PreprintarXiv licencearXiv
Regular closed gradient ranges and asymptotic gradient values
Abstract
We prove that the gradient range of every nonzero compactly supported C1 function on ℝn is the closure of its interior, for every n≥1. More generally, for an arbitrary C1 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 ℝ2n and their images under symplectic homeomorphisms have regular closed momentum projections.
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BibTeX
@misc{hu2026regular,
title = {{Regular closed gradient ranges and asymptotic gradient values}},
author = {Tao Hu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05951},
eprint = {2609.05951},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05951},
}
RIS
TY - UNPB
TI - Regular closed gradient ranges and asymptotic gradient values
AU - Tao Hu
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.05951
UR - https://arxiv.org/abs/2609.05951
AB - We prove that the gradient range of every nonzero compactly supported C^1 function on ℝ^n is the closure of its interior, for every n≥1. 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 ℝ^2n and their images under symplectic homeomorphisms have regular closed momentum projections.
ER -