Full flexibility of the Monge-Amp\`ere system in codimension $d_*-d+1$

TL;DR

This paper demonstrates the density of ,lpha solutions to the Monge-Ampre system in certain dimensions and codimensions, extending previous flexibility results and applying advanced techniques to broader geometric contexts.

Contribution

It generalizes and strengthens existing flexibility results for the Monge-Ampre system and isometric immersions across various dimensions and codimensions.

Findings

,lpha solutions are dense in the space of continuous functions for specific dimensions and codimensions.

The proof scheme extends to local full flexibility of isometric immersions of Riemannian metrics.

The results can be extended to compact manifolds with higher codimension using advanced techniques.

Abstract

We prove that $\mathcal{C}^{1,\alpha}$ solutions to the Monge-Amp\`ere system in dimension $d$ and codimension $k= d_*-d+1$, where $d_*$ denotes the Janet dimension, are dense in the space of continuous functions, for every H\"older exponent $\alpha<1$. Our result strengthens the statement in [Lewicka 2022], obtained for $k = 2d_*$ and based on ideas from [K\"allen 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 $d_* + 1$, 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 $(d+1)d_*-d+1$.