Por favor, use este identificador para citar o enlazar este ítem:
http://cimat.repositorioinstitucional.mx/jspui/handle/1008/984| SOLUTIONS OF THE YAMABE EQUATION BY LYAPUNOV-SCHMIDT REDUCTION | |
| JORGE DAVILA ORTIZ | |
| Acceso Abierto | |
| Atribución-NoComercial | |
| The Yamabe Problem | |
| Given any closed Riemannian manifold (M, g) we use the Lyapunov-Schmidt finite dimensional reduction method to prove multiplicity results for positive solutions of a subcritical Yamabe type equation on (M, g). If (N, h) is a closed Riemannian manifold of constant positive scalar curvature, our result gives multiplicity of solutions for the Yamabe equation on the Riemannian product (M × N, g + ε2h), for ε > 0 small. − am+ng+"2hu + 􀀀 Sg + ε−2Sh u = upm+n−1. (0.0.1) We restrict our study of solutions to functions that only depend onM, u : M → R. Normalizing h, such that Sh = am+n, we have that u solves the Yamabe equation if and only if − ε2gu + Sg am+n ε2 + 1 u = upm+n−1. (0.0.2) We define a functional J" on a space of Sobolev H", such that its critical points, are positive solutions to our equation 0.0.2. The method of Lyapunov-Schmidt will allow us to reduce our poblem to one of finite dimension. Also in this process we will obtain a function function C2, W : M → H". We conclude our problem, proving that the critical points of the function F" : M → R, defined by F" = J" ◦ W, induces critical points to the functional J", that is, solutions to the equation 0.0.2. We have that the critical points of the map F" give solutions to 0.0.2. This allows to apply classical results of Lusternik-Schnirelmann category of M . We obtain the following result. There is εo > 0 such that for ε ∈ (0, εo) the Yamabe equation on the Riemannian product (M × N, g + ε2h) has at least Cat(M) ( Lusternick-Schnirelmann category) solutions which depend only on M. Applying Morse theory, we have that if ε ∈ (0, εo) all the critical points of the function F" : M → R are non-degenerate, then the Yamabe equation on the Riemannian product (M×N, g+ ε2h) has at least b(M) solutions which depend only onM, where bi(M) .= dim(Hi(M,R)) and b(M) .= b1(M) + · · · + bn(M). 4 | |
| 30-05-2019 | |
| Trabajo de grado, doctorado | |
| Inglés | |
| OTRAS ESPECIALIDADES MATEMÁTICAS | |
| Versión aceptada | |
| acceptedVersion - Versión aceptada | |
| Aparece en las colecciones: | Tesis del CIMAT |
Cargar archivos:
| Fichero | Descripción | Tamaño | Formato | |
|---|---|---|---|---|
| TE 709.pdf | 573.39 kB | Adobe PDF | Visualizar/Abrir |