Coarea Inequality for Monotone Functions on Metric Surfaces

We study coarea inequalities for metric surfaces – metric spaces that are topological surfaces, without boundary, and which have locally finite Hausdorff 2-measure ${\mathcal{H}}^2$. For monotone Sobolev functions $u\text{\hspace{0.17em}⁣}:X\to \mathbb{R}$, we prove the inequality \begin{equation*} \int_{ $\mathbb{R}$ }^{} \int_{ u^{-1}(t) } g ,d$\mathcal{H}$^{1} ,dt \leq κ \int_{ X } g ρ ,d$\mathcal{H}$^{2} \quad$\text{for every Borel }g\text{\hspace{0.17em}⁣}:X\to [0,\mathrm{\infty }]\text{,}$ \end{equation} where $\rho$ is any integrable upper gradient of $u$. If $\rho$ is locally $L^2$-integrable, we obtain the sharp constant $\kappa =4\mathrm{/}\pi$. The monotonicity condition cannot be removed as we give an example of a metric surface $X$ and a Lipschitz function $u\text{\hspace{0.17em}⁣}:X\to \mathbb{R}$ for which the coarea inequality above fails.