On the analysis of isometric immersions of Riemannian manifolds

This thesis is a study of problems related to isometric immersions of Riemannian manifolds in Euclidean space. We address three main questions: the weak continuity of geometric invariants along sequences of immersions of Riemannian manifolds; the construction of isometric immersions by weak compactness methods; and the validity and regularity of the Gauss equation. First, we investigate the validity of Cartan's equations for W1,p coframes on surfaces, for all 1 ≤ p ≤ ∞, and employ this to derive a version of the Gauß equation valid for W2,p immersed surfaces in R3. Under some additional regularity hypotheses, a distributional formulation of the Gauß equation on immersed surfaces in R3 is proved, and as a corollary, a new local regularity result is established for isometric immersions of positively curved surfaces. Investigating the weak continuity properties of immersions of Riemannian manifolds, we first prove a general weak continuity result for curvatures of connections with $L^p$ bounds on principal bundles. As a corollary, it is proved that when $p>2$, curvatures are weakly continuous for the weak $W^{2,p}$ convergence of immersions. In some cases, we also recover the weak continuity of the general Gauss equation when $p=2$. Finally, we give a general viscosity framework for constructing isometric immersions in prescribed target spaces under natural boundedness assumptions in $L^p$ spaces. Assuming this set-up, we prove a new weak compactness theorem for approximate solutions of the Gauss--Codazzi--Ricci equations.