Coarea Inequality for Monotone Functions on Metric Surfaces

TL;DR

This paper establishes a coarea inequality for monotone Sobolev functions on metric surfaces, providing sharp constants and demonstrating the necessity of monotonicity through counterexamples.

Contribution

It proves a coarea inequality with sharp constants for monotone Sobolev functions on metric surfaces, extending classical results to a broader metric setting.

Findings

Proves coarea inequality with sharp constant 4/π for monotone functions.

Shows the inequality fails without monotonicity through counterexamples.

Extends classical geometric measure theory results to metric surfaces.

Abstract

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 \kappa \int_{ X } g \rho \,d\mathcal{H}^{2} \quad\text{for every Borel $g \colon X \rightarrow \left[0,\infty\right]$,} \end{equation*} where $\rho$ is any integrable upper gradient of $u$. If $\rho$ is locally $L^2$-integrable, we obtain the sharp constant $\kappa=4/\pi$. 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.