AI 中文总结
本文证明非零紧支撑C^1函数梯度值域为内部闭包，一般C^1函数梯度值域闭包为内部闭包与渐近梯度值之并，并证明紧致光滑拉格朗日子流形及其辛同胚像有正则闭动量投影。
AI 中文摘要
我们证明，对于每个 $n\geq1$，$\mathbb{R}^n$ 上任意非零紧支撑 $C^1$ 函数的梯度值域是其内部的闭包。更一般地，对于任意 $C^1$ 函数，其梯度值域的闭包是值域内部的闭包与渐近梯度值集合的并集。我们还证明，$\mathbb{R}^{2n}$ 中无边界的光滑紧致拉格朗日子流形及其在辛同胚下的像具有正则闭动量投影。
英文摘要
We prove that the gradient range of every nonzero compactly supported $C^1$ function on $\mathbb{R}^n$ is the closure of its interior, for every $n\geq1$. More generally, for an arbitrary $C^1$ function, the closure of its gradient range is the union of the closure of the range's interior and the set of asymptotic gradient values. We also prove that compact smooth Lagrangian submanifolds without boundary in $\mathbb{R}^{2n}$ and their images under symplectic homeomorphisms have regular closed momentum projections.
Comments8 pages, 1 figure
