Full flexibility of the Monge-Ampère system in codimension $d_{\ast }-d+1$

Abstract

We prove that ${\mathcal{C}}^{1,\alpha }$ solutions to the Monge-Ampère system in dimension $d$ and codimension $k=d_{\ast }-d+1$, where $d_{\ast }$ denotes the Janet dimension, are dense in the space of continuous functions, for every Hölder exponent $\alpha <1$. Our result strengthens the statement in [Lewicka 2022], obtained for $k=2d_{\ast }$ and based on ideas from [Källen 1978] in the context of the isometric immersion system. It also generalizes the result of [Inauen-Lewicka 2025], where full flexibility was established in dimension $d=2$ and codimension $k=2$. The same proof scheme further yields local full flexibility of isometric immersions of $d$-dimensional Riemannian metrics into Euclidean space of dimension $d_{\ast }+1$, generalizing the result in [Lewicka 2025] proved for $d=k=2$. By using techniques of [Conti-De Lellis-Szekelyhidi], the result can be extended to compact manifolds, in codimension $(d+1)d_{\ast }-d+1$.

27 pages, 1 figure