2026/07/17 by Marta Lewicka, Hui Li
#math.AP
We prove that, for a given Cr,β-regular Riemann metric posed on a d-dimensional domain, every short immersion into the Euclidean space ℝd+k, can be uniformly approximated by exact isometric immersions of regularity C1,α for any α<α0=min\(r+β)/(2), (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∈ (1, (d(d+1))/(2)-d+1)∖ \d\, where no corresponding general result was previously available. Our threshold flexibility exponent α0 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 (4α0)/(α0+1).