Space-filling surfaces: sharp H\"older 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 $m\geq 2$ we build $\alpha$-H\"older continuous parameterizations $f:[0,1]^m\rightarrow[0,1]^{m+1}$ with sharp exponent $\alpha=m/(m+1)$. In particular, there exist $(2/3)$-H\"older continuous surjections from squares to cubes. This solves Arnold's problem 1988--5.
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