Systolic inequalities and the Horowitz-Myers conjecture

Let $n$ be an integer with $3\le n\le 7$, let $M$ be a compact manifold of dimension $n$ with boundary $\mathrm{\partial }M$, and let $g$ be a Riemannian metric on $M$ with scalar curvature at least $-n(n-1)$. Under a topological assumption on $M$, we establish an inequality relating the infimum of the boundary mean curvature to the systole of the boundary $\mathrm{\partial }M$. As a consequence, we obtain a new positive energy theorem, with equality being attained by the Horowitz-Myers metrics.