We prove the existence of embedded minimal surfaces of arbitrary genus g ≥ 3 in any flat 3-torus. In fact, we construct a sequence of such surfaces converging to a planar foliation of the 3-torus. In particular, the area of the surface can be chosen arbitrarily large.

Evolving smooth, compact hypersurfaces in Rn+1 with normal speed equal to a positive power k of the mean curvature improves a certain ‘isoperimetric difference’ for k > n.1. As singularities may develop before the volume goes to zero, we develop a weak level-set formulation for such flows and show that the above monotonicity is still valid. This proves the isoperimetric inequality for n 6 7. Extending this to complete, simply connected 3-dimensional manifolds with nonpositive sectional curvature, we give a new proof for the Euclidean isoperimetric inequality on such manifolds.

A version of the Donaldson-Thomas invariants of a Calabi-Yau threefold is proposed as a conjectural mathematical definition of the Gopakumar-Vafa invariants. These invariants have a local version, which is verified to satisfy the required properties for contractible curves. This provides a new viewpoint on the computation of the local Gromov-Witten invariants of contractible curves by Bryan, Leung, and the author.

Let (Mn, g) be a complete non-compact K¨ahler manifold with non-negative and bounded holomorphic bisectional curvature. We prove that M is holomorphically covered by a pseudoconvex domain in Cn which is homeomorphic to R2n, provided (Mn, g) has uniform linear average quadratic curvature decay.

We prove a general maximum principle at infinity for properly immersed minimal surfaces with boundary in R^3. An important corollary of this maximum principle at infinity is the existence of a fixed sized regular neighborhood for any properly embedded minimal surface of bounded curvature.