Let $\phi$ be a smooth map from a smooth compact surface $S$ (possibly with boundary) to $\mathbb{R}^n$, equipped with the euclidean metric. Suppose that $S$ is equipped with a volume form $\omega$, and that $|\phi^*(vol_{eucl})|\leq |\omega|$.
Does there exist a universal constant $K$ and a riemmannian metric $g$ on $S$ which induces $\omega$, and such that $\phi$ is $K$-Lipschitz? Can we take $K$ arbitrarily close to $1$? $K=1$?