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 Poincaré 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.