Two Monotone Quantities Along the Inverse Mean Curvature Flow in Hn

ABSTRACT In this paper, we derive two monotone quantities along the inverse mean curvature flow (IMCF) in hyperbolic space . First, we prove that the quantity is increasing along the IMCF, which relates the weighted volume enclosed by hypersurface and the weighted total mean curvature of the hypersurface. Second, we prove that the first eigenvalue of ‐Laplacian of the free boundary hypersurfaces supported on the geodesic spheres of is decreasing along the IMCF.