Flexibility of isometric embeddings beyond Borisov's exponent

The problem of isometrically embedding a Riemannian manifold into Euclidean space (that is, preserving the length of curves) is a fundamental problem in differential geometry. Classical results indicate that sufficiently smooth isometries are often unique, up to translation and rotation. However, the Nash–Kuiper theorem presents the seemingly paradoxical abundance (or flexibility) of isometries that are only $C^1$. A natural question then arises: Do these uniqueness and flexibility properties extend to isometries of class $C^{1,\alpha}$ for $0 < \alpha < 1$, particularly when $\alpha$ is close to $1$ (uniqueness) or close to $0$ (flexibility)? Moreover, is there a sharp Hölder exponent $\alpha_0$ that separates these contrasting behaviors?

In 1965, Yu. F. Borsov conjectured that the flexibility of isometric embeddings extends to the class $C^{1,\alpha}$ for $\alpha < \frac{1}{1+n+n^2}$, a claim that was proven true in 2012 by Conti, De Lellis, and Székelyhidi. In this talk, I will discuss a recent extension of their result, which in particular achieves flexibility up to the Onsager exponent $\alpha < \frac{1}{3}$ in two dimensions.