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 $u\colon X \to \mathbb{R}$, 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 $g \colon X \rightarrow \left[0,\infty\right]$,} \end{equation*} where $ρ$ is any integrable upper gradient of $u$. If $ρ$ is locally $L^2$-integrable, we obtain the sharp constant $κ=4/π$. The monotonicity condition cannot be removed as we give an example of a metric surface $X$ and a Lipschitz function $u \colon X \to \mathbb{R}$ for which the coarea inequality above fails.

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