Flexibility of the isometric immersion system in arbitrary dimension and codimension and the energy scaling of prestrained thin films

Abstract

We prove that, for a given $C^{r,\beta }$-regular Riemann metric posed on a $d$-dimensional domain, every short immersion into the Euclidean space ${\mathbb{R}}^{d+k}$, can be uniformly approximated by exact isometric immersions of regularity $C^{1,\alpha }$ for any $\alpha <{\alpha }_0=\min \{\frac{r+\beta }{2},\frac{1}{1+d(d+1)/k}\}$. Our theorem recovers several previously known results as special cases. The novelty thereof lies in providing a unified flexibility statement for arbitrary dimensions $d$ and codimensions $k$, while also treating the so far uncharted range $k\in (1,\frac{d(d+1)}{2}-d+1)\setminus \{d\}$, where no corresponding general result was previously available. Our threshold flexibility exponent ${\alpha }_0$ agrees with that previously obtained for the closely related Monge-Ampère system. As an application, we prove a new estimate in the quantitative immersability of thin prestrained films, setting the scaling exponent of the infimum of non-Euclidean energies in presence of an arbitrary prestrain metric, and in the limit of the film's vanishing thickness, at $\frac{4{\alpha }_0}{{\alpha }_0+1}$.

29 pages. The paper is now self-contained, whereas previously it relied on an external result concerning bounds for the normal frame, which was partially erroneous