We study coarea inequalities for metric surfaces — metric spaces that are topological surfaces, without boundary, and which have locally finite Hausdorff 2-measure
H
2
\mathcal {H}^2
. For monotone Sobolev functions
u
:
X
→
R
u\colon X \to \mathbb {R}
, we prove the inequality
∫
R
∗
∫
u
−
1
(
t
)
g
d
H
1
d
t
≤
κ
∫
X
g
ρ
d
H
2
for every Borel
g
:
X
→
[
0
,
∞
]
,
\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
ρ
\rho
is any integrable upper gradient of
u
u
. If
ρ
\rho
is locally
L
2
L^2
-integrable, we obtain the sharp constant
κ
=
4
/
π
\kappa =4/\pi
. The monotonicity condition cannot be removed as we give an example of a metric surface
X
X
and a Lipschitz function
u
:
X
→
R
u \colon X \to \mathbb {R}
for which the coarea inequality above fails.