Saturating spaces for the squared-gradient nonlinearity on compact Riemannian manifolds via Nash embeddings
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 $f\mapsto|\nabla_g f|_g^2$ 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 $\mathbb{R}^n$ 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.
math.AP↗