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We investigate in detail recent suggestions that codimension-two braneworldsin six dimensional supergravity might circumvent Weinberg's no-go theorem forself-tuning of the cosmological constant. The branes are given finite thicknessin order to regularize mild singularities in their vicinity, and we allow themto have an arbitrary equation of state. We study perturbatively the timeevolution of the solutions by solving the equations of motion linearized arounda static background. Even allowing for the most general possibility of warpingand nonconical singularities, the geometry does not relax to a static solutionwhen the brane stress-energies are perturbed. Rather, both the internal andexternal geometries become time-dependent, and the system does not exhibit anyself-tuning behavior.Comment: Revtex4, 25 pages, matches published versio

Codimension-two branes in six-dimensional supergravity and the cosmological constant problem