In this talk, I will discuss very recent results in substatic manifolds. These manifolds satisfy the curvature conditions

\[

f\ric - \nabla \nabla f + \Delta f g \geq 0,

\]

for a potential function $f$. These metrics naturally arise from the static spacetimes of General Relativity, and constitute a generalization of those with nonnegative Ricci curvature.

After having pointed a surprising conformal relation with $\mathrm{CD}(0,1)$ spaces, I will deal with Bishop-Gromov type theorems, leading in turn to a Willmore-type inequality written in terms of a new Asymptotic Volume Ratio. Time permitting, I will present an application to an Isoperimetric Inequality for a suitably weighted volume. These results are, to our knowledge, new also in model situations.

The talk is based on a work in progress with S. Borghini (Milano-Bicocca).