带边界流形上梯度流的层凸性
Stratified convexity & concavity of gradient flows on manifolds with boundary
摘要：
正如Morse \cite{Mo}所观察到的那样，具有边界的紧致光滑流形$X$上的任何一般向量场$v$都会引起边界$\d X$被紧致子流形$\{\d_j^\pm X(v)\}_{1 \leq j \leq \dim(X)}$分层，其中$\textup{codim}(\d_j^\pm X(v))= j$。我们的主要观察是，这种分层反映了分层凸/分层的边界$\d X$相对于$v$流。我们研究了向量场$v$变形下的分层行为。我们还研究了凸/凹遍历$v$ -流的存在对$X$拓扑的限制。设$v_1$是$v$在$\d X$切丛上的正交投影。我们把边界上的$v_1$ -流的动力学与$X$中的$v$是凸/凹的性质联系起来。这种联系是一个更普遍的现象，我们称之为“穿越场的全息术”的一个例子-这是另一篇论文的主题。
英文摘要：
As has been observed by Morse \cite{Mo}, any generic vector field $v$ on a compact smooth manifold $X$ with boundary gives rise to a stratification of the boundary $\d X$ by compact submanifolds $\{\d_j^\pm X(v)\}_{1 \leq j \leq \dim(X)}$, where $\textup{codim}(\d_j^\pm X(v))= j$. Our main observation is that this stratification reflects the stratified convexity/concavity of the boundary $\d X$ with respect to the $v$-flow. We study the behavior of this stratification under deformations of the vector field $v$. We also investigate the restrictions that the existence of a convex/concave traversing $v$-flow imposes on the topology of $X$. Let $v_1$ be the orthogonal projection of $v$ on the tangent bundle of $\d X$. We link the dynamics of the $v_1$-flow on the boundary with the property of $v$ in $X$ being convex/concave. This linkage is an instance of more general phenomenon that we call "holography of traversing fields"---a subject of a different paper to follow.
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