We study the question of local and global uniqueness of completions, based on null geodesics, of Lorentzian manifolds. We show local uniqueness of such boundary extensions. We give a necessary and sufficient condition for existence of unique maximal completions. The condition is verified in several situations of interest. This leads to existence and uniqueness of maximal spacelike conformal boundaries, of maximal strongly causal boundaries, as well as uniqueness of conformal boundary extensions for asymptotically simple space-times. Examples of applications include the definition of mass, or the classification of inequivalent extensions across a Cauchy horizon of the Taub space-time.

We study the spaces of locally finite stability conditions on the derived categories of coherent sheaves on the minimal resolutions of $A_n$-singularities supported at the exceptional sets. Our main theorem is that they are connected and simply-connected. The proof is based on the study of spherical objects in A. Ishii and H. Uehara, "Autoequivalences of derived categories on the minimal resolutions of $A_n$-singularities on surfaces", J. Differential Geom., 71(3):385–435, 2005, and the homological mirror symmetry for $A_n$-singularities.

A sharp affine isoperimetric inequality is established which gives a sharp lower bound for the volume of the polar body. It is shown that equality occurs in the inequality if and only if the body is a simplex.

Suppose $L$ is a lamination of a Riemannian manifold by hypersurfaces with the same constant mean curvature $H$. We prove that every limit leaf of $L$ is stable for the Jacobi operator. A simple but important consequence of this result is that the set of stable leaves of $L$ has the structure of a lamination.

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 $S^2$.