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

TL;DR

This paper establishes weighted monotonicity theorems for functions with Hessians proportional to the metric in various spaces, leading to new geometric inequalities and invariants for minimal surfaces in hyperbolic and spherical geometries.

Contribution

It introduces a unified approach to weighted monotonicity theorems in different geometries and derives new bounds and invariants for minimal surfaces and submanifolds.

Findings

Derived weighted monotonicity theorems in hyperbolic and spherical spaces.

Established bounds for Graham--Witten renormalised area of minimal surfaces.

Provided new invariants and vanishing results related to minimal surfaces and knot theory.

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.