Type II ancient compact solutions to the Yamabe flow

Type II ancient compact solutions to the Yamabe flow AbstractWe construct new type II ancient compact solutions to the Yamabe flow. Our solutionsare rotationally symmetric and converge, as t→-∞{t\to{-}\infty}, 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 equationand 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 resultgeneralizes to the gluing of k spheres for any k≥2{k\geq 2}, in such a way the configuration of radii of the spheres glued is driven as t→-∞{t\to{-}\infty}by a First order Toda system. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal für die reine und angewandte Mathematik de Gruyter

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Publisher
de Gruyter
Copyright
© 2018 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1435-5345
eISSN
1435-5345
DOI
10.1515/crelle-2015-0048
Publisher site
See Article on Publisher Site

Abstract

AbstractWe construct new type II ancient compact solutions to the Yamabe flow. Our solutionsare rotationally symmetric and converge, as t→-∞{t\to{-}\infty}, 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 equationand 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 resultgeneralizes to the gluing of k spheres for any k≥2{k\geq 2}, in such a way the configuration of radii of the spheres glued is driven as t→-∞{t\to{-}\infty}by a First order Toda system.

Journal

Journal für die reine und angewandte Mathematikde Gruyter

Published: May 1, 2018

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