  We investigate singular geometries which can be associated with warped branes
in arbitrary dimensions. If the brane tension is allowed to be variable, the
extremum condition for the action requires additional constraints beyond the
solution of the field equations. In a higher dimensional world, such
constraints arise from variations of the metric which are local in the usual
four-dimensional spacetime, without changing the geometry of internal space. As
a consequence, continuous families of singular solutions of the field
equations, with arbitrary integration constants, are generically reduced to a
discrete subset of extrema of the action, similar to regular spaces. As an
example, no static extrema of the action with effective four-dimensional
gravity exist for six-dimensional gravity with a cosmological constant. These
findings explain why the field equations of the reduced four-dimensional theory
are not consistent with arbitrary solutions of the higher dimensional field
equations - consistency requires the additional constraints. The characteristic
solutions for variable tension branes are non-static runaway solutions where
the effective four-dimensional cosmological constant vanishes as time goes to
infinity.
