论文标题
封闭的多种多样和球类别上平滑功能的临界点数量最少
The minimal number of critical points of a smooth function on a closed manifold and the ball category
论文作者
论文摘要
由Seifert和Threlfall引入的，平滑功能的隔离关键点的圆柱邻域是Lusternik-Schnirelmann理论中的必不可少的工具。我们猜想，平滑功能的每个孤立临界点都允许圆柱球邻居。我们表明，猜想对于锥形临界点，角膜合理的临界点以及满足Rothe H假设的关键点是正确的。特别是，至少对于那些不是无限退化的关键点，猜想至少是正确的。如果与猜想的主张相反，则有一些孤立的临界点不接受圆柱球邻域，那么我们说这种临界点是异国情调的。我们证明了一种lusternik-schnirelmann型定理，认为在维度的封闭歧管上，平滑函数的临界点的最小数量至少与6个相同，与单人takens在平滑球中填充M的单人takens中的最小数量相同。
Introduced by Seifert and Threlfall, cylindrical neighborhoods of isolated critical points of smooth functions is an essential tool in the Lusternik- Schnirelmann theory. We conjecture that every isolated critical point of a smooth function admits a cylindrical ball neighborhood. We show that the conjecture is true for cone-like critical points, Cornea reasonable critical points, and critical points that satisfy the Rothe H hypothesis. In particular, the conjecture holds true at least for those critical points that are not infinitely degenerate. If, contrary to the assertion of the conjecture, there are isolated critical points that do not admit cylindrical ball neighborhoods, then we say that such critical points are exotic. We prove a Lusternik-Schnirelmann type theorem asserting that the minimal number of critical points of smooth functions without exotic critical points on a closed manifold of dimension at least 6 is the same as the minimal number of elements in a Singhof-Takens filling of M by smooth balls with corners.
