A horospherical torus about a cusp of a hyperbolic manifold inherits a Euclidean similarity structure, called a cusp shape. We bound the change in cusp shape when the hyperbolic structure of the manifold is deformed via cone deformation preserving the cusp. The bounds are in terms of the change in structure in a neighborhood of the singular locus alone. We then apply this result to provide information on the cusp shape of many hyperbolic knots, given only a diagram of the knot. Specifically, we show there is a universal constant C such that if a knot admits a prime, twist reduced diagram with at least C crossings per twist region, then the length of the second shortest curve on the cusp torus is bounded. The bound is linear in the number of twist regions of the diagram.

The mean curvature flow is an evolution process under which a submanifold deforms in the direction of its mean curvature vector. The hypersurface case has been much studied since the eighties. Recently, several theorems on regularity, global existence and convergence of the flow in various ambient spaces and codimensions were proved. We shall explain the results obtained as well as the techniques involved. The potential applications in symplectic topology and mirror symmetry will also be discussed.

We establish a boundary maximum principle for free boundary minimal submanifolds in a Riemannian manifold with boundary, in any dimension and codimension. Our result holds more generally in the context of varifolds.

In this article we introduce higher order Bergman functions for bounded complete Reinhardt domains in a variety with possi- bly isolated singularities. These Bergman functions are invariant under biholomorhic maps. We use Bergman functions to deter- mine all the biholomorhic maps between two such domains. As a result, we can construct an infinite family of numerical invari- ants from the Bergman functions for such  domains in An variety (x, y, z) ∈ C3 : xy = zn+1 . These infinite family of numerical invariants are actually a complete set of invariants for either the set of all bounded strictly pseudoconvex complete Reinhardt domain in An variety or the set of all bounded pseudoconvex complete Reinhardt domains with real analytic boundaries in An variety. In particular the moduli space of these domains in An variety is constructed explicitly as the image of this complete family of numerical invariants. It is well known that An variety is the quo- tient of cyclic group of order n + 1 on C2. We prove that the moduli space of bounded complete Reinhardt domains in An vari- ety coincides with the moduli space of the corresponding bounded complete Reinhardt domains in C2. Since our complete family of numerical invariants are computable, we have solved the biholo- morphically equivalent problem for large family of domains in C2.

This article considers the quasi-local conserved quantities with respect to a reference spacetime with a cosmological constant. We follow the approach developed by the authors in [25,26,7] and define the quasi-local energy as differences of surface Hamiltonians. The ground state for the gravitational energy is taken to be a reference configuration in the de Sitter (dS) or Anti-de Sitter (AdS) spacetime. This defines the quasi-local energy with respect to the reference spacetime and generalizes our previous definition with respect to the Minkowski spacetime. Through an optimal isometric embedding into the reference spacetime, the Killing fields of the reference spacetime are transplanted back to the surface in the physical spacetime to complete the definitions of quasi-local conserved quantities. We also compute how the corresponding total conserved quantities evolve under the Einstein equation with a cosmological constant.