Minimal Submanifolds And Calibrations

In this paper, we use calibration to study the structure of submanifolds and discuss the relation between calibrations and submanifolds. We can costruct an exterior n-form ξ for every hypersurface M in Euclidean space Rn+1. The submanifold M is minimal if and only if is closed (dξ = 0), then ξ is calibration and M is ξ-submanifold. In this way, we show that minimal hypersurfaces of Euclidean spaces can all be determined by calibrations locally. On the other hand, given calibration ξ in Rn+l, if it statisfies Frobenius condition, there is ξ-submanifold through every point. We know that each ξ- submanifold is homologically volume minimizing in Rn+1, so that every minimal hypersurface in Rn+1 is stable locally.For any harmonic function on Rn, we can define a calibration. By this way. we find a minimal hypersurface in Rn(n > 4) which is complete and between two parallel hyperplanes.