We describe a general geometrical construction of representations of fundamental groups of 3-manifolds into PU(2, 1) and eventually of spherical CR structures defined on those 3-manifolds. We construct branched spherical CR structures on the complement of the figure eight knot and the Whitehead link. They have discrete holonomies contained in PU(2, 1,Z[ω]) and PU(2, 1,Z[i]) respectively.

We derive pointwise curvature estimates for graphical mean curvature flows in higher codimensions. To the best of our knowledge, this is the first such estimates without assuming smallness of first derivatives of the defining map. An immediate application is a convergence theorem of the mean curvature flow of the graph of an area decreasing map between flat Riemann surfaces.

Luttinger surgery is used to produce minimal symplectic 4- manifolds with small Euler characteristics. We construct a minimal symplectic 4-manifold which is homeomorphic but not diffeomorphic to CP2#3CP 2 , and which contains a genus two symplectic surface with trivial normal bundle and simply-connected complement. We also construct a minimal symplectic 4-manifold which is homeomorphic but not diffeomorphic to 3CP2#5CP 2 , and which contains two disjoint essential Lagrangian tori such that the complement of the union of the tori is simply-connected. These examples are used to construct minimal symplectic manifolds with Euler characteristic 6 and fundamental group Z, Z3, or Z/p ⊕ Z/q ⊕ Z/r for integers p, q, r. Given a group G presented with g generators and r relations, a symplectic 4-manifold with fundamental group G and Euler characteristic 10 + 6(g + r) is constructed.

A discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying Gauss-Bonnet formula. Furthermore, the hyperbolic polyhedral metric with given curvature can be obtained using a discrete Yamabe flow with surgery. In particular, each hyperbolic polyhedral metric on a closed surface with negative Euler characteristic is discrete conformal to a unique hyperbolic metric.

In this paper, we establish various L 2 -estimates for the exterior differential operator on p-convex Riemannian manifolds in the sense of Harvey and Lawson. As applications, we establish a Carleman type estimate which is uniform with respect to both weight functions and domains, and we also obtain topological restrictions for a Riemannian manifold to be p-convex.