Title: Weighted monotonicity theorems and applications to minimal surfaces of the hyperbolic space
Abstract: We show that there is a weighted version of monotonicity theorem corresponding to each function on a Riemannian manifold whose Hessian is a multiple of the metric tensor. Such function appears in the Euclidean space, the hyperbolic space $\mathbb{H}^n$ and the round sphere $\mathbb{S}^n$ as the distance function, the Minkowskian coordinates of $\mathbb{R}^{n,1}$ and the Euclidean coordinates of $\mathbb{R}^{n+1}$.
In $\mathbb{H}^n$, we show that the time-weighted monotonicity theorem implies the unweighted version in \cite{Anderson82}. Applications include upper bounds for Graham--Witten renormalised area of minimal surfaces in term of the length of boundary curve and a complete computation of Alexakis--Mazzeo degrees defined in \cite{Alexakis.Mazzeo10}.
An argument on area-minimising cones suggests the existence of a minimal surface in $\mathbb{H}^4$ bounded by the Hopf link $\{zw=\epsilon > 0, |z|^2 +|w|^2 = 1\}$ other than the pair of disks. We give an explicit construction of a minimal annulus in $\mathbb{H}^4$ with this property and obtain by the same method its sister in $\mathbb{S}^4$.
A weighted monotonicity theorem is also proved in Riemannian manifolds whose sectional curvature is bounded from above.