4 problems

Let $\alpha\in(0,1)$ and let $k>n$. An $\alpha$-Hölder continuous embedding is an embedding $F:\mathbb{B}^k\subset\mathbb{R}^k\to\mathbb{H}_n$ satisfying a Hölder estimate of expon…

Let $\Omega$ be an open subset of $\mathbb{R}^k$, let $n$ be a positive integer, and let $C^{\gamma}(\Omega,\mathbb{H}_n)$ denote the maps that are $\gamma$-Hölder with respect to…

Let $\mathbb R^2$ and $\mathbb H_1$ denote the Euclidean plane and the first Heisenberg group, respectively. A map is a $C^\alpha$-embedding if it is an embedding with Hölder expon…

Let $g_1$ be the metric on $S^3$ at parameter $t=1$, and let $g_\infty$ be the limit Carnot–Carathéodory metric on $S^3$. A degree 1 $C^\alpha$ map $F$ from $(S^3,g_1)$ to…