Infinity-harmonic functions in the plane: Regularity by injectivity

It has been a long standing conjecture that the $ \infty $-harmonic functions in the plane have a 1/3-H\"older continuous gradient. It \emph{is} known that solutions are $ C^1 $ and that the gradient is locally $ \alpha $-H\"older, but $ \alpha $ comes without any positive lower bound. Aronsson's solution $ x^{4/3} - y^{4/3} $ shows that no better general regularity is possible. In the plane there is also a connection between the $ \infty $-Laplace equation and the one-dimensional heat equation, observed already by Aronsson himself. I shall show that this link can be accessed under a certain injectivity condition on the gradient, and that the caloric structure then is enough to prove the 1/3-H\"older continuity. Of course, an injective gradient is by no means a \emph{necessary} condition, as seen by smooth solutions such as the planes and cones.