  (See the complete and formatted abstract in the paper). We report on VLBI
monitoring of the low-luminosity AGN (LLAGN) in M81 at 1.7, 2.3, 5, and 8.4GHz.
These observations are phase-referenced to the supernova SN1993J (located in
the same galaxy) and cover from late 1993 to late 2005. The source consists at
all frequencies of a slightly resolved core and a small jet extension towards
the north-east direction (position angle of ~65 degrees) in agreement with
previous publications. We find that the position of the intensity peak in the
images at 8.4GHz is very stable in the galactic frame of M81 (proper motion
upper limit about 0.010 mas per year). We confirm previous reports that the
peaks at all frequencies are systematically shifted among them, possibly due to
opacity effects in the jet as predicted by the standard relativistic jet model.
We use this model to estimate the magnetic field in the jet and the mass of the
central black hole. We obtain a black-hole mass of ~2.e+7 solar masses,
comparable to estimates previously reported, but the magnetic fields obtained
are thousands of times lower than previous estimates. We find that the
positions of the cores at 1.7, 2.3, and 5GHz are less stable than that at
8.4GHz and evolve systematically, shifting southward at a rate of several tens
of micro-arcsec per year. The evolution in the jet orientation seems to be
related to changes in the inclination of the cores at all frequencies. These
results can be interpreted as due to a precessing jet. The jet precession also
seems to be related to a flare in the flux densities at 5.0 and 8.4GHz, which
lasts ~4 years. A continued monitoring of the flux density and the jet
structure evolution in this LLAGN will be necessary to further confirm our jet
precession model.
