Regularity condition by mean oscillation to a weak solution of the 2-dimensional Harmonic heat flow into sphere

We show a regularity criterion to the harmonic heat flow from 2-dimensional Riemannian manifold M into a sphere. It is shown that a weak solution of the harmonic heat flow from 2-dimensional manifold into a sphere is regular under the criterion (T)integral(0) parallel to del u(tau)parallel to(B) (M) (2)(Or) d tau where B M O-r is the space of bounded mean oscillations on M. A sharp version of the Sobolev inequality of the Brezis-Gallouet type is introduced on M. A monotonicity formula by the mean oscillation is established and applied for proving such a regularity criterion for weak solutions as above.