arXiv · math/0505622
We construct several non-trivial examples of CAT(1) spaces by using the idea of free construction.
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Koichi Nagano. 2005-08-27. Free construction of CAT(1) spaces. https://arxiv.org/abs/math/0505622
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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 $α$-Hölder continuous parameterizations $f:[0,1]^m\rightarrow[0,1]^{m+1}$ with sharp exponent $α=m/(m+1)$. In particular, there exist $(2/3)$-Hölder continuous surjections from squares to cubes. This solves Arnold's problem 1988--5.
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