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.

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We establish an analytic interpretation for the index of certain transversally elliptic symbols on non-compact manifolds. By using this interpretation, we establish a geometric quantization formula for a Hamiltonian action of a compact Lie group acting on a non-compact symplectic manifold with proper moment map. In particular, we present a solution to a conjecture of Michèle Vergne in her ICM 2006 plenary lecture.

In this paper, we provide a priori $L^2$ error estimates for the semi-discrete discontinuous Galerkin method and the local discontinuous Galerkin method for one- and two-dimensional nonlinear Hamilton-Jacobi equations with smooth solutions. With a special Gauss-Radau projection, the optimal error estimates on rectangular meshes are obtained.