Full flexibility of isometric immersions of metrics with low Hölder regularity in Poznyak theorem's dimension

Abstract

A classical result by Poznyak asserts that any smooth $2$-dimensional Riemannian metric $g$, posed on the closure of a simply connected domain $\omega \subset {\mathbb{R}}^2$, has a smooth isometric immersion into ${\mathbb{R}}^4$. Using techniques of convex integration, we prove that for any $2$-dimensional $g\in C^{r,\beta }$, an isometric immersion of regularity $C^{1,\alpha }(\bar{\omega },{\mathbb{R}}^4)$ for any $\alpha <\min \{\frac{r+\beta }{2},1\}$, may be found arbitrarily close to any short immersion. The fact that this result's regularity reaches $C^{1,1-}$ for $g\in C^2$, which is referred to as "full flexibility", should be contrasted with: (i) the regularity $C^{1,1/3-}$ achieved by Cao, Hirsch and Inauen for isometric immersions into ${\mathbb{R}}^3$ and the lack of flexibility (rigidity) of such isometric immersions with regularity $C^{1,2/3+}$ proved by Borisov and then by Conti, de Lellis and Szekelyhidi; (ii) the regularity $C^{1,1-}$ obtained byt Källen for isometric immersions into higher codimensional space ${\mathbb{R}}^8$; and (iii) the regularity $C^{1,\frac{1}{1+d(d+1)/k}-}$ proved by the author in the general case of $d$-dimensional metrics and $(d+k)$-dimensional immersions for the closely related Monge-Ampère system.

59 pages, 2 figures. The paper is now self-contained, rather than relying on an external result concerning bounds for the normal frame, which was partially erroneous