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

Curvature estimates for minimal hypersurfaces via generalized longitude function