We construct new type II ancient compact solutions to the Yamabe flow. Our solutions
are rotationally symmetric and converge, as , 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 , in such a way the configuration of radii of the spheres glued is driven as by a First order Toda system.
Convergence of the Yamabe flow for arbitrary initial energy,
J. Differential Geom. 69 (2005), 217–278.
M. del Pino, J. Dolbeault and M. Musso,
“Bubble-tower” radial solutions in the slightly supercritical Brezis–Nirenberg problem,
J. Differential Equations 193 (2003), no. 2, 280–306.
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.
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
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