Some ODE solutions for the fractional Yamabe problem

Abstract

We construct some ODE solutins for the fractional Yamabe problem in conformal geometry. The fractional curvature, which is a generalization of the usual scalar curvature, is defined from the conformal fractional Laplacian, which is a non-local operator construced on the conformal infinity of a conformally compact Einstein manifold.

On one hand, we consider the hyperbolic manifold $\mathbb S^1(L)\times \mathbb R^3$ and study the nonuniqueness of solutions for the fractional Yamabe problem. On the other hand, we look at the existence of radial solutions for the Yamabe problem in Euclidean space with an isolated singularity at the origin. Both equations are fractional order ODE for which new tools need to be developed.. La geometria conforme estudia les transformacions que preserven angles. Això fa que les nocions de curvatura més importants es puguin descriure mitjançant equacions en derivades parcials ellíptiques. El projecte consisteix en aprofondier en aquesta relació, fent servir eines de les EDP per a resoldre problemes geomètrics