  Small realistic Majorana neutrino masses can be generated via a Higgs triplet
$(\xi^{++}, \xi^+, \xi^0)$ without having energy scales larger than $M_*={\cal
O}(1)$ TeV in the theory. The large effective mass scale $\Lambda$ in the
well-known seesaw neutrino-mass operator $\Lambda^{-1} (LL\Phi\Phi)$ is
naturally obtained with $\Lambda\sim M_*^2/\mu,$ where $\mu$ is a {\it small}
scale of lepton-number violation. In theories with large extra dimensions, the
smallness of $\mu$ is naturally obtained by the mechanism of ``shining'' if the
number of extra dimensions $n\ge 3.$ We study here the Higgs phenomenology of
this model, where the spontaneous violation of lepton number is treated as an
external source from extra dimensions. The observable decays $\xi^{++} \to
l_i^+l_j^+$ will determine directly the magnitudes of the $\{ij\}$ elements of
the neutrino mass matrix. The decays $\xi^+ \to W^+ J^0$ and $\xi^0 \to Z J^0$,
where $J^0$ is the massless Goldstone boson (Majoron), are also possible, but
of special importance is the decay $\xi^0 \to J^0 J^0$ which provides stringent
constraints on the allowed parameter space of this model. Based on the current
neutrino data, we also predict observable rates of $\mu-e$ conversion in
nuclei.
