【作者】 王庆；

【导师】 周建伟；

【作者基本信息】 苏州大学 ， 基础数学， 2004， 硕士

【摘要】 本文利用calibration这一研究子流形的有力工具进一步了解子流形的结构并讨论它们与极小子流形之间的关系．我们证明了对于欧氏空间Rⁿ⁺¹中每一超曲面M，可以构造η-微分式ζ，而超曲面极小的条件恰是ζ为闭形式，即dζ=0的条件。这时ζ是calibration，而M是ζ的积分子流形，从而证明了欧氏空间中的极小超曲面局部都可以由calibration决定。反之，给定Rⁿ⁺¹上的calibrationζ，如果它满足Frobenius可积条件，则过每一点有ζ的积分子流形M，我们知道ζ的积分子流形在其同调类中体积最小，自然M也局部是Rⁿ⁺¹中的稳定极小子流形。 本文还证明了对于定义在欧氏空间中Rⁿ上的某些调和函数，自然地可以定义calibration，利用这一方法我们找到一个夹在两个超平面中的完备极小超曲面。更多还原

【Abstract】 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.更多还原