We consider complete nonpositively curved surfaces with one end twice continuously differentiably immersed in Euclidean three space. If such a surface is embedded near infinity and has square integrable second fundamental form then it must lie between two parallel planes.

We consider the L² gradient flow for the Willmore functional. In [5] it was proved that the curvature concentrates if a singularity develops. Here we show that a suitable blowup converges to a nonumbilic (compact or noncompact) Willmore surface. Furthermore, an L^∞ estimate is derived for the tracefree part of the curvature of a Willmore surface, assuming that its L² norm (the Willmore energy) is locally small. One consequence is that a properly immersed Willmore surface with restricted growth of the curvature at infinity and small total energy must be a plane or a sphere. Combining the results we obtain long time existence and convergence to a round sphere if the total energy is initially small.

The purpose of this note is to introduce a new method for proving the existence of Sasakian-Einstein metrics on certain simply connected odd dimensional manifolds. We then apply this method to prove the existence of new Sasakian-Einstein metrics on S² × S³ and on (S² × S³)#(S² × S³). These give the first known examples of nonregular Sasakian-Einstein 5-manifolds. Our method involves describing the Sasakian-Einstein structures as links of certain isolated hypersurface singularities, and makes use of the recent work of Demailly and Kollár who obtained new examples of Kähler-Einstein del Pezzo surfaces with quotient singularities.

We show that for each control metric (i.e., Carnot-Caratheodory metric), there is an equivalent metric which has the maximal expected degree of smoothness. The equivalent metric satisfies the natural differential inequalities with respect to the vector fields used to define the metric. This generalizes the regularity of the usual Euclidean metric in Rⁿ. There are also corresponding differential inequalities for scaled "bump functions" supported on balls associated to these metrics. The smooth metrics and bump functions are particularly useful in problems of harmonic analysis in situations where the given metrics arise.

First we prove a version of the Strong Half-Space Theorem for minimal surfaces with bounded curvature in ℝ³. With the techniques developed in our proof we give criteria for deciding if a complete minimal surface is proper. We prove a mixed version of the Strong Half-Space Theorem. Turning to 3-dimensional manifolds of bounded geometry and positive Ricci curvature, we show that complete injectively immersed minimal surfaces with bounded curvature are proper and as a corollary we have a Half-Space Theorem in this setting. Finally we show an application of the maximum principle for nonproper minimal immersions in ℝ³.

Let W/C be a degeneration of smooth varieties so that the special fiber has normal crossing singularity. In this paper, we first construct the stack of expanded degenerations of W. We then construct the moduli space of stable morphisms to this stack, which provides a degeneration of the moduli spaces of stable morphisms associated to W/C. Using a similar technique, for a pair (Z, D) of smooth variety and a smooth divisor, we construct the stack of expanded relative pairs and then the moduli spaces of relative stable morphisms to (Z, D). This is the algebro-geometric analogue of Donaldson-Floer theory in gauge theory. The construction of relative Gromov-Witten invariants and the degeneration formula of Gromov-Witten invariants will be treated in the subsequent paper.