# 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

# Type II ancient compact solutions to the Yamabe flow

Journal für die reine und angewandte Mathematik, Volume 2018 (738): 71 – May 1, 2018
71 pages

/lp/de-gruyter/type-ii-ancient-compact-solutions-to-the-yamabe-flow-v0VB4MdYC4
Publisher
de Gruyter
© 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

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create folders to

Export folders, citations