  In the asymptotically locally hyperbolic setting it is possible to have
metrics with scalar curvature at least -6 and negative mass when the genus of
the conformal boundary at infinity is positive. Using inverse mean curvature
flow, we prove a Penrose inequality for these negative mass metrics. The
motivation comes from a previous result of P. Chru\'sciel and W. Simon, which
states that the Penrose inequality we prove implies a static uniqueness theorem
for negative mass Kottler metrics.

Differential Geometry