Weighted monotonicity theorems and applications to minimal surfaces in hyperbolic space

Manh Tien Nguyen

Published 2021-05-26Version 1

We show that there is a weighted monotonicity theorem corresponding to each function on a Riemannian manifold whose Hessian is a multiple of the metric tensor. Such functions appear in Euclidean space, 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 ), the time-weighted monotonicity theorem is compared with the unweighted version due to Anderson in \cite{Anderson82_CompleteMinimalVarieties}. 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_RenormalizedAreaProperly}. An argument on area-minimising cones suggests existence of a minimal surface in ( \mathbb{H}^4 ) bounded by the Hopf link ( \{(z,w)\in \mathbb{C}^2: 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. We also obtain its sister in ( \mathbb{S}^4 ) by the same method. A weighted monotonicity theorem is also proved in Riemannian manifolds whose sectional curvature is bounded from above.