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In the last two decades,much progress has been made on the regularity theory of harmonic maps between manifolds,for instance,one can see [GG1], [GG2], [SU],[E],[B] etc. Since the methods used in these papers are variations of Schauder's device of freezing the coefficients they need assume that the metrics are at least uniformly continuous while in [SU],[E],[B] they assume g G C(M). If the image of a map is contained in a convex ball; the metric of the domain is only bounded measurable,Hildebrandt-Widman proved the map is Holder continuous(see[H-W]). On the other hand,M.Gromov and R. Schoen established the existence and regularity theory of harmonic maps into singular space(see [GS]). It may be more interesting,m view of analysis,to consider the same problem when the domain is singular. In this paper, we shall consider this problem and prove partial regularity results of energy minimizing maps from manifolds with bounded measurable Riemannian metrics. We will consider the more complicated case,for example, the domain is a algebraic variety, in elsewhere. Suppose M is an n-dimensional differentiable manifold, g is a metric on M ,then g can be represented by a positive definite matrix (gij) in each coordinate chart (T>^x) of M.If g^ merely belongs to L^, R), then g is called bounded measurable Riemannian metric,or simply L^-metric.Let (iV, h) be a smooth compact Riemannian manifold, by the Nash embedding theorem, we can always assume (iV, h) is isometrically embedded in some Euclidean space R^. In order to discuss harmonic maps from (M, g) to (iV, h) , we work in the seperable Hilbert space L^ (M, R) , the set of maps u : M \—> R^, whose component functions have first derivatives in L. As in the smooth case,

A partial regularity result of harmonic maps from manifolds with bounded measurable Riemannian metrics