  We explore brane induced gravity on a 3-brane in six locally flat dimensions.
To regulate the short distance singularities in the brane core, we resolve the
thin brane by a cylindrical 4-brane, with the geometry of 4D Minkowski $\times$
a circle, which has an axion flux to cancel the vacuum pressure in the compact
direction. We discover a large diversity of possible solutions controlled by
the axion flux, as governed by its boundary conditions. Hence brane induced
gravity models really give rise to a {\it landscape} of vacua, at least
semiclassically. For sub-critical tensions, the crossover scale, below which
gravity may look 4D, and the effective 4D gravitational coupling are sensitive
to vacuum energy. This shows how the vacuum energy problem manifests in brane
induced gravity: instead of tuning the 4D curvature, generically one must tune
the crossover scale. On the other hand, in the near-critical limit, branes live
inside very deep throats which efficiently compactify the angular dimension. In
there, 4D gravity first changes to $5D$, and only later to $6D$. The crossover
scale saturates at the gravitational see-saw scale, independent of the tension.
Using the fields of static loops on a wrapped brane, we check the perturbative
description of long range gravity below the crossover scale. In sub-critical
cases the scalars are strongly coupled already at the crossover scale even in
the vacuum, because the brane bending is turned on by the axion flux. Near the
critical limit, linearized perturbation theory remains under control below the
crossover scale, and we find that linearized gravity around the vacuum looks
like a scalar-tensor theory.
