2012/09/25 by Manuel del Pino, Panagiota Daskalopoulos, del Pino, Manuel +3
Mathematics · #53C44 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1209.5479
openalex publication_date 2012/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 techniques in this article may be generalized to the gluing of n spheres.