  Using the (dynamical) density-matrix renormalization group method, we study
the low-energy physics of three-leg antiferromagnetic Heisenberg model where
the periodic boundary conditions are applied in the rung direction. We confirm
that the spin excitations are always gapped as long as the intra-ring couplings
form a regular triangle. From precise finite-size-scaling analyses of the spin
gap and dimerization order parameter, we also find that the spin gap is
collapsed by very small asymmetric modulation of the intra-ring couplings.
Moreover, the dynamical spin structure factors on the intra- and inter-leg
correlations are calculated. It is demonstrated that the low-lying structure of
the inter-leg spectra is particularly affected by the asymmetric modulation.
