A Nash-Kuiper theorem for isometric immersions beyond Borisov's exponent

Given any short immersion from an $n$-dimensional bounded and simply connected domain into ${\mathbb{R}}^{n+1}$ and any Hölder exponent $\alpha <(1+n^2-n{)}^{-1}$, we construct a $C^{1,\alpha }$ isometric immersion arbitrarily close in the $C^0$ topology. This extends the classical Nash–Kuiper theorem and shows the flexibility of $C^{1,\alpha }$ isometric immersions beyond Borisov’s exponent. In particular, for $n=2$, the regularity threshold aligns with the Onsager exponent $1\mathrm{/}3$ for the incompressible Euler equations. Our proof relies on three novelties that allow for the cancellation of leading-order error terms in the convex integration scheme: a new corrugation ansatz, an integration by parts procedure, and an adapted algebraic decomposition of these errors.