PreprintCC BY 4.0arXiv
Saturating spaces for the squared-gradient nonlinearity on compact Riemannian manifolds via Nash embeddings
Abstract
We study the saturation property on smooth compact Riemannian manifolds without boundary. Starting from a finite-dimensional space of smooth functions, one considers successive iterations of the nonlinear map f↦|∇g f|g2 and convex combinations. We extend to the general setting of compact manifolds existing results on particular geometries, namely the torus, spheres and Lie groups. Our approach relies on the existence of an isometric embedding of the manifold into ℝn due to Nash. As an application, we obtain global approximate controllability in arbitrarily small time for the bilinear Schrödinger equation on any connected compact Riemannian manifold, with finitely many potentials.
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BibTeX
@misc{remi2026saturating,
title = {{Saturating spaces for the squared-gradient nonlinearity on compact Riemannian manifolds via Nash embeddings}},
author = {Buffe Rémi and Duca Alessandro and Parada Hugo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29952},
eprint = {2610.29952},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29952},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Saturating spaces for the squared-gradient nonlinearity on compact Riemannian manifolds via Nash embeddings
AU - Buffe Rémi
AU - Duca Alessandro
AU - Parada Hugo
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29952
UR - https://arxiv.org/abs/2610.29952
AB - We study the saturation property on smooth compact Riemannian manifolds without boundary. Starting from a finite-dimensional space of smooth functions, one considers successive iterations of the nonlinear map f↦|∇_g f|_g^2 and convex combinations. We extend to the general setting of compact manifolds existing results on particular geometries, namely the torus, spheres and Lie groups. Our approach relies on the existence of an isometric embedding of the manifold into ℝ^n due to Nash. As an application, we obtain global approximate controllability in arbitrarily small time for the bilinear Schrödinger equation on any connected compact Riemannian manifold, with finitely many potentials.
ER -