Weighted monotonicity theorems and applications to minimal surfaces of $\mathbb {H}^n$ and $S^n$

Abstract:

We prove that in a Riemannian manifold $M$, each function whose Hessian is proportional the metric tensor yields a weighted monotonicity theorem. Such function appears in the Euclidean space, the round sphere ${S}^n$ and the hyperbolic space $\mathbb {H}^n$ as the distance function, the Euclidean coordinates of $\mathbb {R}^{n+1}$ and the Minkowskian coordinates of $\mathbb {R}^{n,1}$. Then we show that weighted monotonicity theorems can be compared and that in the hyperbolic case, this comparison implies three $SO(n,1)$-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 $\mathbb {H}^n$ and a quantification of how antipodal a minimal submanifold of $S^n$ has to be in term of its volume.