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.
Funding source: National Science Foundation
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.
 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. 10.1016/S0022-0396(03)00151-7Search in Google Scholar
 R. S. Hamilton, Lectures on geometric flows, unpublished manuscript, (1989). Search in Google Scholar
 R. Mazzeo, F. Pacard and D. Pollack, Connected sums of constant mean curvature surfaces in Euclidean 3 space, J. reine angew. Math. 536 (2001), 115–165. 10.1515/crll.2001.054Search in Google Scholar
 R. Mazzeo, D. Pollack and K. Uhlenbeck, Connected sum constructions for constant scalar curvature metrics, Topol. Methods Nonlinear Anal. 6 (1995), no. 2, 207–233. 10.12775/TMNA.1995.042Search in Google Scholar
 R. Schoen, The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation, Comm. Pure Appl. Math. 41 (1988), 317–392. 10.1002/cpa.3160410305Search in Google Scholar
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