Type II ancient compact solutions to the Yamabe flow

Panagiota Daskalopoulos 1 , Manuel del Pino 2 ,  and Natasa Sesum 3
  • 1 Department of Mathematics, Columbia University, 2990 Broadway, New York, USA
  • 2 Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
  • 3 Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, USA
Panagiota Daskalopoulos
  • Corresponding author
  • Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027, USA
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, Manuel del Pino
  • Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
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and Natasa Sesum

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

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The Journal für die reine und angewandte Mathematik is the oldest mathematics periodical still in existence. Founded in 1826 by August Leopold Crelle and edited by him until his death in 1855, it soon became widely known under the name of Crelle’s Journal. In the 190 years of its existence, Crelle’s Journal has developed to an outstanding scholarly periodical with one of the worldwide largest circulations among mathematics journals.

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