent parameters in our ansatz. We use perturbation theory, via fixed point arguments, based on sharp estimates on ancient solutions of the approximated linear equation and careful estimation of the error terms which allow us to make the right choice of parameters. Our technique may be viewed as a parabolic analogue of gluing two exact solutions to the rescaled equation, that is the spheres, with narrow cylindrical necks to obtain a new ancient solution to the Yamabe flow. The result generalizes to the gluing of k spheres for any k ≥ 2, in such a way the configuration of radii of the spheres glued is driven as t → -∞ by a First order Toda system.

Original languageEnglish
Pages (from-to)1-71
Number of pages71
JournalJournal für die reine und angewandte Mathematik (Crelles Journal)
Volume2018
Issue number738
Early online date14 Oct 2015
DOIs
Publication statusPublished - 1 May 2018

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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