Uniform boundedness principles for Sobolev maps into manifolds

Abstract Given a connected Riemannian manifold N , an m -dimensional Riemannian manifold M which is either compact or the Euclidean space, p ∈ [ 1 , + ∞ ) and s ∈ ( 0 , 1 ] , we establish, for the problems of surjectivity of the trace, of weak-bounded approximation, of lifting and of superposition, that qualitative properties satisfied by every map in a nonlinear Sobolev space W s , p ( M , N ) imply corresponding uniform quantitative bounds. This result is a nonlinear counterpart of the classical Banach–Steinhaus uniform boundedness principle in linear Banach spaces.