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.

We consider minimal surfaces M which are complete, embedded, and have finite total curvature in R3, and bounded, entire solutions with finite Morse index of the Allen-Cahn equation
u+ f(u) = 0 in R3. Here f = −W0 with W bi-stable and balanced, for instance W(u) = 1 4 (1 − u2)2. We assume that M has m  2 ends, and additionally that M is non-degenerate, in the sense that its bounded Jacobi fields are all originated from rigid motions (this is known for instance for a Catenoid and for the Costa-Hoffman- Meeks surface of any genus). We prove that for any small  > 0, the Allen-Cahn equation has a family of bounded solutions depending on m − 1 parameters distinct from rigid motions, whose level sets are embedded surfaces lying close to the blown-up surface M := −1M, with ends possibly diverging logarithmically from M. We prove that these solutions are L1-non-degenerate up to rigid motions, and find that their Morse index coincides with the index of the minimal surface. Our construction suggests parallels of De Giorgi conjecture for general bounded solutions of finite Morse index.

We prove that a compact Riemannian 3-symmetric space is globally symmetric if every Jacobi field along a geodesic vanishing at two points is the restriction to that geodesic of a Killing field induced by the isotropy action or, in particular, if the isotropy action is variationally complete.

One of the natural generalizations of conformal structure on a two dimensional surface is a conformally flat structure on an n-manifold. In higher dimensions, not every manifold admits such a structure and it is a difficult problem to give a good classification of conformally flat manifolds. Recall that conformally flat Riemannian manifolds are manifolds whose metrics are locally conformally equivalent to the Euclidean metric. Kuiper [Kul, Ku2] was the first to study the global properties of these manifolds. He classified those compact conformally flat manifolds with abelian fundamental group. For dimensions greater than two, the Liouville theorem tells us that the conformal transformations of S" are determined locally and are given by M6bius transformations. Hence by a standard monodromy argument, a simply connected conformally flat manifold (with dimension> 3) has a conformal immersion into S" which is unique up to

We construct balanced metrics on the family of non-K¨ahler Calabi-Yau threefolds that are obtained by smoothing after contracting (−1,−1)-rational curves on a K¨ahler Calabi-Yau threefold. As an application, we construct balanced metrics on complex manifolds diffeomorphic to the connected sum of k  2 copies of S3 × S3.