The cosmological constant and warped extra dimensions

We study the behavior of a general gravitational action, including quadratic terms in the curvature, supplemented by a compact scalar field in 4+1 dimensions. The generalized Einstein equation for this system admits solutions which are compact in one direction and Poincare invariant in the remaining directions. These solutions do not require any fine-tuning of the parameters in the action-including the cosmological constant-only that they should satisfy some mild inequalities. Some of these inequalities can be expressed in a universal form that does not depend on the number of extra compact dimensions when the scenario is generalized beyond 4+1 dimensions.