Infinitesimal Hilbertianity for manifolds with Sobolev regular Riemannian metrics through uniform lower capacity bounds

We prove that metric measure spaces arising from $2$-manifolds equipped with Riemannian metrics $g$ such that $g,g^{-1}\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{\infty }}\cap W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,p}$ are infinitesimally Hilbertian. Along the way, we establish uniform lower $1$-capacity bounds of several $1$-dimensional subsets of ${\mathbb{R}}^2$.