Simple geodesics on a punctured surface

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.

math.MG↗

arXiv2026

The disjoint disks property for Busemann $G$-spaces

We prove that every finite-dimensional Busemann $G$-space of dimension at least five has the disjoint disks property (DDP). For a sufficiently small metric sphere $L=S(c,r)$, we show that every embedded arc contained in an exact distance level is a homotopical $Z_2$-set in $L$. It follows that $L$ has the disjoint arc-disk property and the disjoint homotopies property. Daverman's product theorem then gives DDP for $L\times\mathbb R$, and a local avoidance argument at the center yields DDP for the ambient $G$-space. Since finite-dimensional Busemann $G$-spaces are generalized manifolds, in dimensions at least five the remaining obstruction to the Busemann conjecture is the resolution problem.

math.MG↗

arXiv2026

Every Compact Metric Space Is Isometrically Embeddable into the Gromov-Hausdorff Space

Let $(\mathcal{M},d_{\mathrm{GH}})$ denote the Gromov-Hausdorff space of isometry classes of nonempty compact metric spaces. We prove that every nonempty compact metric space is isometrically embeddable into $(\mathcal{M},d_{\mathrm{GH}})$. More precisely, for every $D>0$ and every nonempty compact metric space $K$ with $\operatorname{diam} K\le D$, we realize the space of all $1$-Lipschitz functions on $K$ with values in $[0,D]$ as a family of metrics on a fixed Cantor space. Under this realization, the Gromov-Hausdorff distance agrees exactly with the uniform distance between functions, and each resulting metric space has diameter at most $76D$. We also construct finite approximations for which the Gromov-Hausdorff distance is given by an exact formula, together with a uniform approximation estimate.

math.MG↗