  On some specified convex supporting sets of spheres, we find a generalized
longitude function whose level sets are totally geodesic. Given an arbitrary
(weakly) harmonic map into spheres, the composition of the generalized
longitude function and harmonic map satisfies an elliptic equation of
divergence type. With the aid of corresponding Harnack inequality, we establish
image shrinking property and then the regularity results are followed. Applying
such results to study the Gauss image of minimal hypersurfaces in Euclidean
spaces, we obtain curvature estimates and corresponding Bernstein theorems.
