Space-filling surfaces: sharp Hölder continuous parameterizations from squares to cubes

Following a hint of Semmes, we employ Stong’s bijections between integer lattices to construct space-filling surfaces, which are higher-dimensional analogues of space-filling curves. For each $m\ge 2$ we build $\alpha$-Hölder continuous parameterizations $f:[0,1{]}^m\to [0,1{]}^{m+1}$ with sharp exponent $\alpha =m\mathrm{/}(m+1)$. In particular, there exist $(2\mathrm{/}3)$-Hölder continuous surjections from squares to cubes. This solves Arnold’s problem 1988–5.