Conditions for existence of single valued optimal transport maps on convex boundaries with nontwisted cost

We prove that if $\mathrm{\Omega }\subset {\mathbb{R}}^{n+1}$ is a (not necessarily strictly) convex, $C^1$ domain, and $\mu$ and $\bar{\mu }$ are probability measures absolutely continuous with respect to surface measure on $\mathrm{\partial }\mathrm{\Omega }$, with densities bounded away from zero and infinity, whose $2$-Monge-Kantorovich distance is sufficiently small, then there exists a continuous Monge solution to the optimal transport problem with cost function given by the quadratic distance on the ambient space ${\mathbb{R}}^{n+1}$. This result is also shown to be sharp, via a counterexample when $\mathrm{\Omega }$ is uniformly convex but not $C^1$. Additionally, if $\mathrm{\Omega }$ is $C^{1,\alpha }$ regular for some $\alpha$, then the Monge solution is shown to be Hölder continuous.