中心仿射超曲面的热方程

The Heat Equation in Centroaffine Hypersurfaces

假设初始流形是仿射空间中的局部严格凸的紧致无边的光滑超曲面 ,坐标原点在曲面凹的一侧 ,位置矢量与曲面横截 ,利用欧氏支撑函数 ,得到中心仿射超曲面的热方程的解在任何有限时间区间内都存在 ,并且保局部严格凸性及位置矢量与解曲面的横截性 ,当时间趋于无穷大时 。

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 Making use of the Euclid supporting function,the solution of the heat equation of centroaffine hypersurface exists in any finite time interval,and preserves the locally strict convexity and the transversality of the position vector to the solution hypersurface,and shrinks to a point as t→∞