中心仿射几何中的热流问题

Heat Flow Problem in Centroaffine Geometry

假设初始流形是仿射空间中的局部严格凸的紧致无边光滑超曲面，坐标原点在曲面凹的一侧，位置矢量与曲面横截，则中心仿射超曲面的发展方程ｘｔ＝－Ｋ１ｎ＋２ｘ的解在一个最大有限时间区间［０，Ｔ＊）内存在，并且保局部严格凸性及位置矢量与解曲面的横截性，在有限时间后解曲面收缩于一点．

Let initial manifold be a locally strictly convex,compact without boundary,smooth hypersurface whose position vector is transversal to the hypersurface in affine space.The origin is in the concave side of the hypersurface.Then the solution of the evolution equation t·x=-K 1n+2x of centreaffine hypersurface exists in a maximum finite time interval [0,T *) ,and preserves the locally strict convexity and the transversality of the position vector to the hypersurface,and shrinks to a point as tT * .