Abstract; A Nash-Kuiper theorem for $C^{1,1/5}$ isometric immersions of disks

A Nash-Kuiper theorem for $C^{1,1/5}$ isometric immersions of disks

In a joint work with Dominik Inauen and Laszlo Szekelyhidi we
prove that, given a $C^2$ Riemannian metric $g$ on the
$2$-dimensional disk, any short $C^1$ embedding can be
uniformly approximated with $C^{1,\alpha}$ isometric embeddings
for any $\alpha < \frac{1}{5}$. The same statement with $C^1$
isometric embeddings is a groundbreaking result due to
Nash and Kuiper. The previous Hoelder threshold, 1/7, was first
announced in the sixties by Borisov. If time allows I will also
discuss the connection with a conjecture of Onsager on weak
solutions to the Euler equations.