2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/229609We show that for two dimensional manifolds M with negative Euler characteristic there exists subsets of the space of smooth Riemannian metrics which are invariant and either parabolic or backwards-parabolic for the 2nd order RG flow. We also show that solutions exists globally on these sets. Finally, we establish the existence of an eternal solution that has both a UV and IR limit, and passes through regions where the flow is parabolic and backwards-parabolic.High Energy Physics - TheoryGeneral Relativity and Quantum CosmologyDifferential Geometrytext