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Type II ancient compact solutions to the Yamabe flow

  • Panagiota Daskalopoulos EMAIL logo , Manuel del Pino and Natasa Sesum


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 k2, in such a way the configuration of radii of the spheres glued is driven as t- by a First order Toda system.

Award Identifier / Grant number: 0604657

Award Identifier / Grant number: 1266172

Award Identifier / Grant number: 0905749

Award Identifier / Grant number: 1056387

Funding statement: P. Daskalopoulos has been partially supported by NSF grants 0604657 and 1266172. M. del Pino has been supported by grants Fondecyt 1150066, Fondo Basal CMM, Millenium Nucleus CAPDE NC130017. N. Sesum has been partially supported by NSF grants 0905749 and 1056387.


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Received: 2013-6-10
Revised: 2015-2-13
Published Online: 2015-10-14
Published in Print: 2018-5-1

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