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We show that for two dimensional manifolds M with negative Eulercharacteristic there exists subsets of the space of smooth Riemannian metricswhich are invariant and either parabolic or backwards-parabolic for the 2ndorder 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 UVand IR limit, and passes through regions where the flow is parabolic andbackwards-parabolic.

The 2nd order renormalization group flow for non-linear sigma models in 2 dimensions