In this paper we develop methods to extend the minimal hypersurface approach to positive scalar curvature problems to all dimensions. This includes a proof of the positive mass theorem in all dimensions without a spin assumption. It also includes statements about the structure of compact manifolds of positive scalar curvature extending the work of\cite {sy1} to all dimensions. The technical work in this paper is to construct minimal slicings and associated weight functions in the presence of small singular sets and to show that the singular sets do not become too large in the lower dimensional slices. It is shown that the singular set in any slice is a closed set with Hausdorff codimension at least three. In particular for arguments which involve slicing down to dimension 1 or 1 the method is successful. The arguments can be viewed as an extension of the minimal hypersurface regularity theory to this setting of minimal slicings.

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

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.

By adapting the test functions introduced by Choi-Daskaspoulos \cite{c-d} and Brendle-Choi-Daskaspoulos \cite{b-c-d} and exploring properties of the $k$-th elementary symmetric functions $\sigma_{k}$ intensively, we show that for any fixed $k$ with $1\leq k\leq n-1$, any strictly convex closed hypersurface in $\mathbb{R}^{n+1}$ satisfying $\sigma_{k}^{\alpha}=\langle X,\nu \rangle$, with $\alpha\geq \frac{1}{k}$, must be a round sphere.

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.