Sobolev spaces of isometric immersions of arbitrary dimension and co-dimension

Abstract

We prove the $$C^{1}_{\mathrm{{loc}}}$$ regularity and developability of $$W^{2,p}_{\mathrm{{loc}}}$$ isometric immersions of n-dimensional flat domains into $${\mathbb {R}}^{n+k}$$ where $$p\ge \min \{2k, n\}$$ . We also prove similar rigidity and regularity results for scalar functions of n variables for which the rank of the Hessian matrix is a.e. bounded by some $$k<n$$ , again assuming $$W^{2,p}_{\mathrm{{loc}}}$$ regularity for $$p\ge \min \{2k,n\}$$ . In particular, this includes results about the degenerate Monge–Ampere equation, $$\mathrm{{det}} D^2 u = 0$$ , corresponding to the case $$k=n-1$$ .