We define the contact homology for Legendrian submanifolds in standard contact (2n + 1)-space using moduli spaces of holomorphic disks with Lagrangian boundary conditions in complex n-space. This homology provides new invariants of Legendrian isotopy which indicate that the theory of Legendrian isotopy is very rich. Indeed, in [4], the homology is used to detect infinite families of pairwise non-isotopic Legendrian submanifolds which are indistinguishable using previously known invariants.

We show that there is an integral homology class in a Kähler-Einstein surface that can be represented by a lagrangian two-sphere, but that a minimizer of area among lagrangian two-spheres representing this class has isolated singularities with non-flat tangent cones.

Given a compact Riemannian manifold (Mⁿ, g), with positive Yamabe quotient, not conformally diffeomorphic to the standard sphere, we prove a priori estimates for solutions to the Yamabe problem. We restrict ourselves to the dimensions where the Positive Mass Theorem is known to be true, that is, when n ≤ 7. We also show that, when n ≥ 6, the Weyl tensor has to vanish at a point where solutions to the Yamabe equation blow up.