PreprintCC BY 4.0HAL
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 gradient squared nonlinear map 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 an Euclidean space 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{buffe2026saturating,
title = {{Saturating spaces for the squared-gradient nonlinearity on compact Riemannian manifolds via Nash embeddings}},
author = {Rémi Buffe and Alessandro Duca and Hugo Parada},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05829373v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Saturating spaces for the squared-gradient nonlinearity on compact Riemannian manifolds via Nash embeddings
AU - Rémi Buffe
AU - Alessandro Duca
AU - Hugo Parada
PY - 2026
DA - 2026/10/21
PB - HAL
UR - https://hal.science/hal-05829373v1
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 gradient squared nonlinear map 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 an Euclidean space 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 -