Isometric Immersions with Controlled Curvatures

We $\delta$-approximate strictly short (e.g. constant) maps between Riemannin manifolds $f_0:X^m\to Y^N$ for $N>>m^2\mathrm{/}2$ by $C^{\mathrm{\infty }}$-smooth isometric immersions $f_{\delta }:X^m\to Y^N$ with curvatures $curv(f_{\delta })<\frac{\sqrt{3}}{\delta }$, for $\delta \to 0$