We construct a solution to inverse mean curvature flow on an asymptotically hyperbolic 3-manifold which does not have the convergence properties needed in order to prove a Penrose–type inequality. This contrasts sharply with the asymptotically flat case. The main idea consists in combining inverse mean curvature flow with work done by Shi–Tam regarding boundary behavior of compact manifolds. Assuming the Penrose inequality holds, we also derive a nontrivial inequality for functions on S2.

We discuss our recent work [4] in which gravitational radiation was studied by evaluating the Wang-Yau quasi-local mass of surfaces of fixed size at the infinity of both axial and polar perturbations of the Schwarzschild spacetime, \`{a} la Chandrasekhar [1].

In this paper we study constant mean curvature surfaces  in a product space, M2 × R, where M2 is a complete Riemannian manifold. We assume the angle function ν = hN, @ @t i does not change sign on . We classify these surfaces according to the infimum c() of the Gaussian curvature of the projection of . When H 6= 0 and c() ≥ 0, then  is a cylinder over a complete curve with curvature 2H. If H = 0 and c() ≥ 0, then  must be a vertical plane or  is a slice M2 ×{t}, or M2 ≡ R2 with the flat metric and  is a tilted plane (after possibly passing to a covering space). When c() < 0 and H > p −c()/2, then  is a vertical cylinder over a complete curve of M2 of constant geodesic curvature 2H. This result is optimal. We also prove a non-existence result concerning complete multigraphs in M2 × R, when c(M2) < 0.

In this paper we investigate the life-span of classical solutions to the hyperbolic geometric flow in two space variables with slow decay initial data. By establishing some new estimates on the solutions of linear wave equations in two space variables, we give a lower bound of the life-span of classical solutions to the hyperbolic geometric flow with asymptotic flat initial Riemann surfaces.

Let f : X .仺 Y be a holomorphic map of complex manifolds, which is proper, K丯ahler, and surjective with connected fibers, and which is smooth over Y \ Z the complement of an analytic subset Z. Let E be a Nakano semi-positive vector bundle on X. In our previous paper, we discussed the Nakano semi-positivity of Rqf(KX/Y . E) for q . 0 with respect to the so-called Hodge metric, when the map f is smooth. Here we discuss the extension of the induced metric on the tautological line bundle O(1) on the projective space bundle P(Rqf(KX/Y . E)) 乬over Y \ Z乭 as a singular Hermitian metric with semi-positive curvature 乬over Y 乭. As a particular consequence, if Y is projective, Rqf(KX/Y . E) is weakly positive over Y \ Z in the sense of Viehweg.