Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Author: Hu, Tao
Institution: The University of Texas at Austin
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212066296