Full flexibility of isometric immersions of metrics with low H\"older 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 $g\in\mathcal{C}^{r,\beta}$, an isometric immersion of regularity $\mathcal{C}^{1,\alpha}(\bar\omega,\mathbb{R}^4)$ 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 $\mathcal{C}^{1,1-}$ for $g\in \mathcal{C}^2$, which is referred to as "full flexibility", should be contrasted with: (i) the regularity $\mathcal{C}^{1,1/3-}$ 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 $\mathcal{C}^{1, 2/3+}$ proved by Borisov and then by Conti, de Lellis and Szekelyhidi; (ii) the regularity $\mathcal{C}^{1,1-}$ obtained byt K\"allen for isometric immersions into higher codimensional space $\mathbb{R}^{8}$; and (iii) the regularity $\mathcal{C}^{1,\frac{1}{1+d(d+1)/k}-}$ proved by the author in the general case of $d$-dimensional metrics and $(d+k)$-dimensional immersions for the closely related Monge-Amp\`ere system.