Weighted monotonicity theorems and applications to minimal surfaces of ℍⁿ and 𝕊ⁿ

Abstract

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.