Systolic inequalities and the Horowitz-Myers conjecture

TL;DR

This paper proves a new inequality linking boundary systole and mean curvature for certain Riemannian manifolds, leading to a positive energy theorem with equality for Horowitz-Myers metrics.

Contribution

It establishes a novel systolic inequality for manifolds with scalar curvature bounds, advancing understanding of geometric inequalities and energy theorems.

Findings

Derived a systolic inequality for manifolds with scalar curvature ≥ -n(n-1)

Proved a new positive energy theorem with equality for Horowitz-Myers metrics

Extended geometric inequalities to higher-dimensional manifolds with boundary

Abstract

Let $n$ be an integer with $3 \leq n \leq 7$, and let $g$ be a Riemannian metric on $B^2 \times T^{n-2}$ with scalar curvature at least $-n(n-1)$. We establish an inequality relating the systole of the boundary to the infimum of the mean curvature on the boundary. As a consequence, we obtain a new positive energy theorem where equality holds for the Horowitz-Myers metrics.