Monotonicity formulas for parabolic flows on manifolds
Monotonicity formulas for parabolic flows on manifolds
复制标题
DOI：
10.4310/cag.1993.v1.n1.a7
复制
发表时间：
1993
影响因子：
0.7
通讯作者：
R. Hamilton
中科院分区：
文献类型：
--
作者：
R. Hamilton
最近，Michael Struwe[S]和Gerhard Huisken[HU2]独立地导出了欧氏区域上调和映射热流和欧氏空间中超曲面平均曲率流的单调公式。本文证明了如何将这些结果推广到一般紧流形上的流动，并给出了杨-Mills热流的类似单调性公式。关键部分是[H]中给出的标量热方程正解的矩阵Harnack估计。在[GrH]中，作者证明了如何使用单调性公式来证明调和映射热流中快速形成的奇点对同态收缩孤子是渐近的；在其他情况下也可能得到类似的结果，就像Huisken在[HU2]中对欧氏空间中的平均曲率流所做的那样。我们只获得了一类特殊度量的严格单调性，但通常有一个足够小的误差项来给出相同的效果。(陈雨梅和Michael Struwe[CS]对流形上的错误给出了不同的方法。)这类特殊的度量是那些Ricci平行的(使得Dirjk=0)并且具有弱正截面曲率的度量(使得对于所有向量V和W，RijkiViWjVkWt>；0)。例如，如果M是平坦的、球面的或复射影空间，或它们的乘积，或乘积与有限自由等距群的商，则这是成立的。在每种情况下，我们考虑紧致流形M上的抛物型方程在某个有限时间区间0<；t<；T的解，我们设k是任意的
Recently Michael Struwe [S] and Gerhard Huisken [Hu2] have independently derived monotonicity formulas for the Harmonic Map heat flow on a Euclidean domain and for the Mean Curvature flow of a hypersurface in Euclidean space. In this paper we show how to generalize these results to the case of flows on a general compact manifold, and we also give the analogous monotonicity formula for the Yang-Mills heat flow. The key ingredient is a matrix Harnack estimate for positive solutions to the scalar heat equation given in [H]. In [GrH] the authors show how to use the monotonicity formula to prove that rapidly forming singularities in the Harmonic Map heat flow are asymptotic to homothetically shrinking solitons; similar results may be expected in other cases, as Huisken does in [Hu2] for the Mean Curvature flow in Euclidean space. We only obtain strict monotonicity for a special class of metrics, but in general there is an error term which is small enough to give the same effect. (Chen Yummei and Michael Struwe [CS] give a different approach to the error on manifolds.) The special class of metrics are those which are Ricci parallel (so that DiRjk = 0) and have weakly positive sectional curvature (so that RijkiViWjVkWt > 0 for all vectors V and W). This holds for example if M is flat or a sphere or a complex projective space, or a product of such, or a quotient of a product by a finite free group of isometries. In each case we consider a solution to our parabolic equation on a compact manifold M for some finite time interval 0 < t < T, and we let k be any