We prove that in a Riemannian manifold
M
M
, each function whose Hessian is proportional the metric tensor yields a weighted monotonicity theorem. Such function appears in the Euclidean space, the round sphere
S
n
{S}^n
and the hyperbolic space
H
n
\mathbb {H}^n
as the distance function, the Euclidean coordinates of
R
n
+
1
\mathbb {R}^{n+1}
and the Minkowskian coordinates of
R
n
,
1
\mathbb {R}^{n,1}
. Then we show that weighted monotonicity theorems can be compared and that in the hyperbolic case, this comparison implies three
S
O
(
n
,
1
)
SO(n,1)
-distinct unweighted monotonicity theorems. From these, we obtain upper bounds of the Graham–Witten renormalised area of a minimal surface in term of its ideal perimeter measured under different metrics of the conformal infinity. Other applications include a vanishing result for knot invariants coming from counting minimal surfaces of
H
n
\mathbb {H}^n
and a quantification of how antipodal a minimal submanifold of
S
n
S^n
has to be in term of its volume.