We prove that a stable minimal hypersurface of an open ball which is immersed away from a closed (singular) set of finite codimension 2 Hausdorff measure and weakly close to a multiplicity 2 hyperplane must in the interior be the graph over the hyperplane of a 2-valued function satisfying a local C1, estimate. This regularity is optimal under our hypotheses. As a consequence, we also establish compactness of the class of stable minimal hypersur- faces of an open ball which have volume density ratios uniformly bounded by 3−δ for any fixed δ ∈ (0, 1) and interior singular sets of vanishing co-dimension 2 Hausdorff measure.

The holonomy of the ambient metrics of Nurowski’s conformal structures associated to generic real-analytic 2-plane fields on oriented 5-manifolds is investigated. It is shown that the holonomy is always contained in the split real form G2 of the exceptional Lie group, and is equal to G2 for an open dense set of 2-plane fields given by explicit conditions. In particular, this gives an infinitedimensional family of metrics of holonomy equal to split G2. These results generalize work of Leistner-Nurowski. The inclusion of the holonomy in G2 is established by proving an ambient extension theorem for parallel tractors in the context of conformal geometry in general signature and dimension, which is expected to be of independent interest.

In the ¯rst Heisenberg group H1 with its sub-Riemannian structure generated by the horizontal subbundle, we single out a class of C2 non-characteristic entire intrinsic graphs which we call strict graphical strips. We prove that such strict graphical strips have vanishing horizontal mean curvature (i.e., they are H-minimal) and are unstable (i.e., there exist compactly supported deformations for which the second variation of the horizontal perimeter is strictly negative). We then show that, modulo left-translations and rotations about the center of the group, every C2 entire Hminimal graph with empty characteristic locus and which is not a vertical plane contains a strict graphical strip. Combining these results we prove the conjecture that in H1 the only stable C2 Hminimal entire graphs, with empty characteristic locus, are thevertical planes.

We prove the mean curvature flow of the graph of a symplectomorphism between Riemann surfaces converges smoothly as time approaches infinity.

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.